General Nth order integrals of the motion

0.28 MB·English

by  S. Post

General Nth order integrals of the motion
S. Post1 and P. Winternitz2‡
1 DepartmentofMathematics,UniversityofHawai’iatM¯anoa
2625McCarthyMall,Honolulu(HI)96822,USA
2 D´epartement deMath´ematiques etdeStatistiqueandCentredeRecherches
Math´ematiques, Universit´edeMontr´eal.
5 CP6128,SuccursaleCentre-Ville,Montr´eal(QC)H3C3J7,Canada
1
E-mail: [email protected], [email protected]
0
2
Abstract. The general form of an integral of motion that is a polynomial
b of order N in the momenta is presented for a Hamiltonian system in two-
e
dimensional Euclidean space. The classical and the quantum cases are treated
F
separately,emphasizingboththesimilaritiesandthedifferencesbetweenthetwo.

1 The main application will be to study Nth order superintegrable systems that
1 allow separation of variables in the Hamilton-Jacobi and Schro¨dinger equations,
respectively.

]
h
p
- PACSnumbers: 02.30.Ik45.20.Jj
h
t
a 1. Introduction
m
[ The purpose of this article is to provide a framework for systematically studying

finite-dimensional integrable and superintegrable systems with integrals of motion
3
that are polynomials of arbitrary finite order, N, in the momenta. In the process,
v
1 we also establish some basic properties of the integrals of the motion and study some
7 differences between the integrals in classical and quantum mechanics.
4 We restrict ourselves to a two-dimensional real Euclidean plane and to
0
Hamiltonians of the form
0
. H =p2+p2+V(x,y). (1)
1 1 2
0 Inclassicalmechanicsp andp arethecomponentsofthelinearmomentum,towhich
1 2
5
we add for use below the angular momentum
1
: L =xp −yp .
v 3 2 1
i In quantum mechanics, p and p (as well as H and L ) will be Hermitian operators
X 1 2 3
with
r
a pˆ =−i~∂ , pˆ =−i~∂ . (2)
1 x 2 y
In classical mechanics an Nth order integral of the motion can be written as
N N−k
X = f (x,y)pjpN−k−j, f (x,y)∈R, (3)
j,k 1 2 j,k
k=0 j=0
X X
‡ On Sabbatical at Dipartimento di Matematica e Fisica, Universit`a di Roma Tre, Viadella Vasca
Navale84,00146, Roma,Italy
General Nth order integrals of the motion 2
or simply
X = f (x,y)pjpN−k−j, (4)
j,k 1 2
j,k
X
with f = 0 for j < 0, k < 0 and k+j > N. The leading terms (of order N) are
j,k
obtained by restricting the summation to k =0. In quantum mechanics we also take
the integral of the form (3) (or (4)) but the p are operators as in (2) and we must
i
symmetrize in order for X to be Hermitian (see Section 3 below).
We recall that a Hamiltonian with n degrees of freedom in classical mechanics
is integrable (Liouville integrable) if it allows n integrals of motion (including the
Hamiltonian)thatarewelldefinedfunctionsonphasespace,areininvolution(Poisson
commute) and are functionally independent. The system is superintegrable if it
allows more than n integrals that are functionally independent and commute with
the Hamiltonian. The system is maximally superintegrable [49] if it allows 2n−1
functionally independent, well defined integrals, though at most n of them can be in
involution.
In quantum mechanics the definitions are similar. The integrals are operators in
the envelopingalgebraofthe HeisenbergalgebraH ∼{x ,...,x ,p ,...,p ,~}that
n 1 n 1 n
are either polynomials or convergent power series. A set of integrals is algebraically
independent if no Jordan polynomial (formed using only anti-commutators) in the
operators vanishes.
The best known superintegrable systems are the Kepler-Coulomb system [4,12]
with the potential V = α/r and the harmonic oscillator [25,50] with V = αr2.
According to Bertrand’s theorem [5], the only rotationally invariant potentials in
whichallboundedtrajectoriesareclosedarepreciselythesetwopotentials. Atheorem
proven by Nekhoroshev [51] states that, away from singular points, if a Hamiltonian
systeminndimensionsismaximallysuperintegrable(2n−1integralsofmotion)then
all bounded trajectories are periodic. It follows that there are no other rotationally
invariant maximally superintegrable systems in a Euclidean space E .
n
Asystematicstudyofothersuperintegrablesystemsstartedwiththeconstruction
of all quadratically superintegrablesystems in E and E [13,39,63]. Superintegrable
2 3
systems with second-order integrals of motion are by now well understood both in
spaces of constant curvature and in more general spaces [9–11,27–32,34,58,59].
