Pure quantum integrability

Pure quantum integrability Jarmo Hietarinta arXiv:solv-int/9708010v2 9 Sep 1997 Department of Physics, University of Turku FIN-20014 Turku, Finland November 14, 2018 Abstract The correspondence between the integrability of classical mechanical systems and their quantum counterparts is not a 1-1, although some close correspondencies exist. If a classical mechanical system is integrable with invariants that are polynomial in momenta one can construct a corresponding commuting set of differential operators. Here we discuss some 2- or 3-dimensional purely quantum integrable systems (the 1-dimensional counterpart is the Lame equation). That is, we have an integrable potential whose amplitude is not free but rather proportional to h̄2 , and in the classical limit the potential vanishes. Furthermore it turns out that some of these systems actually have N + 1 commuting differential operators, connected by a nontrivial algebraic relation. Some of them have been discussed recently by A.P. Veselov et. al. from the point of view of Baker-Akheizer functions. 1 Introduction In classical mechanics the most common definition of integrability is that of Liouville integrability: Suppose we have a system with N degrees of freedom, that is, we have N coordinates qi and N conjugate momenta pj , with Poisson brackets [qi , pj ] = δij . This system is said to be Liouville integrable [1], if there are N functions Fk (p, q) such that 1) [Fi , Fj ] = 0, ∀i, j, and 2) the Fi are functionally independent and 3) sufficiently regular. 1 One of the F ’s is the Hamiltonian that gives the dynamics. Weaker and stronger types of integrability have also been used (e.g. partial integrability, algebraic integrability and super-integrability). When the idea of integrability is applied to quantum mechanics the properties of commutativity and functional independence are mutually exclusive: According to a theorem of von Neumann [2], if a pair of self-adjoint operators A and B commutes, then there is a third self-adjoint operator C such that A = f (C), B = g(C). This problem is avoided if we concentrate on differential operators, which anyway are the most interesting from practical point of view. Thus, even if A = i∂x and B = i∂y commute, the operator C mentioned above is not a differential operator. In this paper we consider only differential operators and property 3) above is then automatically satisfied, but we should relax the independence requirement 2): we should only ignore cases where one of the commuting operator can be given as a polynomial of the others. In this article we study the existence of operators commuting with a standard type two-dimensional Hamiltonian (Schrödinger operator)
