Quantized Neumann problem, separable potentials on Sn and the

Transcription

Quantized Neumann problem, separable potentials
and the Lam6 equation
on S”
David Gurarie
Case Western Reserve University Cleveland, Ohio 44106
(Received 20 January 1995; accepted for publication 2 May 1995)
The paper studies spectral theory of Schrodinger operators H= fi2A + V on the
sphere from the standpoint of integrability and separation. Our goal is to uncover
the fine structure of spec H, i.e., asymptotics of eigenvalues and spectral clusters,
determine their relation to the underlying geometry and classical dynamics and
apply this data to the inverse spectral problem on the sphere. The prototype model
is the celebrated Neumann Hamiltonian p2+ V with quadratic potential V on Sn.
We show that the quantum Neumann Hamiltonian (Schrodinger operator H) remains an integrable and find an explicit set of commuting integrals. We also exhibit
large classes of separable potentials {V} based on ellipsoidal coordinates on S”.
Several approaches to spectral theory of such Hamiltonians are outlined. The semiclassical problem (small h) involves the EKB(M)-quantization
of the classical
NeumannJcIow along with its invariant tori, Maslov indices, etc., all made explicit
via separation of variables. Another approach exploits Sdckel-Robertson separation of i.he quantum Hamiltonian and reduction to certain ODE problems: the Hill’s
and the generalized Lam; equations. The detailed analysis is carried out for S2,
where the ODE becomes the perturbed classical Lame equation and the Schriidinger eigenvalues are expressed through the Lame eigendata. 0 1995 American
Institute
of Physics.
TABLE OF CONTENTS
I. INTRODUCTION ..........................................................
II. QUANTIZED NEUMANN PROBLEM. ......................................
A. Classical conserved integrals. .............................................
B. Quantization ...........................................................
................
III. SEMICLASSICAL EIGENVALUES AND EBK-QUANTIZATION.
A. EBK-quantization .......................................................
B. Ellipsoidal action variables. ..............................................
C. Invariant Lagrangians, singular projections and Maslov indices. .................
IV. ELLIPSOIDAL COORDINATES AND STACKEL-ROBERTSON
SEPARATION ....
A. Ellipsoidalco~rdin_ates ...................................................
B. Laplacian in ellipsoidal coordinates. .......................................
C. Stackel-Robertson separation for Laplacians and Schrodinger operators. ..........
D. The Neumann Hamiltonian on S”. .........................................
E. Reduction to the Matheau and Lame equation. ...............................
1. Boundary conditions ..................................................
2 ., The Matheau-Hill problem. ...........................................
V. LAME EQUATION AND THE S2 SPECTRAL PROBLEM. ......................
A. Lame eigenmodes .......................................................
B. Perturbedeigenvalues ....................................................
C. Comments .............................................................
VI. APPENDICES ...........................................................
A. Integrability of the Neumann system. ......................................
5356
5360
5360
5360
5362
5362
5363
5365
5369
5369
5371
5372
5374
5376
5376
5377
5379
5379
5382
5384
5384
5384
0022-2488/95/36(1 O)l5355/37l$6.00
5355
0 1995 American Institute of Physics
J. Math. Phys. 36 (iO), October 1995
Downloaded 11 Aug 2005 to 128.117.47.43. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp
5356
David Gurarie: Quantized
Neumann problem
B. Orthogonality of ellipsoidal coordinates. ................................
C. Hamilton-Jacobi equation and StEkel separation. ........................
1. General procedure......................‘.........................
2. The Liouville case. ..............................................
3. Sdckel Hamiltonians. ............................................
4. Stackel form of the Liouville and Neumann problems. .................
5. Integration and classical periods. ...................................
.........................................
VII. RESEARCHBIBLIOGRAPHY
...
...
...
...
...
...
...
...
5385
5386
5386
5387
5387
5389
5390
5390
I. INTRODUCTION
The classical Neumann problem represents a harmonic oscillator restricted on the unit sphere
S”={x:IxI=
l} in Wn+*, or a spherical pendulum in the field of linear force F=Ax. Its Hamiltonian h= i(p’+ V) is made of the standard kinetic energy T= -b2, the Euclidean metric-tensor
on T* (Sn)CW2n+2 and quadratic potential V=Ax.x
with diagonal matrix A=diag( LY,;...;a!,,+ ,).
The classical Neumann problem is well known to be integrable’ the commuting integrals being
fk(x;p)=xi+,Tk &;k=
I
l;...n
(1)
(see Ref. 2). Here {J;j=Xipj-Xjpi}
denote the components of the angular momentum J=xAp.
One can show that Hamiltonians {fk} Poisson commute on the phase-space R2n+2. Furthermore,
they obey the relations,
C
fi(X;P)=X2,
while their weighted combination with weights { Cyj} gives the Hamiltonian itself
These relations allow one to reduce the Neumann system from the phase-space R2n+2 to a
cotangent bundle T* (S”)CR2”+2, defined by two constraints: x2= 1 and x-p=O.
The reduced
system remains integrable. The latter is far from obvious, as there are no general reasons for a
constrained system to remain integrable. Indeed, the trivial integrable Hamiltonian p2 on W”
restricted on a hypersurface X gives rise to a geodesic flow, which would be typically chaotic
rather than integrable.
In Sec. II we shall see that classical Hamiltonian h, along with conserved integrals cfi}, could
be naturally quantized to produce a SchrGdinger operator
H=-$A+V
with potential V= iAx-x on §‘.
In the spectral theory of Schrtidinger operators one is interested in eigenvalues {X} of H, and
their connection to geometry (potential, metric, etc.). Such geometric data encoded in X’s could
eventually lead to solution of the inverse problem and characterization of isospectral classes of
{V}. The n-sphere Schriidinger theory provides an interesting example of the inverse problem, that
was subject of many works (see references in Ref. 3). Though a great deal is known about spectra
of Schriidinger operators on S”, we are still far from resolving such basic problems, like rigidity
(whether the isospectral class consists of all rotations of V by the group SO(n + l)), or the inverse
spectral problem. Both results however, were established in special cases.4-6 An essential feature
of the n-sphere Schrijdinger theory is clustering of eigenvalues of H, about the unperturbed and
J. Math. Phys., Vol. 36, No. 10, October 1995
Neumann
problem
5357
highly degenerate spectrum of the Laplacian spec( A) = { hl= I( I + II - 2)). Each cluster A, consists
of spectral shifts {pr,}, localized near unperturbed X, , that result from breaking the underlying
SO( n + 1 )-symmetry of A by perturbation V,
l;...d,},
fl,={X,+/~~~:rn=
Here dl denotes the multiplicity of the free (Laplace) eigenvalue hl , i.e., dimension of the degree
1 spherical harmonics on R”+ ’. Clusters AZ’are well separated, dist( A, ;A/+ r) =@(I), while their
size remains bounded or decreases with Z5
@‘(I),
IA 1I={
B(P),
for even/generic V
for odd V
’
The large-l asymptotics of spectral shifts {,ulClm}could be described by cluster-distribution
measures {d v[= ( 1ld,) 1 S( X - pu.,,)} .7 It turns out that the Z-asymptotics of measures {d v~} can be
expressed through the Radon and related transfomsof potential V.4-8 Furthermore, in case of
zonal (axisymmetric) potentials {V} asymptotic expansion was established for individual spectral
shifts {p/m).4 The argument of Ref. 4 was based on the auxiliary SO( n)-symmetry of zonal
potentials and the corresponding commuting integrals (like angular momentum J,=xp,-yp,
on
S*), that allows one to “quantize” the joint {H”;J,}-spectrum
by pairs of integers {(Z;m):
- ISrn==Z}. Then the ml-th spectral shift was shown to be
p,,~a(~)+fb(q)+~c(~)+...
as 1403.
(2)
Two quantum numbers acquire a transparent meaning here with Z measuring the principal
and m standing for the quantized angular momentum .Z,[ $I= m @. Coefficients
are certain functions on [O; 11, depending on V. Precisely, a = “Radon transform
{dx);W);c(x))
of V” (i.e., V integrated along great circles %S’), while b and c involve more complicated
polynomial expressions in V and its derivatives integrated along {Y}.~,~
Semiclassical expansion (2) could be also viewed as resulting from the effective (“averaged”)
perturbation Vefi = q(J,la)
. In other words, operators H = - A + V and Heff= -A+ Veff prove
to be “almost unitary equivalent.” Hence we can interpret fractions {m/Z} in (2) as representing
quantized values of two commuting operators J, and J-h.
Let us remark that the approach of
Ref. 4 did not use the EKB-quantization of the underlying classical flow explicitly (rather certain
“symmetry-reduction” procedures), so it gave no clue as to the semiclassical structure of spectral
shifts {plm} in nonzonal cases.
The present paper aims to uncover such semiclassical structure of {,uUlm} and to outline an
approach to the inverse problem and the isospectrality in the context of integrable or separable
potentials V on S”. The simplest example is furnished by the quantum Neumann operator:
cluster-number
H=A+V;V=~
C
aiXf
on Sn.
We remark that quadratic potentials on 5” share one essential feature of zonal {V}, namely,
integrability both on the classical level (due to C. Neumann) and the quantum level (see Sec. II
and Ref. 9). The integrability works in the quantum case via explicit integrals {fk} of (l), and the
EKB-quantization of the classical flow. We pursue this approach in Sets. II-III. The final result
(Sec. III) gives semiclassical eigenvalues for the Neumann operator H(h) = li2A + V, expressed
through a system of hyperelliptic integrals. Thus we get the semiclassical eigenvalues {Xr,(fi)}