Second-ordersuperintegrabilityisrelatedtomulti-separabilityintheHamilton-Jacobi
equations and the Schr¨odingerequation. The superintegrable potentials are the same
in classical and quantum mechanics. When commuted amongst each other, the
integrals of motion form quadratic algebras [6–8,15,16,28,34,35].
Superintegrablesystemsinvolvingonethird-orderandonelowerorderintegralof
motion in E have been studied more recently [17,18,48,52,62]. The connectionwith
2
multiple separation of variables is lost. The quantum potentials are not necessarily
the same as the classical ones, i.e. quantum potentials that depend on the Planck
constant appear. Some of the quantum potentials obtained involve elliptic functions
orPainlev´etranscendents. Theintegralsofmotionformpolynomialalgebrasandthese
canbeusedtocalculateenergyspectraandwavefunctions[41,43,48]. Arelationwith
supersymmetryhasbeenestablished[42,44,46,56]. Itwasrecentlyshownthatinfinite
families of two-dimensional superintegrable systems exist with integrals of arbitrary
order [26,33,36,44,53,54,60,61]. Superintegrable systems not allowing separation of
variables have been constructed [37,38,55]
General Nth order integrals of the motion 3
The present article is to be viewed in the context of a systematic study of
integrable and superintegrable systems with integrals that are polynomials in the
momenta, especially for those of degree higher than two. Here we concentrate on the
properties of one integral of order N in two-dimensional Euclidean space.
The remainder of the article is organized as follows. Section 2 is devoted to N-
th order integrals of motion in classical mechanics and includes a derivation of the
classicaldeterminingequations. The determining equationsforquantumintegralsare
derived in Section 3. They are shown to have the same form as the classical ones,
up to quantum corrections. These corrections are shown to be polynomials in the
squareofthePlanckconstant~andarepresentedexplicitly. Thegeneralformulasare
specializedtolowordercasesN=2,...,5inSection4. Differentpossiblequantization
procedures are compared and special cases are considered in Section 5. Conclusions
and a future outlook are presented in Section 6.
2. Classical Nth order integrals of the motion
Let us consider the classical Hamiltonian (1) and the Nth order integral (4) where
f (x,y)arerealfunctions. SinceX of(4)isassumedtobeanintegralofthemotion,
j,k
it must Poisson commute with the Hamiltonian
{H,X} =0. (5)
PB
The commutation relation (5) leads directly to a simple but powerful theorem.
Theorem 1 A classical Nth order integral for the Hamiltonian (1) has the form
[N2]N−2ℓ
X = f pjpN−j−2ℓ, (6)
j,2ℓ 1 2
ℓ=0 j=0
X X
where f are real functions that are identically 0 for j,ℓ<0 or j >N−2ℓ, with the
j,2ℓ
following properties:
(i) The functions f and the potential V(x,y) satisfy the determining equations
j,2ℓ
0=2 ∂ f +∂ f − (j+1)f ∂ V+(N−2ℓ+2−j)f ∂ V .(7)
x j−1,2ℓ y j,2ℓ j+1,2ℓ−2 x j,2ℓ−2 y
(cid:18) (cid:19) (cid:18) (cid:19)
(ii) As indicated in (6), all terms in the polynomial X have the same parity.
(iii) The leading terms in (6) (of order N obtained for ℓ = 0) are polynomials of
order N in the enveloping algebra of the Euclidean Lie algebra e(2) with basis
{p ,p ,L }.
1 2 3
Proof We calculate the Poissoncommutator (5) for H as in (1) and X as in (4)
∂f ∂f
{H,X} = −2 j,kpj+1pN−k−j −2 j,kpjpN−k−j+1
PB ∂x 1 2 ∂y 1 2
j,k
X
∂V ∂V
+j f pj−1pN−k−j +(N −k−j) f pjpN−k−j−1. (8)
∂x j,k 1 2 ∂y j,k 1 2
The first two terms are of order N −k+1, the second of order N −k−1. We shift
j →j−1 in the first term and j →j+1 in the third to obtain
∂f ∂f
0= −2 j−1,k + j,k pjpN−k−j+1
∂x ∂y 1 2
j,k (cid:18) (cid:19)
X
∂V ∂V
+ (j+1) f +(N −k−j) f pjpN−k−j−1. (9)
∂x j+1,k ∂y j,k 1 2
(cid:18) (cid:19)
General Nth order integrals of the motion 4
The terms of order N +1 in the momenta, in (9), correspond to k =0 and are
∂f ∂f
j−1,0 j,0
+ =0. (10)
∂x ∂y
Eq. (10) is the condition for the highest order terms of X to Poisson commute with
the free Hamiltonian H = p2+p2, thus proving the third statement in Theorem 1.