H = − 21 h̄2 ∂x2 + ∂y2 + V (x, y). (1) This may be considered as arising from the classical system H = 12 (p2x + p2y ) + V (x, y) by the usual correspondence pα → −ih̄∂α . (The classical integrability of such systems has been reviewed in [3].) The transition from classical to quantum integrability seems to be always possible, sometimes one just has to add correction terms to the Hamiltonian and to the operator commuting with it, but these correction terms are always O(h̄2 ) [4]. Recently the existence of a differential operator commuting with (1) has been discussed at length in [5] (along with multi-dimensional generalizations). In that work strong assumptions were made on the symmetries of the potential (invariance under the Weyl group), but it turns out that the purely quantum integrable (PQI) systems discussed here do not have such symmetry properties. An important difference between classical and quantum integrability lies in the question of independence as mentioned above. Thus it will be meaningful and interesting to have N + 1 commuting operators, even if they must be algebraically related (sometimes this is called algebraic integrability). This can be already illustrated for 1-dimensional systems. 2 They are always integrable (the Hamiltonian commutes with itself) but some systems are more integrable that others. For example [6] the Lame operator L = − 12 h̄2 d2 + h̄2 P(x), 2 dx (2) where P is the Weierstrass elliptic function, commutes with I = −ih̄3 d d3 + 23 ih̄3 , P(x) , 3 dx dx (3) where the bracket {., .} stands for an anticommutator. The operators L and I are algebraically related, I 2 = 8L3 − 21 h̄4 g2 L + 41 h̄6 g3 , (4) (where gi are the constants characterizing P) and this relation can be used to express the wave-function in terms of other elliptic functions. The potential h̄2 P(x) is the only one that has a third order commuting operator, for a fifth order operator generalizations are possible [6]. The classification of commuting ordinary differential operators was studied already in the 1920’s [7] and more recently this problem has been studied from the point of view of soliton theory. The corresponding two dimensional problem (i.e., the existence of three commuting differential operators) was studied in [8]. In the one-dimensional case the algebraically integrable cases are (trivially) special cases of normal (classical) integrability, and in [8] it was conjectured that this holds also in higher dimensions, but it turns out not to be the case. In Section 2 we present results from a search for a third order differential operator, commuting with H of (1). We find that there are indeed some PQI systems, whose potential is proportional to h̄2 and therefore vanishes in the classical limit. In section 3 we show, that the integrable systems found in Section 2 also have a fourth order commuting operator, and that the three operators are polynomially related. In Section 4 we consider fourth order differential operators commuting with the Hamiltonian (1) with a potential that is a generalization of the rational of four term Calogero-Moser potential. Again some PQI potentials are found. 