for the “small Planck” problem, but such results are not directly applicable to the high energy
asymptotics {X,,} as Z-+a at h=@(l).
J. Math. Phys., Vol. 36, No. IO, October 1995
5358
Neumann problem
Another essential feature of the Neumann problem has to do with its separability (separation
of variables), both on the classical level (Hamilton-Jacobi equation) and the quantum Schrodinger
problem. The separation exploits a special orthogonal coordinates system on S’, known as ellipsoidal or sphero-conal. It is defined by a family of confocal quadrics on R”’ ‘, depending on
We shall briefly review the ellipsoidal coordinates in
(n+ 1) real parameters cur<‘~~<*.*<c~~~r.
Sec. IV and Appendix C. They will be shown to belong to a wider class of separable coordinate
systems studied by Stackel,‘““l whose origins go back to Liouville (cf. Ref. 12). A Stackel system
is given by a metric tensor (kinetic energy form) of the type h(x;p) = C g”(x)p?, that allows a
complete separation of variables and explicit integration in quadratures. The separation procedure
works identically both on the classical level-the
geodesic flow of h given by the HamiltonianJacobi equation h(x;V’) = 0, and on the quantum level, i.e., the Laplace-Beltrami
operator
A= i 2 ai&g”dj.
&
Furthermore, each Stackel system gives rise to a family of integrable/separable potentials. In
the §“-case they take the form
(3)
.; V,(x,)},
the Neumann Hamiltonian correspondwith arbitrary one-variable functions { Vl(xl);..
ing to a special choice of {Vi}.
Separation of variables reduces the spectral problem for “Stlckel” Schrodinger operators with
potentials (3) to a system of singular ODES on adjacent intervals [“j ; crjt r]CR. An interesting
feature of the ellipsoidal reduction is that all ODES are given by a single differential operator
,=a*,(
ig
&}a+%
&-a
generalized Lame operator.
(4)
That contrasts the standard (polar/spherical) separation where different expressions for e5,0appear.
Operator L has (n + 1) regular-type singular points at { Crj}, and a possible irregular singularity at {w} depending on separation constants {qj}. The sphere problem requires all solutions
L[ $j] =0 on intervals [aj ; aj+ r] to be regular at both end points. The regularity condition
imposes a system of algebraic constraints on separation constants {qj}, whose solution could in
principle provide a quantized (discrete) set of parameters { qj( m) :m = (m 1 ; . . . ;m,) E T}, hence the
quantized (exact) spherical eigenvalues,
xm=C
ajqj(m)-
However, the resulting algebraic system is difficult
in dimensions higher than 1. As for the S’-Neumann
substitution) into the standard Matheau equation: 8-X
case, S*, poses a more challenging problem. Here (4) is
the celebrated Lam.&-Ince operator
S=a*-ii
sn*(x;k),
or
to write down explicitly let alone to solve
problem it is easily converted (via trig.
cos* IT?+,!?on [O; 275-I.However, the 2D
converted into another well-known model
-!+a*--@,
with the Weierstrass g-function as potential. The S*-Schrodinger problem leads to a perturbation
of 5?, which in the Neumann case V=eAx.x
on S* turns into the so-called Lami wave equation
L=a*-h
Neumann problem
sn*(x;k)+e
sn4(x;k).
5359
(5)
The terminology points to another source of (5)-&e
reduced wave (Helmholtz) operator
A+2 in ellipsoidal coordinates on R3. The Lame-Ince operator 3 is well-known in the finitezone potential theory on R,13,14as the first and the simplest example of a multizone potential.
Precisely, quantizing the coupling parameter X to a discrete set of values {X = I( I + 1) ; I = 1; 2 ; . . .}
yields operators %! = d* - I( Z+ 1 )Q having precisely Z zones/gaps in the continuous spectrum on
L* (R). The (2Z+ 1) end-points of the gaps {Ef, :Osrns 21) correspond to periodic and antiperiodic eigenvalues on x.
Furthermore, potential hg and its perturbations Xp + V, resulting from the S2-Schrodinger
problem turn into double-periodic
meromorphic
(elliptic) functions on C. Any such operator, like
Lame’s z or its perturbation L= !%+ V can be considered on both the real and the imaginary
solutions of the
periods in d= (Sec. V). So one can look for the double periodickmtiperiodic
eigenvalue problem. Indeed, the double periodicity condition results from separation on S* and is
directly linked to double-periodicity of the Jacobi map Q:C-+S*, that implements the ellipsoidal
coordinate change,
Q>:u+iu+
x=sn(u;k)sn(u;k)
y=cn(u;k)cn(u;k).
z=dn(u;k)dn(u;k)
(6)
Let us elucidate the point by drawing analogy between the S* and the torus case. The torus is
obtained by a trigonometric map Q:R2/~*+TXT,
so R* forms an infinite-sheet cover of I’*, the fundamental regions being shifted rectangles
{OCX,YC 1). Similarly, ellipsoidal coordinates on §* implemented by the Jacobi functions (6),
define a (2 K;2 K’)-double periodic conformal map from “C modulo period lattice” onto the
“sphere with a cut.” Once again C makes up an infinite-sheet cover of S*. The resulting eigenfunctions, L[ $1 = E r+4extend analytically as periodic/antiperiodic functions in both the real and the
imaginary directions.
Our last Sec. V exploits the Lame eigenvalues and the double-periodicity of the Lame problem for the asymptotic analysis of spectral shifts {pI,} on S*. We observe that the role of the
quantized ratios {m/Z} in the zonal case (2) is played now by the Lame eigenvalues {ET}.
The appearance of double-periodic problems and the link to the finite-zone theory on R
suggests a possible approach to nontrivial isospectral deformations on S*, by analogy with the well
studied torus case.15 Indeed, generic potentials on ‘I’* are well known to be spectrally rigid, but
separable-type potentials V= V,(x) + V,(y) clearly allow large (w-D) isospectral deformations,
defined by higher KdV-flows of VI and V, . Since the ellipsoidal change (6) makes S* to resemble
“torus,” one wonders whether similar constructions (deformations) could be implemented on S*.
We made a preliminary study along these lines, based on the finite-zone potential theory.13 It
produced large families of partially isospectrul deformations, but also indicated that the finitezone constraint might be too restrictive to get the complete solution (globally isospectral nontrivial
deformations). The complete answer hinges on some unresolved issues in the theory of infinitezone double periodic potentials “elliptic solitons” (cf., Ref. 16) and would require further study.
The original version of the article appeared in March 1993. Soon afterwards we learned about
the recent work’ by Toth who studied similar problems and proved some of our results (Sec. II).
The current revised version completes and clarifies several points left open in the original manuscript, particularly Maslov indices in the semiclassical quantization and the role of the LiouvilleSdckel separation (Sec. III).
5360
David Gurarie: Quantized Neumann problem
II. QUANTIZED NEUMANN PROBLEM
A. Classical conserved
integrals
The classical Neumann system defines a Hamiltonian flow on the phase-space T” (S”) of the
unit sphere S’={lxl=
1) in IX”” g’rven by Hamiltonian h(x;p) =p* + V(x), with quadratic potential
V(x)=c
qx:=Ax.x
Here matrix A=diag(ar;...a,+,
) for an arbitrary increasing sequence 0 c aI 6 cy2&
“-Can+*.
So one can view the Neumann Hamiltonian either as a constrained oscillator, or a
spherical pendulum in the linear field of force F=Ax. Mose? gave a concise account on the
Neumann problem and showed many interesting connections to the classical Jacobi problem
(geodesics on ellipsoids, see also Ref. l), spectral theory of finite-zone Sturm-Liouville operators
on R and the periodic KdV problem. The Neumann and Jacobi problems are known to be integrable, their commuting integrals defined by polynomial functions
fitxiP)=xf+,Ti
&
(Neumann),
I
I
(7)
.
fi(x;P)=Pf+,&
&
I
(Jacobi).
I
Here J= (Jij> =xAp=(xipi--xipi)
denotes the angular momentum. The Neumann integrals
(fi} are easily verified to obey the relations
Those would allow one to constrain the Neumann system from W*“‘* to T* (SO), and to
express the constraint Hamiltonian as the weighted sum of n + 1 commuting integrals (only n of
them ut ; . . .f,} being independent, as f n f 1 = 1 - Zl f i on S”). The involutivity/integrability
of the
constrained system is explained in Appendix A. Our goal here is to quantize the classical Hamiltonian h along with its integrals vi}.
B. Quantization
We want to assign quantum Hamiltonians, operators on L2 (Sn) to classical observables
df,(X;p)} (7) defined on the phase-space T* (Y). As above we do it first on the extended
phase-space W*”t *. Here the procedure is straightforward
fi(X;p)-tFi(X;id)=X:+,~i
A;
I
I
where J=xX iV represents the angular momentum operator. The a-weighted sum of operators
{Fi} becomes the quantum Hamiltonian
H= V(x) + jJj*;
with quadratic potential
V= C
LyiXF.
5361
Neumann problem
Next we want to restrict operators {Fi} and H originally defined on the extended quantum
(Hilbert) space L* (W+ ’) to functions on the unit sphere L* (Sn), in a way that respects all
classical (Poisson) brackets. That could be accomplished via the following general
Proposition
1: If vector fields { & ;. . . &} on R” leave a submanifold ,& invariant, then the
operators with funcalgebra of differential operators A, generated by {$} and all multiplication
tions Hx)} can be restricted on -85, hence on the quantum (Hilbert) space L2 (AS).
It remains only to verify the proper commutation relations of operators {Fi}. We start with the
simplest 2D case (n= 1). Here the operators are F, =x2+( lla)J* and F,= y2-( lla)J2, with
a= a, - fx2. Their commutator
CFI; F2]=;
[J2;x2+y2]=o.
As for the Hamiltonian
H=alF,+a2F2=J2+a1~2+~2y2~S~=d2+,
cos’ B+a2,
it turns into the standard Matheau differential operator on [O; 27r] with periodic boundary conditions.
Next we consider the 3D case. Introducing parameters a= CY~
- a2,; /?= CY~-CY~;operators {Fi}
assume the form
F,=x’+
J*
L+
J;
a+P
J*
$
F2=y2-
;
d+
F3+-
-- 5;
-J,2
ff+p
P
.
(9)