0 1 2
From the form of (9), we see that even and odd terms in X are independent. This
proves statement 2 of the theorem. Finally, we shift k →k+2 in the second term of
(9). The coefficient of pjpN−k−j+1 (after the shift) must vanish independently for all
1 2
(j,k) and we obtained the determining equations (7).
Corollary 1 The classical integral (6) can be rewritten as
⌊N2⌋N−2ℓ
X = A LN−m−npmpn+ f pjpN−j−2ℓ, (11)
N−m−n,m,n 3 1 2 j,2ℓ 1 2
0≤m+n≤N ℓ=1 j=0
X X X
where A are constants.
N−m−n,m,n
Proof: Asnotedabove,thedeterminingequationsforf givenby(10)donotdepend
j,0
on the potential. The solutions of (10) are
N−j j
f = N−n−m A xN−j−n(−y)j−m, (12)
j,0 j−m N−n−m,m,n
n=0m=0(cid:16) (cid:17)
X X
which give the form of the integral (11). Thus for all N the solutions of (7) for ℓ=0
are known in terms of the (N +1)(N +2)/2 constants A figuring in (11).
N−n−m,m,n
Let us add some comments.
(i) Forphysicalreasons(timereversalinvariance)wehaveassumedthatthefunctions
f (x,y) ∈ R from the beginning. This is actually no restriction. If X were
j,k
complex, its realandimaginarypartswouldPoissoncommute withH separately
(for V(x,y)∈R) and hence each complex integral would provide two real ones.
(ii) The number of determining equations (7) is equal to
[N+1]
2 1(N +3)2 N odd
(N −2ℓ+2)= 4 (13)
1(N +2)(N +4) N even.
ℓ=0 (cid:26) 4
X
ForagivenpotentialV(x,y)theequationsarelinearfirst-orderpartialdifferential
equations for the unknowns f (x,y). The number of unknowns is
j,2ℓ
[N+1]
2 (N+1)(N+3) N odd
(N −2ℓ+1)= 4 (14)
1(N +2)2 N even.
ℓ=0 (cid:26) 4
X
As is clear from Corollary 1, the determining equations for f can be solved
j,0
without knowledge of the potential and the solutions depend on (N +1)(N +2)/2
constants. Thus, N +1 of the functions f , namely f , are known in terms of
j,2ℓ j,0
(N +1)(N +2)/2 constants. The remaining system is overdetermined and subject
to further compatibility conditions. If the potential V(x,y) is not a priori known,
then the system (7) becomes nonlinear and V(x,y) must be determined from the
compatibility conditions. We present the first set of compatibility conditions as a
corollary.
General Nth order integrals of the motion 5
Corollary 2 If the Hamiltonian (1) admits X as an integral then the potential
function V(x,y) satisfies the following linear partial differential equation (PDE)
N−1
0= ∂N−1−j∂j(−1)j[(j+1)f ∂ V +(N −j)f ∂ V]. (15)
x y j+1,0 x j,0 y
j=0
X
Proof: This linear PDE is determined by the compatibility conditions for the ℓ = 1
set of determining equations, namely
2(∂ f +∂ f )−[(j+1)f ∂ V +(N −j)f ∂ V]=0.(16)
x j−1,2 y j,2 j+1,0 x j,0 y
Therefore when X is an integral, the functions f exist and satisfy (16) and so the
j,2
potential satisfies (15). This PDE depends only onthe constants A in the highest
j,k,ℓ
order terms of X, (11).