3 2 Third order commuting operator We will consider only those third order differential operators I3 whose leading order part has constant coefficients. The analysis of the leading terms is the same as that of the corresponding classical system; this follows from the Weyl correspondence and the fact that the leading part of the Moyal bracket (which represents the commutator) agrees with the Poisson bracket [4]. Since the second order operator H has no first order derivatives, we may assume that I3 has no second order derivatives, i.e., I3 = (ih̄)3 (c3 ∂x3 + c2 ∂x2 ∂y + c1 ∂x ∂y2 + c0 ∂y3 ) + d11 ∂x + d10 ∂y + d0 . (5) From the leading terms of the commutation condition [I3 , H] = 0 one immediately derives a linear PDE for the potential V c1 Vxxx + (3c0 − 2c2 )Vxxy + (3c3 − 2c1 )Vxyy + c2 Vyyy = 0, (6) where the subscripts stand for partial derivatives. The generic solution of this equation is V = g1 (α1 x + β1 y) + g2 (α2 x + β2 y) + g3 (α3 x + β3 y), (7) where (αi , βi ) are the three solutions of c1 α3 + (3c0 − 2c2 )α2 β + (3c3 − 2c1 )αβ 2 + c2 β 3 = 0. (The non-generic cases need separate study.) The potential (7) is form invariant under coordinate rotations, which just imply changes in the parameters αi , βi . This suggests writing all formulae in a rotationally covariant form. Furthermore, we may assume that the parameters αi , βi satisfy the condition α1 + α2 + α3 = 0, β1 + β2 + β3 = 0. (8) (To obtain this it is only necessary to scale the arguments of the functions g.) The submanifold (8) is invariant under parameter rotations. Instead of x, y and the corresponding partial derivatives let us use the quantities Xi = αi x + βi y, X̌i = βi x − αi y, Li = −ih̄(βi ∂x − αi ∂y ), Ľi = −ih̄(αi ∂x + βi ∂y ), (9) 4 which satisfy [Li , Xj ] = −[Ľi , X̌j ] = ih̄Dij , [Li , X̌j ] = [Ľi , Xj ] = −ih̄∆ij , (10) ∆ij := αi αj + βi βj . (11) where Dij := αi βj − αj βi , Due to (8) we have D ≡ D12 = D23 = D31 = −D21 = −D32 = −D13 . On the manifold (8) we have ∆1i + ∆2i + ∆3i = 0 so that we have the following relations between the diagonal and non-diagonal elements: ∆ii = −(∆ij + ∆ki ), ∆ij = 21 (−∆ii − ∆jj + ∆kk ), (12) where {i, j, k} is a permutation of {1, 2, 3}. Furthermore L1 + L2 + L3 = 0, L21 ∆23 + L22 ∆31 + L23 ∆12 = h̄2 D 2 (∂x2 + ∂y2 ). (13) By a direct computation we find the following expression for I3 I3 = L31 ∆223 + L32 ∆231 + L33 ∆212 ) ( 3 3 X X + 32 D 2 gk ∆jk ) − 2gj ∆jj , Lj , ( j=1 (14) k=1 but there is one remaining condition that can be written as (∆22 − ∆33 )(g1′′′h̄2 ∆11 − 12g1′ g1 ) + (∆33 − ∆11 )(g2′′′h̄2 ∆22 − 12g2′ g2 ) (15) + (∆11 − ∆22 )(g3′′′h̄2 ∆33 − 12g3′ g3 ) − 12 [(g1 g2′ − g1′ g2 )∆12 + (g2 g3′ − g2′ g3 )∆23 + (g3 g1′ − g3′ g1 )∆31 ] = 0. The complete solution of (15) is not known, but several solutions can be found when we note that the Weierstrass elliptic function P satisfies the equations P ′′′ = 12PP ′ , 1 P(u) 1 P(v) P(u)′ P(v)′ = 0, where u + v + w = 0. 1 P(w) P(w)′ 5 (16) Thus if we assume that gi = ai P we can identify the last three terms of (15) as the above determinant, if a1 a2 ∆12 = a2 a3 ∆23 = a3 a1 ∆31 . (17) The first three terms of (15) imply three further equation: (∆22 − ∆33 )(h̄2 ∆11 − a1 ) = 0, (∆33 − ∆11 )(h̄2 ∆22 − a2 ) = 0, (18) (∆11 − ∆22 )(h̄2 ∆33 − a3 ) = 0. From (17) we immediately find that ai = ∆jk X, (i, j, k cyclic), (19) so that in any case the potential will be of the form V ∝ ∆23 P(α1 x + β1 