We apply the standard commutation relations for the so(3)-generators (Pauli matrices) X; Y;Z
as well as their squares [elements of the envelope of so(3)]:
[X; Y]=Z;[Y;
[X2;
Y2]=[Y2;
Z2]=[Z2;
LJ:; J:l-
X]=Y;
X2]=4XYZ+2(Y2-Z*-X’)=N.
Now the commutators of operators {Fi}
rF1’ F21= $
Z]=X;[Z;
of (9) are easily computed
a&P)
[J,“; J,2]+ p(a;p)
[J;;
J;];
all other terms clearly drop out (commuting operators). Hence, we get by (10)
-$-
aca;p,-
p~,:p~
N=O.
I
Remark I: In the limiting
case of coaxial ellipsoids
(cyI=a2) one multiplies
F, ; F2 by
LY= cr,-a, and lets cy go to 0. Then cuF,~,++J~ , and one recovers the standard angular momentum J,-symmetry of such axisymmetric
(zonal) potential V=a(gfy2).
The above argument is easily extended to higher dimensions and orthogonal algebras so(n),
so we skip the details.
5362
Neumann problem
Our main objective is the spectrum of operator H. Existence of the conserved integrals
suggests looking for their joint eigenvalues. However, any direct attempt at diagonalizing operators { Fi} seems utterly hopeless. So we try next to find a semiclassical approximation of spec (H)
based on its classical Hamiltonian flow.
III. SEMICLASSICAL
EIGENVALUES
AND EBK-QUANTIZATION
The word “quantize” could be used in two different meaning. In the previous section it meant
to assigned quantum Hamiltonians (operators) {Fi} to their classical counterparts pi}. Here the
word “quantization” refers to a proper “discretization” of classical integrals to produce eigenvalue spectra.
A. EBK-quantization
We shall follow the general quantization scheme applicable to any classically integrable
Hamiltonian of the form H= h(x; ifiV> on a cotangent phase-space T* (M) of a Riemannian
manifold (configuration space) M (see for instance Refs. 17 and 18). The classical conserved
integrals uj} of the Neumann problem restricted on the unit sphere obey the relation
n+1
2
(11)
ficXzcl.
So the phase-space T* (Y) is foliated into the joint level sets of n independent integrals
h(c)=h(c,;...
C,)={fi(X;p)=Ci
: 1 GiSn}.
Furthermore, each joint level set will be shown to consist of a single product-type invariant
Lagrangian torus ACT* (9) (Appendix C). The latter involves a suitable choice of coordinates
(ellipsoidal) and a representation of the Neumann Hamiltonian in the so-called Stackel form.
To quantize semiclassically Hamiltonians pi} of (7) by the EBK (generalized BomSommerfeld) rules, one picks a basis of fundamental cycles { rj( C) : 1 sj G n} in each Lagrangian
R(c) and writes down a system of algebraic equations
...
Tj(c)=$,,jp.dx=r(2mi+
i
f ind( rj)).
(12)
. . .
Here points {m j} vary over the lattice Z”+ and ind( r) denotes the Maslov (Morse) index of path
rCA(c)
(see Ref. 17). Solution of system (12) yields a quantized set of parameters
{c(m):m=(m,;...;
m,)}, hence a quantized sequence (lattice) of eigenvalues of Hamiltonian
H=f(F,;...F,),
h,-f(cl(m);...;c,(m))+...;mEZn+.
(13)
The accuracy of quasiclassical approximation (13) increases with h+O. To apply the general
EKB-scheme, however, one needs to know the fundamental periods {Tj(C)} of the action form
p . dx on the Lagrangian A(c) and Maslov indices {ind( rj)} over fundamental cycles of A. There
are no general recipes for doing it, and the answer typically would depend on the nature of the
integrable system in question. In our case one can get a closed form representation of periods (12)
due to a peculiar feature of the classical Neumann problem, its algebraic integrability
to’ be
explained below. Another method involves separation of variables in the Hamilton-Jacobi equation of the Neumann system. It yields both the fundamental periods and their Maslov indices.
B. Ellipsoidal
Neumann problem
5363
action variables
Following Refs. 2 and 19 we introduce a new set of commuting action-integrals {pj}
replace cfi}. These are defined by zeros of the rational function
b(z)
= -=o
fi
R(r)=+
a(z)
Z-@j
to
* z=pl;...pn.
The numerator and denominator of R are polynomials
n+l
a(z)=
v
It+1
(Zvaj)ib(Z)=
9
(Z-Pj>.
New variables {pi} together with polynomials a;b give a convenient parametrization of the
Neumann problem. Indeed, all relevant quantities, including Hamiltonians pi ;h} are simply represented through LY’Sand p’s. Namely,
fk=Res
i.e., pi} are residues of the rational function R(z). Constraint E f i = 1 is automatically satisfied by
residues (fi}
of R, by the standard residue Theorem for a pair of polynomials:
and b=b,z”+**a=aoz n+l +***II(z-ai)
The hamiltonian is simply expressed through either one of the two families of conserved integrals
h=C aJi=-$ pi+$ LYE.
The latter follow once again from the residue calculus applied to rational function z( b(z)/
Let us remark that all p’s are real-valued, and in case {ci>O} they lie in the adjacent
intervals separated by {~j} (see Fig. 1)
a(z)).
al<Pl<az<...<a,<P,<LY”.
(18)
It turns out2,19 that each invariant torus A of the Neumann problem represents a (real) Jacobian variety of a complex hyperelliptic curve (Riemann surface)
r=r(a;P)={(z;y):y*=P(Z)]
of genus n defined by polynomial
tl+1
p(Z)=
IJ
CZwaiJc
(Z-Pi)-
(19)
Depending on the mutual location of the (Y- and p-zeros of P, determined by signs of cfi}, we
choose a suitable set of n branch cuts in the complex plane [notice that f,+ t = I- Z; fj is always
positive by (II)]. For example, positive coefficients {cj} yield all (Y’S and p’s intermingled, as in
(IS), so the cuts are made over intervals: A1=[al; P1];...;A,=[a,;
,B,] (Fig. 1).
J. Math. Phys., Vol. 36, No. IO, October 199.5
5364
a:
ia:
A
Neumann problem
A
t3
:
0
FIG. 1. Poles and zeros of R(z) in case of the positive residues.
The fundamental cycles on Lagrangian A will be labeled by fundamental cycles on I?, chosen
as closed loops around branch-cuts {Aj}. Veselov and Novikov” gave an expression of the actionform p edx for general classes of “algebraically integrable” systems, i.e., systems whose invariant
Lagrangians are parametrized by Jacobians of complex algebraic curves (Riemann surfaces) r (see
Ref. 13). Namely,
p.dx=
b(z)dz
-=
Jpo
l-I(z-/3j)dz
=
(20)
JIJ(Zwai)(Z-Pj)
The branch-cuts {Ai} will be chosen so that polynomial P, equivalently fraction b(z)la(z)
inside the square-root (20) remains positive. Hence we get the fundamental periods {Tj} as
functions of variables {pj},
...
Tj(P)=
IAj$ dz*
...
(21)
The latter are to be quantized by the EKB-rules (12) to get a sequence {p(m)}. The resulting
sequence of quantized p’s is then substituted in the classical Hamiltonian h given by (11) to
produce semiclassical eigenvalues of the Schrijdinger operator H,
LW~C%-c l%(m).
(22)
C. Invariant
Lagrangians,
singular
Neumann problem
projections,
5365
and Maslov indices
The complete semiclassical analysis of operators H(h) requires the detail structure of invariant tori AC T* (Y) and Maslov indices of fundamental path { ~j} on A. In general Maslov index
of path yCA depends on singular set C(A) of the projection Pr: (x,p)-+x of A on the coordinate
space M and the intersection of the projected ray Pr(y) with 2. We recall that regular points {x,,}
have the property that projection Pr is one-to-one in the vicinity of (xO,pO), so the patch of
Lagrangian A near (x0 ,pa) is locally represented as the graph-surface of a map x-p(x).
Singular
refers to singular points {(xe,pe)} relative to Pr: A +A! (see Ref. 17), in other words local
mapping x-p(x)
becomes singular at such x0
X=[~o:det(~)=-,
or
det($)=O
at.,].
The index of y at an intersection point with C is equal to the number of coordinates {xi} that
change sign on passing through C along path y and the total index ind($ is obtained by summing
up indices over all intersection points.
There are no general recipes for computing Maslov indices and each case requires a specialized treatment. In the case of Neumann Hamiltonian the analysis is greatly facilitated by introduction of ellipsoidal coordinates on S” and the St&e1 separation procedure, reviewed in Appendices B and C. Here we shall state the final results.
Ellipsoidal coordinates {U , ; . . . ; u,} vary within a set of adjacent intervals
a,<ul<a,<U~<(Yj<“‘<a,<u,<a,+1
(23)
obtained by partitioning W with n + 1 coefficients {LY~} of the basic quadratic form (those could
always be normalized so that LY,=0 and CX~+,= 1). The ellipsoidal coordinates allow a complete
separation of variables in the Hamilton-Jacobi equation of the classical Neumann problem, hence
its integration in quadratures (see, e.g., Ref. 12). Furthermore, invariant Lagrangian tori ACT*
(S”) are factored into the product of one-D tori (circles) yi , each one lying in the ith coordinate
phase-plane in T* (!?). These { ri} form a basis of fundamental cycles on A and their indices are
computed in a straightforward manner.
Precisely, the rational function R(z) = b(z)lu(z)
of (l), or equivalently the polynomial P(z)
= u(z)b(z)
of ( 19) give a convenient parametrization of { ri}, namely,
yi :pf-R(U,)=O*
(24)
Notice, that roots {p, ; . . . ; p,} of polynomial b(z) and fraction R(z) are conserved integrals of the
Neumann flow, that could be used in place of vi} of Sec. 1.
We take a generic torus A(p) and project it down on the configuration space
Y=[a~;
a*]x**-x[a,;
a,+,]
Since the image must cover an open range of variables {ui} (the condition of nondegeneracy),
each yi of (24) is projected down onto a open subinterval of [ cri ; LY~+r]. The projections determine the admissible values of integrals {p,} to ensure the nondegeneracy of A(p). Namely,
rational function R(z)= b/u has to assume positive values in each interval [ai ; ai+r].
Hence our task is to distribute n points {pi} (roots of polynomial b) among n intervals
Iai;
cui+ll, in such a way that R would take on a positive value in each one of them. A simple
combinatorial analysis shows that each [ ai ; oi+ t] could have no more than two p’s. All possible
configurations are illustrated in Fig. 2. They include:
Sign-changing branch (km at two ends). Here yi is projected onto a subinterval [/?; CX]
below the positive part of the branch.
l
5366
b
a
d
l-k