(iii) For N odd, the lowest-order determining equations are
f V +f V =0. (17)
1,N−1 x 0,N−1 y
In particular, for the N = 3 case, the compatibility conditions of (17) with the
determining equations for f lead to nonlinear equations for the potential [18,62].
j,2
3. Quantum Nth order integrals of the motion
Inthissection,weshallpresentatheoremanalogoustotheclassicaloneoftheprevious
section, but applied instead to quantum systems. That is, we would like to showthat
givenanNthorderdifferentialoperatorX thatisformallyself-adjointwithrespectto
the Euclidean metric and that commutes with a given Hamiltonian, then the number
of independent functions and determining equations are equal to that of the classical
caseandthedeterminingequationsarethesameasintheclassicalcase,uptoquantum
corrections which are polynomial in ~2. Specifically, we have the following theorem.
Theorem 2 A quantum Nth order integral for the Hamiltonian (1) has the form
1⌊N2⌋N−2ℓ
X = {f ,pˆjpˆN−2ℓ−j}, (18)
2 j,2ℓ 1 2
ℓ=0 j=0
X X
where f are real functions that are identically 0 for j,ℓ<0 or j >N−2ℓ, with the
j,2ℓ
following properties:
(i) The functions f and the potential V(x,y) satisfy the determining equations
j,2ℓ
0=M , (19)
j,2ℓ
with
M ≡2(∂ f +∂ f )
j,2ℓ x j−1,2ℓ y j,2ℓ
− (j+1)f ∂ V +(N −2ℓ+2−j)f ∂ V +~2Q ,
j+1,2ℓ−2 x j,2ℓ−2 y j,2ℓ
where Q is a quantum correction term given by
j,2ℓ(cid:0) (cid:1)
Q ≡ 2∂ φ +2∂ φ +∂2φ +∂2φ (20)
j,2ℓ x j−1,2ℓ y j,2ℓ x j,2ℓ−1 y j,2ℓ−1
ℓ−22n+3
(cid:0) (cid:1)
− (−~2)n j+m N−2ℓ+2n+4−j−m (∂m∂2n+3−mV)f
m 2n+3−m x y j+m,2ℓ−2n−4
n=0m=0 (cid:16) (cid:17)(cid:16) (cid:17)
X X
2ℓ−1 n
− (−~2)⌊(n−1)/2⌋ j+m N−2ℓ+n+1+j−m (∂m∂n−mV)φ ,
m n−m x y j+m,2ℓ−n−1
n=1m=0 (cid:16) (cid:17)(cid:16) (cid:17)
X X
General Nth order integrals of the motion 6
where the φ are defined for k >0 as
j,k
ℓ 2b−ǫ(−~2)b−1
φ = j+a N−2ℓ+2b−j−a ∂a∂2b−ǫ−af , (21)
j,2ℓ−ǫ 2 a 2b−ǫ−a x y j+a,2ℓ−2b
b=1 a=0 (cid:16) (cid:17)(cid:16) (cid:17)
X X
with ǫ=0,1. In particular φ =0, hence Q =0 so the ℓ=0 determining are
j,0 j,0
the same as in the classical case.
(ii) As indicated in (18), the symmetrized integral will have terms which are
differential operators of the same parity.
(iii) The leading terms in (18) (of order N obtained for ℓ = 0) are polynomials of
order N in the enveloping algebra of the Euclidean Lie algebra e(2) with basis
{pˆ ,pˆ ,Lˆ }.
1 2 3
Notice that the parity constraint on the integral (18) reduces the number of
possible functional coefficients by about half. Indeed, if the integral is expanded out
with the derivatives moved to the left then the integral would be of form
X = f −~2φ pˆjpˆN−2ℓ−j −i~ φ pˆjpˆN−2ℓ+1−j. (22)
j,2ℓ j,2ℓ 1 2 j,2ℓ−1 1 2
ℓ,j ℓ,j
X(cid:0) (cid:1) X
Thus, for a general, self-adjoint Nth-order integral that commutes with H,
approximately half of the coefficient functions depend only on derivatives of the
functionsf ,thesearetheφ . Ingeneral,thefunctionsφ (21)arepolynomial
j,2ℓ j,2ℓ−1 j,k
in ~2.
We begin the proofs by showing that, modulo lower order integrals of motion, X
can be taken to be self-adjoint.
Lemma 1 Given X an Nth order differential operator that commutes with a self-
adjoint Hamiltonian H, then X can be assumed to be self-adjoint.