y) + ∆31 P(α2 x + β2 y) + ∆12 P(α3 x + β3 y). (20) The solutions of (19,18) can be divided into three groups: Case 1: All ∆ii ’s equal. From (12) we find that the off-diagonal ∆’s and therefore ai ’s are all equal. The equality of ∆ii ’s allows the parameterization αi = 2A cos(θi ), βi = 2A sin(θi ), and from (8) it follows that θ2 = θ1 + 32 π, θ3 = θ1 − 23 π. This is the well known solution with a nonzero classical limit. In addition to the overall rotation there is the freedom of the amplitudes A and a. If we choose θ1 = π/6 we get the familiar form √ √   V3.1 = a P(A( 3x + y)) + P(A(− 3x + y)) + P(−2Ay) . (21) Case 2: Two ∆ii ’s equal. Let us say ∆11 = ∆22 6= ∆33 . Then from (18) X = h̄2 ∆11 /∆23 = −2h̄2 ∆11 /∆33 and a1 = a2 = h̄2 ∆11 , a3 = h̄2 ∆11 (2∆11 − ∆33 )/∆33 . Note the overall h̄2 in the amplitudes ai . One possible parameterization (after fixing the rotational freedom) is V3.2 = h̄2  A2 + B 2  2 2 2 2 2A P(Bx + Ay) + 2A P(−Bx + Ay) + (B − A )P(−2Ay) . 2A2 (22) This is the solution found by Veselov et. al. [9, 10] for the algebraic special case P(x) = x−2 . 6 Case 3: All ∆ii ’s different. Then we must have ai = h̄2 ∆ii by (18), and (19) implies ∆11 + ∆22 + ∆33 = 0. Thus some of the parameters must be complex. One parameterizations is given by √ √ √  V3.3 = h̄2 A2 2(−1 + i 3 sin(θ))P(i 3A cos(θ)x + A(1 + i 3 sin(θ))y) √ √ √ + 2(−1 − i 3 sin(θ))P(−i 3A cos(θ)x + A(1 − i 3 sin(θ))y)  + 4P(−2Ay) . (23) Note that even though some αi , βi must be complex, it is possible to get a real potential V , e.g., if P itself is real. This solution seems to be new. √ Note that Case 2 intersects with Case 1 at B = 3A (a = 4h̄2 A2 ), and with Case √ 3 at B = i 3A (θ = 0 in (23)). At these special points the system should have more symmetries. For Cases 2 and 3 the amplitudes are proportional to h̄2 so they are PQI. Although the amplitudes are fixed, there are two nontrivial degrees of freedom as in case 1, but they are now all in the direction parameters αi , βi . 2.1 Connection with a three dimensional system A Hamiltonian of the form (1,7) (with the term −∆12 ∆23 ∆31 p2z /D 2 added) can also be written as a three dimensional system H=−
h̄2 ∆23 ∂12 + ∆31 ∂22 + ∆12 ∂32 + g1 (q2 − q3 ) + g2 (q3 − q1 ) + g3 (q1 − q2 ), 2 through the linear transformation    x = −(q1 β1 + q2 β2 + q3 β3 )/D,   y = (q1 α1 + q2 α2 + q3 α3 )/D,     z = (q ∆ ∆ + q ∆ ∆ + q ∆ ∆ )/D 2 , 1 13 12 2 23 12 3 13 23 and correspondingly    ∂ = (∂1 β1 ∆23 + ∂2 β2 ∆13 + ∂3 β3 ∆12 )/D,   x ∂y = −(∂1 α1 ∆23 + ∂2 α2 ∆13 + ∂3 α3 ∆12 )/D,     ∂ = ∂ +∂ +∂ . z 1 2 3 7 (24) (25) (26) If we denote µi = ∆jk , (i, j, k cyclic) and use (20) the results can combined as