FIG. 2. Plots of R(z) for different choices of PI ; p2 (solid dots).
Up-looking (even) branch without real roots. Here yi covers the entire interval [pi ; “i+ I]
Down-looking even branch with two real roots {pj ;fij+ 1}, that mark the end-points of the
projected ‘y.
Let us remark that the up-looking branch with two real roots is excluded, as its presence
would force another branch to be strictly negative. Hence there would be no real projection of A
in one of the adjacent coordinate intervals [“j ; aj+ ,I-the degeneracy condition.
Figure 2 illustrates the 2-D case. There are 10 ways to distribute roots PI; p2 between 3 a’s,
hence 10 types of separable tori. The first 6 of those (a-f) are nondegenerate, whereas the remaining 4 are degenerate.
Let us remark that roots {/?,} determine the singular set X(A), made of the product-type
“walls” Zi :ui= ,Bi . Indeed, the Jacobian matrix dpldx is diagonal (Appendix C) with entries
l
l
3X
(/[
R;IK
..
ap=
WJR,
I*
Here R, abbreviates as above R( ui). So X(A) is made either of poles {ai} or zeros {pi} of R.
However, the former {ayi} are fictitious singularities, due to parameterization of S” by variables
{Ui}. Precisely, coordinate intervals {LY~<u,<LY,+,} are mapped into great circles X:+X:+, = 1 on
Neumann problem
5367
FIG. 3. Projection of the @-type coordinate path y onto the sphere.
i), and they are mapped into spherical ellipsi if one of uj# ~j
(see Fig. 3). Depending on the projection range of yi inside interval [ (Yi ; “i+ r] there are three
possible cases
If Pr( n) covers [ ai ; pi+ r], then no singularities (walls) are encountered in the ith direction, i.e., path yi is projected regularly into two symmetric halves of the “latitudinal” coordinate
ellipse in S” (see Fig. 3 and Fig. 4). Since no singularities are encountered we get ind(y)=O
y is projected onto a subinterval [/?; o] or [a; ,Gj. The subinterval parametrizes a symmetric
arc of the coordinate spherical ellipse with 2 endpoints corresponding to /3 (Fig. 3). Such y
represents a typical Hamiltonian path in a 1-D potential well, each critical point contributes index
1, hence ind(y)=2.
y is projected onto a subinterval [pi ; pi+ ,] of [ LY~; “i+ t]. Once again y is a typical path
in a 1-D potential well, so ind(y)=2.
This completes the analysis of fundamental path and Maslov indices. The 2-D case (Fig. 2)
can be summarize in the Table I.
We shall conclude with the following two comments.
S”, if all but a single uj= ffj (j #
l
l
l
FIG. 4. Ellipsoidal grid on S* made of two families of confocal ellipsi.
5368
Neumann problem
TABLE I. Maslov indices for different types of Lagrangian tori on sz.
Fund.cycles
a,e
Maslov ind
[@ffl
rg
b,f
c
d
[aal
g
j
h,k
;z:;
0
0
2
0
0
2
a
0
2
2
2
1. Formulas (21) express the basic periods {T&p)} in terms of the complete hyperelliptic
integrals, that generalize the standard elliptic integrals of the first and second kind (see example
below). The asymptotic analysis of such integrals, and their dependence on conserved quantities,
elliptic moduli {pi} could be quite involved, particularly at high energies. But formulas (12) and
(22) are fairly straightforward and could be easily implemented numerically to compute semiclassical eigenvalues X,(h) of - h2A + V at low/moderate energy levels.
2. The quantization rules (12) along with the analysis of Maslov indices (Table I) could be
extended to degenerate tori (dim A<n) along the lines of A. Voros, “The WKB-Maslov method
for nonseparable systems,” CNRS Colloq. Int. Geom. Symplect. & Math. Phys. 237, 277-287
(1974). Let us remark that degenerate tori of the Neumann problem cover large (open) regions of
the phase-space. In the above description (Fig. 2) they correspond to function R(z) turning strictly
negative on certain intervals [ ai ; czi+ i ] [cases (g)-(k)]. Hence they should give a significant
input to spec(H), that would require further study.
Turning to the large eigenvalue problem for operator H,= A + V the problem could be reduced in the usual way to a semiclassical one - h2A + h2V, where !i = I/ fi. However, this time
small parameter fi multiplies potential V as well as A, hence the most significant contribution to
X(h) comes as a higher order correction to the unperturbed eigenvalue 1(Z+ 1). Such fine cluster
structure of spec (H,) becomes much more involved, and its exact solution would entail a hardly
tractable problem of “inversion” and asymptotic analysis of hyperelliptic moduli, the period map
of (21). Here we shall illustrate it with the simple 1-D example of S’.
Example:The quantum Neumann Hamiltonian on S’ is the classical Matheau operator on
[O; 27r] with periodic boundary conditions.
H=-d2+((Y2--(Yl)sin2
8+ai.
Fixing a high energy level X and writing the standard Bohr-Sommerfeld
action-integral with potential V= a sin2 0, ~=~a-czi, we get
T=
2TJ(X-a,)I 0
CYsit? dO= dm
I
o“12&?%?
condition for the
di9.
Here small parameter k = m
becomes the modulus of the standard elliptic integral of the
2nd kind E(k). Clearly, function T(X) admits an expansion in powers of k2 = a/X.
The latter can be compared to formula (21) that gives T in terms of the Novikov-Veselov
integral JJ60.
Taking large negative /%a1 and writing
Neumann
5369
problem
T(P)=
la;J(,- ;l;(~2-z)dz,
we get another standard elliptic integral form for T(P),
ds{X(sn-‘(l;k))-cn(sn-‘(l;k))},
of modulus k=z.
Remembering the relation between h and j?= -A+ (or + a2), and comparing modulus k of (25)
with the one of (26) we see that the EKB (Bohr-Sommerfeld) rule is consistent with the VeselovNovikov formula (21).
That concludes our discussion of semiclassical quantization procedures based on the classical
Hamiltonian flow. In the next section we shall adopt an approach based on separation of variables.
Separation will reduce the multi-D spectral problem on S” to certain ODE problems like the
Matheau and the generalized Lamk equations. The connection between the classical and quantum
problems will play an essential role here as well. We shall exploit it to rederive the fundamental
periods and Maslov indices used in semiclassical formulas (12) above.
IV. ELLIPSOIDAL
A. Ellipsoidal
COORDINATES
AND STkKEL-ROBERTSON
SEPARATION
coordinates
Ellipsoidal coordinates in Rn ’ ’ are defined by a family of confocal quadrics
n+1
qz)=F
L=l
Z-~j
(27)
’
A polynomial equation (27) has n + 1 roots
{Zi=Zi(X*;...;X,+l):i=
l;...;n+
1)
that give ellipsoidal coordinates of point x E ET+ ‘. Replacing constant 1 by 0 in the r.h.s. of (27)
we get the family of asymptotic cones to the quadrics
?I+1
R(z)q
ALZ--j
b(z)
a(z)
=o
(28)
.
These cones carve up a coordinate grid on 9, called ellipsoidal or sphero-conal (Fig. 4). So
ellipsoidal coordinates {U r ; . . . ~4~)are zeros of rotational function R(z) with fixed poles {Qi} in the
denominator a(z>=Il)ffl(zCuj), w h ose numerator b(z) = b,IIy(z - ui) has principal coefficient
bo= Z X! = 1 on the unit sphere. Coordinates {Uj} parametrize sphere S” and give convenient
expression to other relevant quantities in terms of polynomials a; b. Namely,
1. Cartesian coordinates are residues of R(z) at {pi};
x&4****[b(z)
1 b(dwq-4)
...
u,)=Res
-=-=
5 Q(Z)
U’(ak)
IT((U,-ai)’
(29)
i ...
2. The “unit sphere” constraint {Z T+‘xf= 1) is automatically satisfied by residues {XT},
provided the leading coefficient b. = 1;
3. quadratic (Neumann) potential is simply expressed through the new variables
5370
III
.;_.
.; ‘;:;:::::_:,.:::
,_.,._
:.:::.:
.‘
,‘;;,.,.
.‘1.1.
,:.:,~
,..::::.:
;;,.,.
1:
:;..
:.
;+;gi
Iv jjiji;jjj
::..,:::::::.
..;;,,‘.’
.:;,,_.
..,.:.:::
.....:.. ;‘
.:::::.:.:.:_:,
I:::.:.:.
.:.:.:::::::.
cy2;j;;;j;;;
;+;;;;;;;;
a3
p3 I
ml
Ic%z/
FIG. 5. Ellipsoidal map from a u, u-coordinate rectangle onto the first cctant of the sphere.
V=C n,xi=-~ Ui+‘~l (Yi.
1
(30)
As in the previous section both relations are easily derived by the Calculus of residues applied
to fractions b(z)la(z)
and z[b(z)la(z)],
respectively. Variables {pi} can be shown to form an
orthogonal coordinate system on S”, each ui varying over interval [ (Y~; (Y~+r] (Appendix B). As
for the Neumann potential V it could be represented in two different ways. One of them is
obtained through the expansion in residues of zb(z)la(z)
v-$ui+y
aiznil
!p.1
1(yi
p(z)/b(z) witi numerator
On the other hand taking residues of fraction
p(Z)=Z”-
(31)
i
C
Qi Znwl+*.’
1
we get
(32)
We remark that the lower order terms of p do not contribute to V, as they sum to 0 by the residue
Theorem. Representations (30)-(32) of V are crucial for the separation procedure described below
both for the (classical) Hamilton-Jacobi equation and the quantum Neumann Harniltonian.
Let us illustrate formulas (30)-(32) in the two-sphere case, parametrized by pairs of variables
aI~u~a2~u~a3
(see Figs. 4 and 5)
‘x=Jm
Y=
j/z.
.z=JZ
(33)
5371
Neumann problem
The Neumann potential on S2 becomes
V(u,u)=
B. Laplacian
u2-(zai)u
-(u+u)+$
in ellipsoidal
ai=-
u2-(Xcq)u
u-v
v-u
-
.
(34)
coordinates
The ellipsoidal Laplacian has diagonal form due to orthogonality (Appendix B), the corresponding metric-tensor being ds2= Z g,,duf. So
*=& $ di&g”di.
The diagonal coefficients are
gii(U)=
“(ui),O
a(ui)
-
’
and g = Il gii denotes the standard determinant of matrix (gij). In the ellipsoidal case on Sn the
determinant becomes
g=-
WI
(35)
al--*a,’
Here and henceforth notation ai ; bi will be used to abbreviate the values of polynomials a(z) and
b’(Z)
at points Z=Ui
The numerator of (35) contains square of the standard Vandermond determinant
W,=
W(Ui ;...
Un)=II
C”iyuj)
i<j
that comes from the product of derivatives {b[}.
The Laplacian in our notations takes the form
A specific example of S2 yields
A= ‘-;;$‘“”
[ di(&33,+d2(
d33dzj.
Let us remark that the products (fractions) inside the square roots are all positive due to opposite
signs of a(u) at points u,; up. The ellipsoidal metric tensor and the corresponding Laplacian
belong to a wider class of separable metric Hamiltonians, studied by St&ckel (see Refs. 10 and 11).
5372
C. Stackel-Robertson
separation
Neumann problem
for Laplacians