Proof. We can always write X = X+X† + X−X†. Using this, we obtain
2 2
X +X† X −X†
0=[ ,H]+[ ,H]
2 2
and its Hermitian conjugate
X +X† X−X†
0=−[ ,H]+[ ,H]
2 2
which together show that the self-adjoint and skew-adjoint part of the operator X
mustsimultaneouslycommute with H. Since, X−X† commuteswith H sowill iX−X†
2 2
and hence it is possible to assume, without loss of generality, that the integral is
already in self-adjoint form. Q.E.D.
Next, we show that a commutator of the form [f,∂j∂1+k−j] can be written as
x y
a sum of lower order anti-commutators which differ in degree by 2. We assume for
convenience that all of the functions are smooth but of course this could be reduced
to require only the number of necessary derivatives, in this case k+1.
Lemma 2 Given a real f ∈C∞(R2), there exist real functions, f ∈C∞(R2), that
m,n
satisfy the equality
k min{j,n+1}
[f,∂j∂1+k−j]= {f ,∂j−m∂k−n−j+m} (23)
x y m,n x y
n=0m=max{0,n−k+j}
X X
and furthermore the functions with odd second index, f , are identically 0.
m,2ℓ+1
General Nth order integrals of the motion 7
Proof. Wedefinethe f inductivelyonn.Noticethatthe summationindicesonthe
m,n
right-hand side of the identity (23) lie within the following regime
Λ={(m,n)|0≤j−m≤j, 0≤k−n−j+m≤k−j+1}. (24)
Furthermore, the index n determines the total degree of the monomial
∂j−m∂k−n−(j−m). We define the identity operator to be ∂0∂0 =Id.
x y x y
Consider the quantity
U =[f,∂j∂1+k−j]− {f ,∂j−m∂k−j−n+m}. (25)
x y m,n x y
n,m∈Λ
X
Expanding out U gives
U =−2 U ∂j−m∂k−n−j+m,
n,m x y
n,m∈Λ
X
with
1
U ≡ f + j k−j+1 ∂m∂1+n−mf (26)
n,m m,n 2 m 1+n−m x y
(cid:18) (cid:16) (cid:17)(cid:16) (cid:17)
n−1
1
+ j−m+a k−n−j+m+1+b−a ∂a∂1+b−af ,
2 a 1+b−a x y m−a,n−1−b
b=0a∈σ(cid:16) (cid:17)(cid:16) (cid:17) (cid:19)
XX
where σ ={max{0,b−n+m}...min{m,k−j−n+b+1+m}}. Hence for a fixed
m,n we have a recurrence relation on the f ’s which will make each U =0 this
m,n m,n
relation is a recurrence on n given by
1
f = − j k−j+1 ∂m∂1+n−mf (27)
m,n 2 m 1+n−m x y
(cid:16) (cid:17)(cid:16) (cid:17)
n−1
1
− j−m+a k−n−j+m+1+b−a ∂a∂1+b−af .
2 a 1+b−a x y m−a,n−1−b
b=0a∈σ(cid:16) (cid:17)(cid:16) (cid:17)
XX
By induction on n, the functions f exist and are real. The first two functions
m,n
(n=0) are
1 1
f =− ∂ f, f =− ∂ f, j >0, 1+k−j >0.
0,0 y 1,0 x
2 2
Ifj =0or1+k−j =0,theneitherf =0orf =0,respectively. Thesummation
1,0 0,0
over a and b ensures that (m−a,n−1−b)∈Λ and so the functions are well-defined
recursively on n for all m,n∈Λ.
Finally, we see that if k is even then k + 1 will be odd and so the operator
[f,∂j∂1+k−j] will be self-adjoint which gives the requirements that the sum
x y
k j
{f ,∂j−m∂k−n−j+m}
m,n x y
n=0m=0
X X
contains only even terms in k−n and hence only terms with n=2ℓ can be non-zero.
Similarly, for k odd, k+1 is even and so both sides must be skew-adjoint and hence
k−n must be odd which again gives the requirement that n=2ℓ. Q.E.D.
Next, we use the previous lemma to show that any self-adjoint operator X can
be put into symmetric form so that the functional coefficients are real.