  2 H = − h̄2 µ1 ∂12 + µ2 ∂22 + µ3 ∂32 + A µ1 P(q2 − q3 ) + µ2 P(q3 − q1 ) + µ3 P(q1 − q2 ) , (27) and the three integrable cases are characterized as follows: 1. The classical case: µ1 = µ2 = µ3 , A free. 2. Elliptic PQI case: µ1 = µ2 , A = −h̄2 (µ1 + µ3 )/µ1 3. Hyperbolic PQI case: µ1 + µ2 + µ3 = 0, A = h̄2 . Note that for the two PQI cases the same expression A = h̄2 [1 − (µ1 + µ2 + µ3 )/µ1] works for the overall amplitude. In [9, 10] the n-dimensional elliptic case was associated to a non-Coxeter configuration called An (m), similar geometric interpretation for the hyperbolic case would be interesting. 3 Fourth order commuting operator In this section we consider the operator I4 = (ih̄) 4 4 X ci ∂xi ∂y4−i + . . . (28) i=0 and now the condition of commutativity [I4 , H] = 0 leads to the necessary condition c1 Vxxxx + 2(2c0 − c2 )Vxxxy + 3(c3 − c1 )Vxxyy + 2(c2 − 2c4 )Vxyyy − c3 Vyyyy = 0. (29) Although this is a fourth order equation it does not allow four arbitrary directions (αi , βi ), because of the way the ci ’s enter in the equation. We will return to this problem later and first consider potential with three terms. 3.1 Three term potential By direct calculation with (7,8) one finds that the invariant can be written as I4 = L41 ∆323 + L42 ∆331 + L43 ∆312 3 X  2 2 Lj , −gj ∆2j+1,j+2 + gk (∆k,j+1∆k,j+2 + 3∆2j+1,j+2) + C(x, y). (30) −D j,k=1 8 Here the indices for ∆ are to be taken modulo 3. The integrability condition for C can be written as (∆23 Ľ1 + ∆31 Ľ2 + ∆12 Ľ3 ) Ω = 0 (31) where Ω = K1 (∆22 − ∆33 )(g1′′′ h̄2 ∆11 − 12g1′ g1 ) + K2 (∆33 − ∆11 )(g2′′′ h̄2 ∆22 − 12g2′ g2 ) (32) + K3 (∆11 − ∆22 )(g3′′′ h̄2 ∆33 − 12g3′ g3 ) − 12 [(g1 g2′ − g1′ g2 )∆12 + (g2 g3′ − g2′ g3 )∆23 + (g3 g1′ − g3′ g1 )∆31 ] , and Ki = 3∆i,i+1 ∆i,i+2 ∆ii ∆i+1,i+2 + 2∆i,i+1 ∆i,i+2 (The denominator of Ki can also be written as [∆23 Ľ1 +∆31 Ľ2 +∆12 Ľ3 , X̌i ].) The expression (32) is the same as (15), except for the extra coefficients Ki , which however play no role if (32) is solved by gi = ai P with (17,18). The integrability condition for C may have other solutions, but we can say at least that all solutions obtained in Section 2 also have a fourth order invariant. However, since H2 is also a fourth order invariant commuting with H we must discuss the independence of the newly found invariants. 3.2 Relations between the invariants Since the system is two dimensional there can be at most two algebraically independent commuting quantities. Indeed on finds relations between the three operators found above, for example in Case 1 we have I4 = −∆11 D 4 H2 , (33) so that in this case the new fourth order invariant is useless. For the other cases the algebraic relation is less trivial and therefore the extra invariant provides additional information. The computations are rather extensive and we have verified only the relation connecting the leading terms of Ij , denoted below by Ij . The results are as follows: 9 For case 2 when ∆22 = ∆11 we have the relation I26 ∆333 (16∆411 + 16∆311 ∆33 − 20∆211 ∆233 − 12∆11 ∆333 + 9∆433 ) + 12I24I4 ∆233 (16∆411 + 8∆311 ∆33 − 16∆211 ∆233 − 2∆11 ∆333 + 3∆433 ) + 32I23I32 ∆233 (12∆411 − 26∆311 ∆33 + 15∆211 ∆233 − ∆433 ) + 48I22I42 ∆11 ∆33 (16∆311 − 9∆11 ∆233 + 2∆333 ) (34) + 384I2I32 I4 ∆11 ∆33 (4∆311 − 9∆211 ∆33 + 6∆11 ∆233 − ∆333 ) + 256I34∆11 (4∆411 − 13∆311 ∆33 + 15∆211 ∆233 − 7∆11 ∆333 + ∆433 ) + 64I43∆211 (16∆211 − 8∆11 ∆33 + ∆233 ) = 0, which is of degree 12 in the derivatives. If ∆11 = ∆33 (intersection with Case 1) this expression simplifies to 9∆411 (∆11 I22 + 4I4 )3 = 0, in agreement with (33). It simplifies also if ∆33 = 4∆11 but this corresponds to the trivial limit B = 0. In case 3 we get the following complicated identity among the leading terms: i64 (a2 − a + 1)4 a2 (a − 1)2 −6 i44 i22 (a2 − a + 1)3 a(a − 1)(a6 − 3a5 + 5a3 − 3a + 1) +12 i44 i23 i2 (a2 − a + 1)3 a(a − 1)(2a6 − 6a5 + 3a4 + 4a3 + 3a2 − 6a + 2) +3 i44 i42 (a2 − a + 1)2 (3a12 − 18a11 + 19a10 + 70a9 − 140a8 − 58a7 + 251a6 − 58a5 − 140a4 + 70a3 + 19a2 − 18a + 3) −2 i34 i43 (a2 − a + 1)3 a(a − 1)(8a6 − 24a5 + 21a4 − 2a3 + 21a2 − 24a + 8) −4 i34 i23 i32 (a2 − a + 1)2 (2a12 − 12a11 − 44a10 + 330a9 − 425a8 − 412a7 + 1124a6 − 412a5 − 425a4 + 330a3 − 44a2 − 12a + 2) +2 i34 i62 (a2 − a + 1)a(a − 1)(35a12 − 210a11 + 283a10 + 510a9 − 1268a8 − 298a7 + 1931a6 − 298a5 − 1268a4 + 510a3 + 283a2 − 210a + 35) −18 i24 i43 i22 (a2 − a + 1)2 a2 (a − 1)2 (a2 − a − 5)(5a2 − 11a + 5)(5a2 + a − 1) −12 i24 i23 i62 (a2 − a + 1)a(a − 1)(13a12 − 78a11 + 54a10 + 445a9 − 786a8 − 384a7 + 1485a6 − 384a5 − 786a4 + 445a3 + 54a2 − 78a + 13) +3 i24 i82 (2a18 − 18a17 + 112a16 − 488a15 + 1012a14 + 28a13 − 3599a12 + 4330a11 + 2433a10 − 7622a9 + 2433a8 + 4330a7 − 3599a6 + 28a5 + 1012a4 10 − 488a3 + 112a2 − 18a + 2) +108 i4 i63 i2 (a2 − a + 1)2 a2 (a − 1)2 (a2 − 4a + 1)(a2 + 2a − 2)(2a2 − 2a − 1) +6 i4 i43 i42 (a2 − a + 1)a(a − 1)(8a12 − 48a11 − 120a10 + 1040a9 − 1323a8 − 1476a7 + 3846a6 − 1476a5 − 1323a4 + 1040a3 − 120a2 − 48a + 8) −12 i4 i23 i72 (a3 − 3a + 1)(a3 − 3a2 + 1)(2a12 − 12a11 + 41a10 − 95a9 + 86a8 + 94a7 − 230a6 + 94a5 + 86a4 − 95a3 + 41a2 − 12a + 2) 2 2 2 2 2 3 2 3 2 +6 i4 i10 2 (a − 2) (a + 1) (2a − 1) (a − a + 1) (a − 1)(a − 2a − a + 1)(a − a − 2a + 1) +729 i83 a4 (a − 1)2 (a2 − a + 1)2 +4 i63 i32 (a2 − a + 1)a(a − 1)(8a12 − 48a11 + 195a10 − 535a9 + 342a8 + 1314a7 − 2544a6 + 1314a5 + 342a4 − 535a3 + 195a2 − 48a + 8) +2 i43 i62 (8a18 − 72a17 + 231a16 − 216a15 − 777a14 + 3507a13 − 6201a12 + 3003a11 + 7470a10 − 13898a9 + 7470a8 + 3003a7 − 6201a6 + 3507a5 − 777a4 − 216a3 + 231a2 − 72a + 8) −4 i3 i92 (a2 − a + 1)2 a(a − 1)(a3 − 3a + 1)(a3 − 3a2 + 1) (5a6 − 15a5 − 3a4 + 31a3 − 3a2 − 15a + 5) 2 4 3 2 2 3 2 2 + i12 2 (a − a + 1) (a − 2a − a + 1) (a − a − 2a + 1) = 0, (35) where we have used the parameterization ∆11 = (1 − a)b, ∆22 = ab, ∆33 = −b, and ij = b1−j Ij . The conditions of Cases 2 and 3 intersect when ∆ii = ∆jj = −2∆kk . The various permutations of this correspond to a = 1/2, −1 and a = 2 and for these special values (35) becomes a square of (34), for a = −1 and 2 we get the relation i62 + 12 i32 i23 − 3 i22 i24 + 36 i2 i23 i4 − 36 i43 + 2 i34 = 0 (36) and for a = 1/2 (i.e., µ1 = µ2 = − 21 µ3 ) i62 − 24 i32 i23 − 12 i22 i24 + 144 i2 i23 i4 − 144 i43 − 16 i34 = 0 (The second form is obtained from the first by i3 → √ (37) −2i3 , i4 → −2i4 .) In fact one can show that the expression (35) can be written as U 2 + (a + 1)2 (a − 1/2)2 (a − 2)2 V = 0, 11 where U and V are some polynomials of a and the ij ’s. Whether (35) factorizes also for some other special values is an open question. 