and Schrijdinger
operators
Sdckel studied metric-type (kinetic energy) Hamiltonians h = +C gij(x)pipj
plete separation of variables for the Hamilton-Jacobi (HJ) equation
that allow a com-
h(x;dS)=E
hence explicit integration (in quadratures) of the corresponding Hamiltonian flow. The details of
the Hamilton-Jacobi method and the Liouville and Stackel separation procedures are reviewed in
Appendix C. Here we shall briefly state the main results. The class of Stackel Hamiltonians is
determined by the so-called Stiickel matrix
u.11(x,) -**
@2,(X2)
***
u=
...
...
_ ~nl(&t) --*
*** ~ln(XJ
*** ~2n(X2)
...
. ..
. *. Ann _
each row of u depending on a single variable (xi-for the ith first row). The Stkkel metric tensor
ds2= &Z gi,dX? and the corresponding (kinetic energy) Hamiltonian h = $2 g”‘p’ (g”= l/gii) are
diagonal with coefficients {g”} determined by the first row entries of the inverse Stackel matrix
i.e., gii=ali.
CT-l=[,ij],
Stackel Hamiltonian can be perturbed by any potential of the form
V= C
with arbitrary one-variable functions { Vi(xi)}.
tonian is
Via”,
So the general form of a separable Stackel Hamil-
h=iC U”(,Df+
Vi).
The separation of the classical HJ equation proceeds as follows. One writes
(38)
and observes that summation (38) gives the first row of u-’ multiplied by the column-vector
{pf+ Vi}. Supplementing (38) with the remaining rows of u-l one can write the entire product as
(39)
where the r.h.s. is made of separation constants (conserved integrals) {Cj}, the first one being
energy c t = E. Multiplying both sides by u one ends up with a sequence of separated ODE
equations
( 8iS)2 + Vi= C
i
CjUij(Xi)
for the ith variable xi. Hence follows the explicit (quadrature) solution of the HJ equation in
St;ickel’s coordinates, Eq. (75) of Appendix C.
Neumann problem
5373
Robertson extended the Stickel’s separation method to the corresponding quantum objects,
Laplacians and Schriidinger operators
A=~~
&
di~~iidi
and H=-A+V
with separable potentials V. He assumed the Stackel form of the metric-tensor, i.e., diagonal
entries
for a suitable matrix a, plus an extra condition on the product of coefficients. Namely, the inverse
determinant
factors into the product of one-variable functions (fi}.
depends on Xi only,
Equivalently, the log-derivative
of g”&
It turns out that the corresponding Laplacian (Schrodinger) eigenvalue-problem
also allows a separation of variables, i.e., eigenfunction (I, factors into the product of one variable
functions $I= H rcli(Xi), that satisfy certain ODES. The separation proceeds similar to the classical
HJ case. One writes
$ fW=~$l
g”~ 1 [dT+fiai+
Vi][9i]=X,
with coefficients pi} of (40), and observes that the 1.h.s. is equal to the first entry of the matrix
product G-I by the column-vector made of the ODE-expressions (l/#,)HJ
pi]. AS above we
augment it by the remaining rows and rewrite the product via inverse Stackel matrix
(41)
As above the r.h.s. is made of separation constant {ci} the first of which gives the eigenvalue
(energy) of the quantum Hamiltonian c t = X. Multiplying both sides of (41) with Stackel u we end
up with an ODE in variable ui for each factor Gi of
d:+fidi+
Vi+ C
j
Cj’+ij(Xi)
[@i]=O.
I
5374
Neumann problem
Suitable boundary conditions should be introduced for (42), depending on the geometry of the
original problem. Those in turn will impose some algebraic constraints on separation constants
{cj}, sitting in the potential part of each Hi. Solution to the resulting algebraic system should give
a suitable set of value to {ci}, in particular, the eigenvalues c,=X of H.
D. The Neumann
Hamiltonian
on S”
We shall now implement the Stackel-Robertson separation in ellipsoidal coordinates on the
sphere Y’. The metric coefficients in ellipsoidal variables {Uj} were shown to be
gii=--i ai
with
w:
g=det(gii)=
G.
bl = b’(ui), as in Sec. JVB, and remembered that II bi
Here we abbreviated ai=a(ui),
= Wi-the
square of the Vandermond determinant. Hence follows the ith diagonal entry of the
metric tensor
gii&=
-f.fjJ IIjUj’
The primed product II’... indicates that the ith factor is dropped. Now coefficient fi of the reduced
ODE (43) becomes
IL+1
fi=~
~=di
1
log a= i kst
A.
1
k
One could directly verify that the StEkel matrix in ellipsoidal coordinates on S is given by
n-2
n-1
-Ul
al
-Ul
al
...
...
...
...
1
an,
n-2
n-l
-u2
CT=
1
al
1
-
...
-u2
a2
a2
...
...
g-2
n
J-1
n
-
-
a,
a2
a,
;
Ui=U(Ui).
Next we recall the Neumann potential expanded in a’s and u’s (3 1)
“+I
v=c
ai-+
Ui=-~
1
$jC;
1
in the numerator. We observe that the ith term of the
with a polynomial p(u)=u”+(X~~~)u”-’
sum can be written as g”Vi in the StEckel convention (43). Hence, follows (remembering that
g ii = ai/bl)
V.-Pi _ Pt”i)
1 Ui
U(Ui)’
Combining Vi with the separation potential derived from the ith row of matrix u via (3.15) we get
vi+2
cjaij(ui)=
Neumann
problem
n-1$C2Uy-2+*-+C,
couy+clui
a(ui)
i
5375
=-
c(ui)
a(ui)
*
Polynomial C(U) of degree n in the numerator has the leading coefficients co= e-coupling constant of H= - A + EV, while all other {Cj : 1s jG,} are Stackel-Robertson separation constants.
The first of them
gives the Schrodinger eigenvalue. Now we can write down the reduced Neumann ODE,
Rational function C(u)la(u)
can be expanded in simple fractions (poles {a,}),
CC”i)
-=
U(Ui)
‘+l
ck=l
4k
Ui-Lyk’
with residues
The sum of residues Z;’ ’ qk= 1, or E, depending on whether V is taken as Zaixf or LX.-* . SO
there are II independent variables {qi}, uniquely determined by coefficients {cj} of C. The Neumann spectral problem is thus reduced to a single ordinary differential operator depending on
2n+ 1 parameters {‘Y~<(Y~<.**<LY,+~}
and {ql;...;qn;qn+l},
subject to X qi=l (or E),
H=dZ+[
;g $-}a+2 &
(45)
considered on different intervals [ czi ; ~i+l]. So each operator Hi of (44) coincides with H
restricted on the ith interval with proper boundary conditions at the end-points. Those boundary
conditions together with the basic algebraic constraint C qi= 1 should give a system of (n + 1)
algebraic equations for unknown separation parameters {qi}. Once {qi} are found the eigenvalue
X of the original Y-problem is computed by
‘=x
( ai-qiI&
mj)zx
aiqi*
(46)
Remark: Formula (46) closely resembles an expansion of the Neumann Hamiltonian h in
terms of the conserved integrals (fi} (Sec. II) h = Z aifi . In this respect {qj} behave like the “joint
eigenvalues” of the commuting quantum operators {Fi}. It is unclear whether such a coincidence
reveals some hidden connections between PDE operators {Fi} on the sphere (Sec. II) and the
reduced ODE-operators {Hi}, as there is no apparent direct relation between both.
Stiickel
separable
potentials
on S”.
Each differential operator Hi can be augmented by a potential function Vi( ui) that comes from
a separable Stackel Hamiltonian (37) of the form
5376
V(u1;...,
sUn)
=
5
1
nj(“in;(u,-uj)
aj)
vi(“i)=C
2
I
For n=2
Vj.
J
we get
v=-
4u1)
V1(u1)+
u1-u2
4u2)
U2-UI
V2(Uz)i
where u(u) = (U - al) (u - a,) (u - as). As we already mentioned the quadratic Neumann potential is a particular example corresponding to identical functions {Vi}
V,(u)=
E. Reduction
---=Vn(u)=‘G
where
,=u~+(
c
ai)(ln-I.
to the Matheau and Lam& equation
An ODE of the form
has n + 1 regular singular points at { ‘Yk} and a possible irregular singularity at 00,provided co # 0.
In special cases we get
n = 1 (the one-sphere). Here H is reduced to the standard hypergeometric (Matheau) equation by a trigonometric substitution.
n=2 (the two-sphere). Here
l
l
(47)
is related to the well-known algebraic Lami equation.
The Lame equation corresponds to co=0 (unperturbed Hamiltonian). Furthermore, special
values of coefficient cl =m(m+ 1) yield Lamb polynomials as regular solutions. Those in turn
give rise to ellipsoidal spherical harmonics on S2 (eigenfunctions of the Laplacian). The nonzero
leading coefficient co results from the quadratic (Neumann) perturbation. It makes {w} an irregular
singular point of H, and turns it into the Lams wave equution.3826 Let us note that coefficient co= E
in the numerator results from the “ellipsoidal” separation of the reduced wave operator A+E on
w n’ ‘, as well as the “sphero-conal” separation of the Neumann operator A+ 4x.x on 3’.
In the next section we shall discuss the perturbed Lame problem in more detail. Here let us
just mention that bringing parameters { Lyi} into the standard form: cu,=O; CY~=k- ’; LYE=1, and
substituting u =sn2(x;k), algebraic equation (47) is converted into the Jacobi conoidal form
d2
;ir;l-k2(c2+cl
1. Boundary
sn2 x+c,, sn4 x).
conditions
The n-sphere problem requires each solution ~i( U;q) of the reduced ODE Hi[ Ji] = 0 on
[ ayi ; Qi+ i] to be regular at both end-points of the interval. We remark that each singular point LYE
of operator H has two independent solutions: a regular series $(z) in variable z=x- a,, and a
singular one 4(z) = & X “regular series”, both depending on separation constants {qj}. We
pick two pairs of solutions {@l;~t} at lyi and {&;a
at CZ,+~,and express one pair in terms of the
other via the transfer matrix:
Neumann problem
5377
The entries of the transfer matrix depend on parameters (4) only, the off-diagonal entry being
here {***;**. } denotes the Wronskian of two solutions. To get a regular solution at both end-points
(i.e., qb2=At,bl) coefficient B must be 0. That gives an algebraic relation for parameters {qj},
hence a system of n algebraic equations.
...
Bi(S)=Btai
;ai+l;q)=O.
...
(48)
Those being supplemented with the basic constraint Z qi= 1 (or E) the resulting algebraic
system should in principle yield a discrete (quantized) set of separation constants {qj(m)}, hence
the quantized eigenvalues {A,} of the Neumann Hamiltonian. However, Eqs. (48) are hard to
write down in any closed form, let alone solve. So next one tries to bring ODE (45) to some
conventional form amenable to such analysis. The first natural choice is the Hill operators on W
with periodic potentials.
2. The Matheau- Hill problem
Equation (45) with two singular points (n = 1) could be converted to the standard Matheau