General Nth order integrals of the motion 8
Lemma 3 Givenageneral, self-adjoint Nthorderdifferentialoperator,X,thereexist
real functions f such that
j,k
1
X = {f ,pˆjpˆN−k−j} (28)
2 j,k 1 2
k j
XX
Proof. Given a self-adjoint Nth-order differential operator X, we can always move all
of the functional coefficients to the left to obtain
X = F (−i~)N−k∂j∂N−k−j, (29)
j,k x y
k j
XX
F ≡0, ∀ j <0,k<0,j >k,k>N,
j,k
where the functions F are possibly complex; we write F = FR +iFI . Since X
j,k j,k j,k j,k
is self-adjoint, it can be expressed as
1 1
X = (X +X†)= {FR ,∂j∂N−k−j}+i[FI ,∂j∂N−k−j] (−i~)N−k. (30)
2 2 j,k x y j,k x y
k j
XX(cid:0) (cid:1)
Next,weusethepreviouslemmatoshowthatforagivenj,kthereexistsrealfunctions,
call them gj,k such that
m,n
⌊N−2k−1⌋ j
i[FI ∂j∂N−k−j](−i~)N−k = {(−1)n~2n+1gj,k ,pˆj−mpˆN−k−1−2n−j+m}.
j,k x y m,2n 1 2
n=0 m=0
X X
Forsimplicity,wedefinedgj,k ≡0whenever(m,2n)∈/ Λ(24). Thus,thereexistreal
m,2n
functions f such that X can be written as
j,k
1
X = {f ,pˆjpˆN−k−j}. (31)
2 j,k 1 2
k j
XX
where
[(k−1)/2] j
f =FR + (−1)n~2n+1gj+n,k−1−2n
j,k j,k m,2n
n=0 m=0
X X
Q.E.D.
Proof of Theorem 2 First, we prove that, modulo lower-order integrals of the
motion, X containsonly evenor odd terms. By Lemma 1, modulo lowerorder terms,
X canbe takentobe self-adjoint. Then, byLemma3weknowthatX canbe written
in the symmetrized form
X =Xe+Xo
1 [N2]N−2ℓ
Xe = (−i~)N−2ℓ{f ,∂j∂N−2ℓ−j}
2 j,2ℓ x y
ℓ=0 j=0
X X
1[N2−1]N−2ℓ−1
Xo = (−i~)N−2ℓ−1{f ,∂j∂N−2ℓ−j−1} (32)
2 j,2ℓ+1 x y
ℓ=0 j=0
X X
where the Xe and Xo have the opposite parity under time reversal (complex
conjugation). Thus, since H is completely real, we know that Xe and Xo must
General Nth order integrals of the motion 9
independently commute with H and so, modulo the lower-order integral Xo, X can
be written as
1 [N2]N−2ℓ
X = (−i~)N−2ℓ{f ,∂j∂N−2ℓ−j}. (33)
2 j,2ℓ x y
ℓ=0 j=0
X X
Next, we consider the determining equations for the functions f and the
j,k
potential V making X a quantum integral. Take X and H as above both self-adjoint
and write
1+N1+N−k
[X,H]≡ M ∂j∂1+N−k−j(−i~)2+N−k. (34)
j,k x y
k=0 j=0
X X
WewillshowthatthetermsM canbewrittenasdifferentialconsequencesofthe
j,2ℓ+1
M andthattherequirements0=M aregivenby(19),whicharethedetermining
j,2ℓ j,2ℓ
equationsforthe system. We beginbyshowingthatthe termsM canbe written
j,2ℓ+1
as differential consequences of the M . We know that [X,H] is skew-adjoint and so
j,2ℓ
we have the requirement that
1+N1+N−k
M ∂j∂1+N−k−j(−i~)1+N−k †+M ∂j∂1+N−k−j(−i~)1+N−k =0.(35)
j,k x y j,k x y
k=0 j=0
X X (cid:0) (cid:1)
Now,from(33)weknowthat(−i)NX willbeevenwithrespecttocomplexconjugation
(time reversal) and so we can see that [(−iN)X,H] will be real. Thus, since we have
the equality
1+N1+N−k
[(−iN)X,H]= (i)NM ∂j∂1+N−k−j(−i~)2+N−k,
j,k x y
k=0 j=0
X X
we see that both side will be real differential operators which implies that the even
terms, M , are completely imaginary and the odd terms, M , are real. Hence,
j,2ℓ j,2ℓ+1
we can compute (35)
1+N1+N−k
0= M ∂j∂1+N−k−j(−i~)2+N−k †+M ∂j∂1+N−k−j(−i~)2+N−k
j,k x y j,k x y
k=0 j=0
X X (cid:0) (cid:1)
= [M ,∂j∂1+N−2ℓ−j(−i~)2+N−2ℓ]+{M ,∂j∂N−2ℓ−j(−i~)1+N−2ℓ}. (36)
j,2ℓ x y j,2ℓ+1 x y
j,ℓ
X
The coefficient of ∂j∂N−2ℓ−j(−i~)1+N−2ℓ in (36) gives
x y
0=M
j,2ℓ+1
ℓ 2n+1(−~2)n
+ i~ j+m N−2ℓ+2n+1−j+m ∂m∂2n+1−mM
2 m 2n+1−m x y j+m,2ℓ−2n
n=0m=0 (cid:16) (cid:17)(cid:16) (cid:17)
X X
ℓ−12n+2(−~2)n
−~2 j+m N−2ℓ+2n+2−j+m ∂m∂2n+2−mM .