equation via a trigonometric substitution z= (LYE- LY~)COS
2x. That is the form of the 1-D Neumann problem. It turns out that higher-D ordinary differential operators Hi could also be converted
to the Hill’s form by a combination of the conjugation and the change of variables. Given a
differential operator H = ( 1lp)apd+ . . . conjugation with 6 brings H into a symmetric (selfadjoint) form
log p)‘2+ ; (log p)”
&H+=d”-(+(
P
In our case (45) function
p has the
product-form
P=n
(Z-aj)1'2,
made of different factors pi = dw.
Each factor pi will eliminate the ith partial fraction
1/(z - ai) in the first order coefficient of H and leave the rest of them intact. Making a particular
choice,
Pcj+fl+,
Pj”II”
Pj;
operator H is brought into the form
$+A
-
1
2 i Z-Lyj
-
1
+ Z-~yi+l
d+;J
i
l
(Z-Ly/)
z+c
E,
I
(49)
with coefficients
5378
&c 16
Neumann problem
.-!ai-LYj*
j+i
Here the double-primed products and sums II”; C” indicate the omitted pair of factors (or summands) {(z-~yi);(z-ai+i)}.
The trigonometric substitution z = b co.? x with b = bi= cri+ t - LY~(length of the ith interval),
transforms (49) into the Hill’s operator d*+ W, with a periodic potential on [O; 7;1,
m
w(x;q)=di2+wo+~
w,
cos(2mx).
Fourier coefficients { W,} are linear combinations of separation constants {qj}, that could be
computed explicitly in terms of geometric parameters {~j}.
Turning to the boundary conditions we notice that regularity at the singular end-points
} for operator Hi (49) corresponds to the choice of even Hill’s eigenfunctions
Iai
iai+*
$=X0” A,,, cos(2mx), made of even Fourier modes {cos(2mx)}, while singular solutions are
transformed into odd ones: I++ ZF B, cos(2m-k 1)x (see Ref. 21). In the simplest 1-D case H is
reduced to the standard Matheau operator
H=d~+2b(ql+q2)cos
with
2x+2b(q2-ql),
b=a2-aI,
i.e., Hill’s potential V=A cos(2x) + B.
After operators {H,} are converted to the Hill’s form, we proceed to rewrite the original
spectral problem in terms of Hill’s eigendata for each Hi
Hi=d2+
V’(q)+
U’(x;q);
where
lJi=F
Wi(q)cos(2mx).
For the sake of convenience the zero Fourier mode V= WO is separated from the rest of the
series and the remaining terms are combined into a single function U(x;q).
All Fouriercoefficients {V’; Ui} are linear combinations of separation constants {qi} that could be computed
in the closed form, but the expressions become fairly cumbersome, so we will not bring them here.
Now we can choose an n -tuple of even eigenvalues E2,,,, ( U’) ; E2,*( U2) ; . . . E2m,( U”) for each of
operators Hi ; . . . H, , and write a system of algebraic equations for {qi}
V’
(q)=Ez,,(U’)
V2(q)
...
=E2m2(
I
U2)
(50)
VY4)=E2mnW”)
Z7+‘qj=l(or
E)
Solution of system
(50) could in principle yield a quantized set of separation constants
hence the quantized eigenvalues of the Schrodinger problem with an
arbitrary separable potential V on §” in the form
{qj(m):m=(ml;...,
am,)},
Let us conclude this section with two comments.
Neumann problem
5379
1. System (50) could be compared to (48) or to the semiclassical equations (22) of Sec. III.
Unlike the Neumann case (22) it applies to arbitrary separable potential. It is more explicit than
the former two, the 1.h.s. being made of certain linear functions of q’s, while the r.h.s. involves the
set of “even eigenvalues” of the given Hill potentials, also depending linearly on {qi}. Eigenvalues of the Hill’s problem {E,(u)}
have well-known asymptotic expansion to any orde?1-23
Em(u)=m2+
lJ,+
2
1
I
(U-U,)2f.”
.
However, the data {qi} enters {E,} in such a way that higher order corrections depend on
higher powers of U, hence higher powers of “large parameters {qi}.” It is not clear whether
system (50) could be truncated (approximated) by a polynomial one, and then how one should deal
with the resulting polynomial (nonlinear) equations to extract a meaningful asymptotic information on the structure of {A,}.
2. The apparent drawbacks of the Hill approximation (50) could be linked to the underlying
geometric deficiency. Indeed, the 1-D Matheau-Hill operator represents the exact S’-Neumann
problem. So system (50) could be viewed as coming from some sort of “circular (1-D)” approxi; “i+ t}. The
mations of the rz-sphere Hamiltonian A -I- V, based on adjacent pairs of parameters { CX~
higher-D Laplacians however, may not be well represented or approximated by the 1-D “circular”
objects. So the next natural step would be to consider adjacent triplets of parameters
{ai- liar iai+l } and the resulting Lumk operators. Geometrically that would correspond to “approximating” higher-D spherical Laplacians by the S2-ones. Indeed, the Lame equation itself
comes from the S2-Laplacian, the separable Schriidinger problem being reduced to the perturbed
Lame equation. In the next section we shall carry out such analysis in the 2-D case. The “triplet”
Lame representation shows clear advantages over the Hill’s representation (50). It reveals in
particular the cluster structure of spec (A + V), and yields asymptotics of individual spectral shifts
{prrn} in terms of potential V and the Lame eigendata. It also suggests a possible approach to
iso-spectral deformations on S2 via the finite and infinite-zone potential theory on R.
V. LAME EQUATION AND THE s2 SPECTRAL
PROBLEM
A. Lam& eigenmodes
From now we shall focus on the 2-sphere case and use the Jacobi form of ellipsoidal coordinates on S2. The latter are defined by three real parameters CY,<CX~<CY~
and relative distances:
a = a2 - at and b = cy3- (Ye. The coefficients {a; b} determine elliptic moduli
and its dual
k’=
The elliptic change of variables,
z=sn2(x;k)
on
[O; a]
and z=sn2(y;k’)
on
LO;bl,
takes an algebraic Lame equation
H=d2+
L
2
-+
1
i z--a1
-+-
1
z-“2
1
z-“3
d-
ez2+(X-Ea)Z+C
4(z-
oi)(z-
a2)k-
+...
a3)
(51)
into a pair of perturbed Lame operators
L=Jz+E-A
sn2-t-V(sn2),
5380
Neumann problem
FIG. 6. The real and imaginary periods of the double-periodic Lam; problem.
L’=d$+E--X
sn’2i-V’(sn’2).
Henceforth we shall abbreviate our notations for the cnoidal Jacobi functions
sn=k sn’(x;k);
sn’=k’
sn’(y;k’).
The resulting potential A sn2+ V is double-periodic with real and imaginary half-periods given
by the complete elliptic integrals of the 1st kind
K=
I
Tl2
0
dt
&7&z-i
?rl2
and K’=
I
0
dt
J 1 -kf2
sin2 t ’
One can show24 that the regular boundary condition of the algebraic Lami operator (51)
corresponds to 2K, 2K’-periodic or antiperiodic conditions for two equations
L[@l=O;
L’[fy]=O.
Double periodicity will be shown to impose strong algebraic constraints on spectral parameter
E and coupling constant X of (52). We shall start with the unperturbed case, i.e., Laplacian A on
S2 in ellipsoidal coordinates. There (52) turns into a pair of unperturbed Lame operators
3?=SA=~~i-E-X
sn2,
in fact a single Lame operator considered on two periods (see Ref. 24): the real one [O; 2K], and
the imaginary one [K; K+ 2iK’] (Fig. 6). Both operators share an identical set of parameters: the
coupling (separation) constant E and the spherical eigenvalue X.
The double-periodic solutions of the Lame equation are known to exist only for a discrete set
of the coupling constant {X = I( I + 1): I= 1; 2 ; . . .}. For any such I spectral parameter E takes on
2Z+ 1 values {EL(k):m=O;l;...;
2Z+ l}, half of which (2 or If 1, depending on parity of I)
corresponds to periodic (cno-idal) eigenfunctions {Ecr} of the Lame operator =!Zl (53) while the
I II II IJ
AAAUbAUAA
II
5381
Neumann problem
II
I1 j&
Ah
I
A
I
A
FIG. 7. A typical spectral structure of the periodic potential on the real line: the periodic eigenvalues (dark triangles) are
interlaced with the antiperiodic ones (white triangles).
other half gives antiperiodic sno-idal ones {Es;“} (Fig. 7). The eigenvalues could also be labeled
{EL} (even) and {El} (odd), depending on the parity of the corresponding eigenfunctions with
even and odd eigenvalues intermingled.
Here we adopt the Ince’s notations for the Lame functions,20’24 those should be read
ellipsoidal-cn
or ellipsoidal-sn, by analogy with the Jacobi elliptic functions. Each Lame function
{Ec;“;EsT} can be written as a product of sna cnp dnY (with exponentials CT,p, y=O; 1) times a
La& polynomial
P(sn*) of degree $( l- CY- p- y). Furthermore, the m th eigenfunction of operator .Z on the real period (53) is analytically extended through the (1-m)-th eigenfunction of X’
on the imaginary period. Since index m counts the number of oscillations (zeros) of an eigenfunction (for any Sturm-Liouville
problem) we see that the total number of the real and imaginary
The corresponding eigenfunctions of
zeros remain constant 1 for any Lame function {E$;Es;l}.
the Laplacian, called ellipsoidal (or sphero-conal) harmonics are made of products of each species
with complementary quantum numbers m and 1 -m,
Ec~(x)Ecf-m(y);m=O;l;...;l;
&Y-&Y)=
or
i Es~(x)Es~-m(y);m=1;2...;1;
XE[O;
2K]
(54)
~E[K;
K+2K’i]
The complementary indices m and I- m that appear in (54) could be interpreted in terms of
the oscillation properties of spherical eigenmodes. Namely, the total number of the “horizontal”
and “vertical” oscillations in the x; y variables (total angular momentum) remains constant 1.
Here one could draw a close parallel between ellipsoidal harmonics { @} and the standard spherical harmonics { Y;“( c$; 8)). The latter are factored in the product of the Fourier mode in 0 and an
associated Legendre function (polynomial) in 4: Yy=e- +“meP;“(~~~ c#J), the role of two Lame
operators being played by the angular momentum idO and the associated Legendre operator
&+cot 4d6-m2/sin2 #A Here the same complementarity relation holds as the number of latitudinal oscillations complements the longitudinal one.
5382
Neumann problem
Remark: To dispel a possible confusion about the “finitude” of the Lame spectrum {E,} let us
make the following comment. Each of two operators: the real Z and the imaginary Z” have
infinite periodic/antiperiodic spectra on R. The double-periodicity requirement, however, imposes
rigid constraints on the joint set of energy-coupling parameters {E,X} due to the analytic continuation property of {Ec;“;E$}. It is hardly surprising then to find out that the “coupled problem”
possesses only a finite (possibly void) set of solutions.
B. Perturbed
eigenvalues
The Lame-Ince operator s gives the well known example of a finite l-zone potential V= I( 1
+ 1)~ on R, its periodic/antiperiodic eigenvalues {E,} marking the ends of stability/instability
intervals (zones/gaps) of the continuous spectrum. Spectra of such operators can be analyzed
within the framework of the l-zone potential theory (see Ref. 13). Ellipsoidal separation of Sec. IV
brings the S2-Laplacian A, the Neumann operator - A + CLY&, as well as more general separable
Hamiltonians {H,} ,
a(u)
V=- u-u
V(u)+
a(u)
G
V’(v);
atz)=tz-a,)(z-n2)(z-LU3);
to the perturbed Lame problem. The eigenvalues and eigenfunctions of the lth cluster of H,,
A, = {A = l(1 + 1) + ,u~,}, would correspond to periodic/antiperiodic solutions of the resulting pair:
sn*+V
L=L,=d*--A
L’=L~=d*-~
sn!*+V’
on real period
[O; 2K],
on imaginary period
[K;
K+2iK’].
To abbreviate further our notations we shall call unperturbed Lame potentials
qo=l(l+
l)sn*;
6q=q-qo=pqo+
V;
and qL=l(l+
1)sn”;
while their perturbations
and Sq’=q’--qA+pq,$+
V’.
The perturbed eigenvalues of L and L’ will be denoted by
{E,=E,(Sq):O~m~21}
and {E~=E,(Sq’):O~m~21}.
The cluster number is fixed here, so we drop a superscript 1 in E, . The unperturbed LameInce eigenvalues {Ei(O)} and {E/(O)} obey the relation
E,(O)=E;-,(O);m=0;1;...;1.
(55)
Precisely, one breaks 21f 1 eigenvalues {E,} into the periodic and antiperiodic halves and let
1. So (55) pairs eigenvalues of E and E’ of the like
parity: periodic-to-periodic, antiperiodic-to-antiperiodic. Perturbed eigenfunctions retain the basic
properties of the Lame functions, namely,
m vary in the range: 1 Cm c 1, or OcrnS
L[Jr,l=&,hn~
L’[ I,$-,]=
E,!-,t,b;-,
.
(56)
Though Ic,and @’need not analytically extend each other, unless perturbation potentials are given
by the same function V= V’, two sets of spectral parameters: eigenvalues {E} and coupling
constant X = l(l+ 1) + p (i.e., separation constant and spherical eigenvalue) are identical in both
Neumann problem
equations (56). The latter allows one to write an equation for spectral shifts {p = Pi,}
the perturbed eigenvalues {E, ;I?;-,},
namely,
E,t/.qo+
VI=&-m&j;,+
V’).
5383
in terms of
(57).
For any fixed 1 and m (57) gives the mth shift in the lth cluster pCLlrn.Formula (57) could be
compared to the results of Sets. III-IV. It also reduces the multi-D spectral problem to certain
ODE (Sturm-Liouville)
operators. The main advantage of (57) however is that each spectral shift
pl,,, is produced from a single algebraic equation (57) rather than a coupled nonlinear system.
Equations (57) could be expanded in terms of small coupling parameter e, i.e., perturbation
H,= -A f EW with a separable potential
w=- a(u) v(u)+ -a(u) V’(u)
U-U
u-u
on S2 to produce shifts {pUlm(c)} to any degree of accuracy in powers of E. Here we shall do it in
the linear approximation
As above E, , E;-, denote the mth and (I- m) -th eigenvalues of L and L’ , respectively, and
all variational derivatives { SEI6q; SE’/Sq} are evaluated at e=O, in other words the unperturbed
“Lame points” qo= X sn* and q; = X snf2.
Formula (59) could be further specified by recalling that any eigenvalue problem Lq[ $I= EI)
has derivative SEISq equal to the square of the corresponding (unperturbed) eigenfunction 14’. In
our case JI coincides with the mlth Lame function,24
eriodic)
@=Ec;“(z;k)(p
So for any (.$small
or
Ecy( z; k) (antiperiodic).
perturbation we get
(60)
Here (*+*I.**) and (***I*..)’ refer to the L*-inner products on the real and complex periods [O; 2K];
[ 0; 2 K’ 1. Formula (60) gives linearized spectral shifts { ,cQ~} of H, explicitly represented in terms
of the Lame eigenfunctions {(y;} and perturbation parameters {V; V’}.
As a corollary we obtained a “linearized” version of local spectral rigidity (cf. Ref. 4)
for large classes of separable potentials on S*, like polynomial and other pairs {V= P(sn*);
V’ = P’ (sn’2). Precisely,
Proposition
2: Given any separable potential
W on S* (58) with polynomial
functions
{V=P(u);
V’ =P’(v)} the linearized spectral shifts {ptm} (60) of Schrodinger operator H= -A+eW
uniquely determine W.
Indeed, the latter depends on finitely many parameters (coefficients), that enter both functions
{V; V’} linearly. Hence (60) turns into a system of linear equations for unknown coefficients for
P and P’ that could possess a unique solution.
Remark: Formula (60) suggests that the unperturbed Lame. eigenvalues {E, = EL} should play
the role of rationals {m/l} in the zonal case of Ref. 4, so one should be able to represent
/~~-A(Efn)f-.
with suitable function coefficients {A(x);...}.
The explicit form of such A
5384
Neumann problem
would require further analysis. As for the asymptotics of {EL} some partial results are known for
small EGX .3,24,25Those could be derived by the standard WKB analysis of the large-coupling
problem, H= d2 + hq, A+ 1. Indeed, the mth eigenvalue of such operator can be expanded as
+ *. -1 where 4” is taken at a critical (minimum) point of the potential well 4. So
CEm - m&g
variational derivative SE,/&
is asymptotic to Const m/A.
Hence at the left/right ends of the
range { 0 s m C I} the Lam; eigenvalues { Ek} behave like fractions, ml Js
= m/l. Of course,
the latter applies only to low eigenvalues E,4X
(resp. m 4 fi), whereas the Lam& spectrum
{Ef, : 0 s m < 21) covers the entire range [ 0; I( 2+ 1 )k2] and is distributed in a nonuniform fashion.
We plan to return to asymptotic analysis of (60) and the nature of function A(E) leading
approximation of {pulm} elsewhere. Here let us just mention that a related asymptotic problem for
the Lam& wave equation:
d2+E-Z(Z+
1)sn2+02sn4
was studied in Ref. 3.
C. Comments
An interesting issue suggested by separable potentials on S2 has to do with possible nontrivial iso-spectral deformations by analogy with the torus case T2. We made some preliminary
study based on the finite-zone theory of Ref. 13 and got partial results. These partial results
however demonstrated limitations of the finite-zone theory, and showed that the complete solution
would involve certain “infinite-zone limits” of finite-zone potentials. Such issues have not yet
been addressed in the infinite-zone theory to our knowledge.
Though large iso-spectral classes on S2 are conceivable, the sphere problem clearly exhibits
more rigidity, than the torus one. Indeed, separable potentials on S2 that allow iso-spectral deformation (if they exist) would be quite exceptional among all (generic) functions W(u;u). As for
generic S” potentials we believe them to be spectrally rigid.
Finally, let us briefly mention the higher-D Neumann-type spectral problems on §“(n 23),
particularly semiclassical asymptotic of {pkm}. Here the reduced (separated) problem becomes a
generalized Lami equation with four (or more) singularities. Unlike the classical Lam; case very
little is known about its solutions. One possible approach would be to develop the analysis based
on the “three-point Lam; approximations” of a generalized Lam& equation.
l
l
l
ACKNOWLEDGMENT
This work was conceived and largely completed during the author’s visit to ETH, Ziirich in
1993. We would like to thank Professor H. Knijrrer, E. Trubowitz, and J. Moser for hospitality and
stimulating discussions on the project and related subjects. We also acknowledge useful comments
and suggestions by A. Veselov.
VI. APPENDICES
A. lntegrability
of the Neumann system
We shall show that the constrained Hamiltonian h= V(x) + IxXp12 with quadratic V=Xcqx?
on the unit sphere coincides with the Neumann’s h, and integrals pi} form a Poisson-commuting
family. Our exposition will follow the elegant method of Ref. 2. Given a diagonal matrix
A=diag(a,;...
cu,) we consider a family of quadratic forms Q, on R” depending on complex
parameter z ,
Q,(x)=(z-A)-‘x.x,
5385
Neumann problem
-
and define a family of Hamiltonians on the phase-space R2”
@,=
where Q,(x;p)
commuting
3e,w+ ~Q,~x)Q,(P)- ~,k~)i,
is the corresponding bilinear form. One verifies directly that family {a),}
{Qz;*,}=O,
for all
z;w
in
is
C.
For any fixed point (x;p) in R 2n function @, becomes meromorphic (rational) in variable z
with poles at { Lyi}. Its residues are precisely integrals (fi}
cjjz=;;:
AZ-~i’
l
This explains involutivity of (fi} on R2”, hence integrability of h = Z ~ifi . Next we want to
constrain system vi} and Hamiltonian h on the unit sphere and its phase-space
T*(S)={(x;p):n2=
l;x.p=O}.
We observe that geometric constraint f = (x2- l)=Z fi is itself a commuting integral, while its
canonical partner g = p 1Vf =x . p has a nonvanishing bracket with f,
cf;g}=IVfj2=1
on
S”.
In such a setup the reduction procedure of Ref. 2 gives a new family of commuting constrained Hamiltonians
fT=fi-&;
h*=h-hf;
(61)
with multipliers
{h;g)
h=m=F;
z4x.x
Pi”
Vi ;gl
Gfig} *
The particular form of multipliers is immaterial here, as long as {h;f ;f i} commute. Geometrically, (61) provides corrections to the original Hamiltonians in such a way that the restricted flows
of h and vi} on the constrained manifold u= 0; g = 0} extend to commuting flows in the ambient
space R2n+2.
B. Orthogonal&y
of ellipsoidal
coordinates
We want to compute gradients of the ellipsoidal coordinate functions
could be expressed through rational functions
Xi(U)
of (29). Those
as
z=
dz;
i=
l;...n.
Now the orthogonality relations for gradients vectors Xi= (** *dxkldui..*)
take the form
5386
Neumann problem
n ---bi(aj)_ b’(ui)
xy=C
j=l A'
a(ui) ’
(62)
(63)
The former (62) results from the residue count of bi(z)la(z),
of bij(z)lU(z),
whose numerator
b(z)
bij(z)=
the latter (63) comes from residues
1
= (Z-Ui)(Z-Uj)
(Z-Ui)2(Z-Uj)2
r-I
k~i;j
(z-4).
It remains to observe that the Ui- and uj-residues cancel each other in bij(z)lU(z),
sum of all a-residues gives the requisite dot-product.