2 m 2n+2−m x y j+m,2ℓ−2n−1
n=0m=0 (cid:16) (cid:17)(cid:16) (cid:17)
X X
For the case, ℓ=0 we obtain
−i~
M = ((j+1)∂ M +(N −j+1)∂ M )
j,1 x j+1,0 y j,0
2
andsobyinductiononℓ,wecanseethattheoddtermsM arelinearcombinations
j,2ℓ+1
of derivatives of the M .
j,2ℓ
General Nth order integrals of the motion 10
ThequantityM canthenbedirectlycomputedusingtheexpansionofX,asin
j,2ℓ
(22)withthe functions φ givenasin(21), toobtain(19). Notice thatthis equation
j,k
agrees with (7) modulo terms which are polynomial in ~2 and which vanish in the
classical limit (~→0).
Finally, as discussedabove,ifthe requirementsM =0 andtheir compatibility
j,2ℓ
equations are satisfied then so too will be M = 0 and so the requirement
j,2ℓ+1
[H,X]=0 will also be satisfied.
To finish the proof of Theorem 2, we show that the highest order terms of X are
in the enveloping algebra of the Lie algebra generated by pˆ ,pˆ and Lˆ ≡ ypˆ −xpˆ .
1 2 3 1 2
The determining equations for the f are the same as in the classical case (10) and
j,0
hencetheirsolutionsarethesame(12). Thus,ifsuchasymmetryoperatorexists,then
the highest order terms agree with those of an Nth-order operator in the enveloping
algebra of the Lie algebra generated by pˆ ,pˆ and Lˆ and so it is always possible to
1 2 3
express the highest order terms as operators in the enveloping algebra, with suitable
modification of the lower order terms. Q. E. D.
Just as in the classicalcase, the highest-order determining equations (10) can be
solveddirectly and the functions f are given by (12). However,as will be discussed
j,0
in Section 4 below, the choice of symmetrization of the leading order terms will lead
to ~2-dependent correction terms in the lower-order functions. The quantum version
of Corollary 2 still holds and, remarkably, the linear compatibility condition is the
same. This does not imply that the quantum and classical potentials are necessarily
the same since further (nonlinear)compatibility conditions exist. They are in general
different in the two cases.
Corollary 3 If the quantum Hamiltonian (1) admits X as an integral then the
potentialfunctionV(x,y)satisfiesthesamelinearPDEasintheclassicalcase,namely
(15).
Proof: The ℓ=1 set of determining equations are
M =0. (37)
j,2
with
M =2(∂ f +∂ f )− (j+1)f ∂ V +(N −j)f ∂ V +~2Q ,
j,2 x j−1,2 y j,2 j+1,0 x j,0 y j,2
with quantum correction term
(cid:2) (cid:3)
Q =2∂ φ +2∂ φ +∂2φ +∂2φ .
j,2 x j−1,2 y j,2 x j,1 y j,1
The functions φ , coming from expanding out the highest order terms are
j,k
j+1 N −j
φ = ∂ f + ∂ f ,
j,1 x j+1,0 y j,0
2 2
2
1 j+a N −j−a
φ = ∂a∂2−af .
j,2 2 a 2−a x y j+a,0
a=0 (cid:18) (cid:19)(cid:18) (cid:19)
X
The linear compatibility condition of (37) is obtained from
N−1
0= ∂N−1−j∂j(−1)j−1M (38)
y x j,2
j=0
X
N−1
= ∂N−1−j∂j(−1)j (j+1)∂ Vf +(N −j)∂ Vf +~2Q .
x y x j+1,0 y j,0 j,2
j=0
X (cid:2) (cid:3)