C. Hamilton-Jacobi
equation
whereas the
and St6ckel separation
1. General procedure
We shall briefly review the separation procedure for the Hamilton-Jacobi (I-II) equation
defined by a classical Hamiltonian h(x;p) on a 2n-phase space 9. The action function S of h
solves a HJ differential equation
h(x;dS)=E-const
(64)
subject to proper boundary conditions. One looks for a family of solutions depending on parameters {ct. , . . . ;c,}-the
constants of integration, the maximal admissible number being n. The
resulting n-parameter solution S(x;c) becomes a generating function of the canonical transform
on 9
@:(-w)+(Y;c),
(65)
where the new canonical variables (y;c) are obtained by solving the system
p=S,h;c),
i y= -S,(x;c)'
(66)
The’first equation (66) could be solved for c=c(x;p)
provided det(S,,) #O. The resulting map
(65) conjugates h to a new Hamiltonian, depending on action variables {Ci} only
h+=E(c).
In some cases the Hamiltonian h itself could be chosen as one of them, e.g., h = E = c 1 . Now the
Hamiltonian flow of h is explicitly resolved in terms of the conserved action variables {ci} and the
canonically conjugate angle variables {yi}. Indeed, algebraic system
dS
yi(X;C)=-z=yy-const
1
i
yields the flow line x=x(t;E,c,y’)
of the initial data {yp}.
t--to=
- g
(x;c)
at a fixed energy E and conserved integrals {ci} as a function
2. The Liouville
5367
Neumann problem
case
Liouville (see Ref. 12) studied Hamiltonian flows that one could integrate explicitly in quadratures. He looked at the standard “kinetic+potential”
Hamihonians and came up with class {h}
defined by two families of functions {gi(xi)}
and {Vi(xi)},
each one depending on a single
variable,
h(X;p)=i 2 k (pF+vi)9
where CT=Cai . The corresponding HJ equation
c
.
CsZi+ vi)
u
=E
clearly allows a separation of variables for the action function S=C Si(xi ;ci). Here each Si
solves an ODE
...
(Sj)2+
Vi-EUi=ci,,
...
(67)
whence
Let us also notice that the conserved integrals {Ci} are subject to the constraint Z ci=O due to
(67). So the Liouville Hamiltonian allows an explicit quadrature solution
(6%
!gq
+-,,
i
J *
with functions fi = ci + Ecri - Vi inside the radicals.
3. Stiickel
Hamiltonians
Sdckel (see Ref. 10) extended the Liouville’s theory to a larger class of “diagonal” metric
Hamiltonians h= $xg”pf, based on the notion of Stiickel matrix
~.1I(Xl)
***
...
~,n(XI)
1
U=I ... ... ... *-e-1... (+nnw
I*
I ~.nl(X,)
a21b2>
...
.a*
~2n(X2)
Entries in each row of u depend on a single variable: xl--for the first row, x,-for
the second
one, etc. The coefficients {g”} of h are given by the first row of the inverse Stackel matrix e-’
5388
where q’,j=Mji(a)/detld,
in terms of minors (cofactors) of u. The reciprocal coefficients
gii= l/g” define the corresponding Stackel metric tensor
ds2=;
c
giidx’.
Stackel metric Hamiltonian (kinetic energy) can be perturbed with any potential of the type
V= C
Via”,
where {Vi(xi)}
are arbitrary single-variable functions as in the Liouville’s
separable StEkel Hamiltonian has the form
case. So a general
The separation of the Stackel’s HJ equation proceeds similarly to the Liouville’s
write
C
a’i[(aiS)2+
Vi]=E
case. We
(72)
and observe that summation Xi... could be viewed as the product of first row of the inverse
Stackel matrix by the column vector v={pF+ Vi}. Supplementing (72) with the remaining rows of
0-I we can write the product as
qv)
$).
(73)
The separation constants {ci} in the r.h.s. can be chosen arbitrarily, the first of them being energy
c, = E. Multiplying both sides by CTwe get a sequence of separated equations
(diS>2+ Vi=C
Cjffij(Xi)
i
an ODE in the ith variable Xi. Hence follows the separation of the HJ equation in St&kel’s
coordinates,
Si(Xi) =
I
&dxi
;
with
fi= C
Cj(Tij(Xi)- Vi
j
(74)
and explicit (quadrature) solution of the flow (68)
(75)
Neumann problem
Furthermore, functions
hi=C j
cr’j(p;+
(summation over the
Vj)
ith
column
of
0)
are easily verified to form a maximal Poisson-commuting family for Stackel Hamiltonians
h = Zsg”pF and E g”(pe + Vi). In particular, the StEckel metric form h gives rise to an integrable
geodesic flow.
4. St&kel
form of the Liouville
and Neumann problems
We shall show that the Liouville and Neumann Hamiltonians belong to the Stackel class, and
exhibit Sdckel matrices for both. Let us remark that the St&ckel class exhosts all possible separable metric-form Hamiltonians in n variables under certain natural conditions (see Ref. 10).
In the Liouville’s case the St&kel matrix has {ai} in the first column, ones on the upper
subdiagonal and {- 1) in the last row,
Cl
1
*.
c2
u=
_ o;,
-1
1
-1
...
Here integrands vi} of (74) are given by fi= aiE + ci SO the conserved vector in the r.h.s. of (73)
is made of (E;c,;...;c,-I).
In the Neumann case entries of (T depend on polynomial a(z) = II (z - ai). We shall abbreviate
the values of t2 at points {U r ; . . . ; un} by ai=a(ui). Then the Stackel matrix becomes
ru1
n-1
n-2
Ul
a2
UIL-l
n
-
...
a2
a2
g-2
n
I a,
1
-
...
1
a,
The corresponding conserved vector (73) is (c, = E; c2 ;. ..; c,) and the integrands assume the
fractional form
Z;f-lC*-jUi
EU1
+-
fi=
ai
ai
the first term representing the metric (kinetic) component of the Hamiltonian (74), the second
resulting from the Neumann potential. The latter represents the value of rational function
R(z)=
b(z)
a(z)
at z= Ui used throughout the paper, particularly Sets. III-IV.
5390
5. Integration
and classical
Neumann problem
periods
We remark that Stickel separation yields explicit momentum variables p =p(x;c)
on the joint
level set of {Ci}, Lagrangian A(c), the fundamental path { ri} on A and periods of the action form
p. dx along { ri}, needed in the semiclassical quantization of Sec. III. Indeed, by (66)
pi+fi
1’
1
So the fundamental paths are coordinate tori
yi:p:=fi
in the
(Xi ;pi)-phase
plane
and the corresponding classical periods
Ti(c)=
fy/-J.dx=2/
&dxi*
In the Neumann case fi=R( Ui), so we rederive the Veselov-Novikov
s Jbladz over the positive branch-cut of rational function R(z) = b/a.
formula for { Ti} as
‘Another closely related classical problem, the geodesic flow on ellipsoid: A-‘x.x=
1, studied by Jacobi, is also integrable. In fact both problems (Neumann and Jacobi) could be transformed one into the other: H. Knijrrer, J. fur die Reine
und Ang. Math. 334, 69-78 (1982).
‘5. Moser, Integrable Hamiltonian systems and spectral theory (Lezioni Fermiane p. 59, 1981).
3F. M. Arscott, Periodic differential equations: an introduction to Math&
Lame’ and allied equations (Pergamon,
London, 1964).
4D. Gurarie, (1) “Inverse spectral Problem for the 2-sphere SchrZdinger operators with zonal potentials,” Lett. Math.
Phys. 16, 313-323 (1988); (2) “Two-sphere Schriidinger operators with odd potentials,” Inverse Problems 371-378
(1990); (3) “Zonal S&&linger operators on the n-sphere: Inverse Spectral Problem and#Rigidity,” Commun. Math.
Phys. 131, 571-603 (1990).
‘V. Guillemin, (1) “Spectral theory of S2: some open questions,” Adv. Math. 42, 283-290 (1981); (2) “Some spectral
results for the Laplacian on the n-sphere,” Adv. Math. 27, 273-286 (1978).
6A. Uribe, (1) “A symbol calculus for a class of pseudodifferential operators on S”,” J. Funct. Anal. 59, 535-556 (1984);
(2) “Band invariants and closed trajectories on S”,” Adv. Math. 58, 3 (1985).
‘A. Weinstein, “Asymptotics of eigenvalue clusters for the Laplacian plus a potential,” Duke Math. J. 44, 883-892
(1977).
*Y. Colin de Verdiere, “Spectre conjoint d’operateurs pseudodifferentials qui commutent,” Math. 2. 171, 5 l-73 (1980).
9J. A. Toth, “The Quantum Neumann problem,” IMR Notices, 1993; “Various quantum mechanical aspects of quadratic
forms,” Preprint, 1993.
“E. G. Kalnins, Separation of variables for Riemannian spaces of constant curvature, Pitman monographs in Pure and
Appl. Math. 28 (Longman, 1986).
“W. Miller, Jr., Symmetry and separation of variables (Addison-Wesley, Reading, MA, 1977).
“A. M. Perelomov, Integrable systems of classical mechanics and Lie algebras (Birkhauser, Boston, 1991).
13B Dubrovin, V. Matveev, and S. Novikov, “Non-linear equations of the KdV-type, finite-zone linear operators and
abelian varieties,” Russ. Math. Sure. 31, 59-146 (1976).
14H. P. McKean and E. Trubowitz, “Hill operators and hyperelliptic function theory in the presence of infinitely many
branch points,” Corn. Pure Appl. Math. 29, 143-226 (1976).
‘sG Eskin, J. Ralston, and E. Trubowitz, “ The multidimensional inverse spectral problem with a periodic potential,”
C&em. Math. 27 (1984).
16E Beokolos and V. Enolskii, “Verdier’s elliptic solitons and the Weierstrass reduction theory,” Func. Anal. Appl. 23,
46-47 (1989).
“V Maslov and M. Fedoryuk, Quasiclassical approximations for the equations of Quantum mechanics (Moskow, Nauka,
1476).
‘*M. Gutzwiller, Chaos in classical and quantum mechanics (Springer, New York, 1990).
19A. Veselov and S. Novikov, “Poisson brackets and complex tori,” Proc. Steklov Inst. 165, 53-66 (1985).
Neumann problem
5391
“E L. Ince, Ordinary differential equations (Dover, New York, 1956).
2’& Magnus and S. Winkler, Hill’s equation (Dover, New York, 1979).
22B. M. Levitan. The inverse Sturm-Liouville Problem (Moskwa, Nauka, 1984).
23J. Pijschel and E. Trubowitz, Inverse spectral theory, Pure and Appl. Math., Vol. 130 1987 (Academic, New York, 1987).
“A. Erdelyi et al., Higher transcendental funciions, Vol. III (Bateman Manuscript Project) (McGraw-Hill, New York,
1953.
2’M. V. Fedoryuk, “The Lame wave equation,” Russ. Math. Surv. 44, 153-180 (1989).

Quantized Neumann problem, separable potentials on Sn and the

Transcription

Similar documents

McGuffey`s 3rd Reader - Eclectic Education Series

our 2015 Spring Newsletter

Issue 4 | September 1, 2015 - University of Colorado Hospital

the conference brochure

2016-2017 Student Handbook - Ss. Neumann and Goretti High School

Cow comfort and lameness on British Columbia dairy farms

the Summer 2011 Newsletter

PDF, 1132 KB

Property Tax (Predial) Government website

Stoichiometry-driven metal-to-insulator transition in