3.3 Four term potential Let us now return to equation (29). Since it is a fourth order linear equation is should have P solutions of type V = 4i=1 g(αi x + βi y). This is indeed the case, but since we really have only four rather than five ci ’s at our disposal (we can use H2 to eliminate c2 , say) there will be a relation between αi , βi . In order to simplify subsequent computations we will use rotational invariance to fix α1 = 0, and scale so that β1 = 1, and αj = 1, when j 6= 1. The geometric relation can then be written as 3β2 β3 β4 + β2 + β3 + β4 = 0. (38) In the following we will also restrict our attention only to g(x) ∝ 1/x2 , i.e., V (x, y) = a1 a2 a3 a4 + + + . 2 2 2 y (x + β2 y) (x + β3 y) (x + β4 y)2 (39) By direct computation we obtain the leading terms of the fourth order operator commuting with the Hamiltonian (1,39), as I4 h̄4 [∂x4 + 4∂x ∂y3 β2 β3 β4 − ∂y4 (β2 β3 + β2 β4 + β3 β4 )] h −4h̄2 ∂x2 , g2 + g3 + g4 = − {∂x ∂y , ((β2 + β3 + β4 )g1 + β2 g2 + β3 g3 + β4 g4 )}  − ∂y2 , ((β2 β3 + β2 β4 + β3 β4 )g1 + β2 (β3 + β4 )g2 +β3 (β2 + β4 )g3 + β4 (β2 + β3 )g4 )} + D(x, y). i (40) This far we can proceed just with (38). The integrability condition for D introduces further conditions, and leads to the following classification: 12 Case 1 Assume that one β vanishes, let us say β4 = 0. Then β3 = −β2 from (38) and we get two integrable potentials: First there is the well known classical one with β2 = −β3 = 1 V4.1 = a2 a2 a1 a1 + + + 2. 2 2 2 y (x + y) (x − y) x (41) We note that there are two degrees of freedom, all in the amplitudes. Case 2 There is another possibility with β3 = −β2 arbitrary:     1 + β22 β24 1 + β22 1 1 1 2 . V4.2 = C + 2 + h̄ + − − x2 y (x + β2 y)2 (x − β2 y)2 8x2 8y 2 (42) This potential has the classical limit V = Ax−2 + By −2 , with is separable and therefore has a quadratic second invariant (in both classical and quantum picture). The result above indicates that in quantum mechanics there exists a non-separable generalization for it. There are now two free parameters, C and β2 , the solution presented in [9] corresponds to the choice C = h̄2 21 (m + 12 )2 , Cβ24 = h̄2 21 (l + 21 )2 Case 3 Next we assume that β4 6= 0, and since due to rotational invariance we could rotate any of the vector to the position y, we may in fact assume that all β’s are nonzero (in other words that no pair of the vectors is orthogonal). From the equations it then follows that ai = h̄2 (1 + βi2 ), which normalizes the vectors. There is one solution of this type, it is defined by (38) and β22 + β32 + β42 − β2 β3 − β3 β4 − β4 β2 = 0, (43) which have the parameterization 1−a 1 − aξ 1 − aξ 2 √ √ √ β2 = , β3 = , β4 = , (44) a3 − 1 a3 − 1 a3 − 1 where ξ is a cubic root of unity 6= 1. This is a new result. For real a > 1 the potential is also real because the g3 and g4 terms are complex conjugates. There is only one free parameter, but this is probably due to the special form 1/x2 , for example if one makes the same restriction in (22,22) one can scale out A. 13 4 Conclusions We have discussed the quantum integrability in two dimensions, and in particular the existence of commuting differential operators of order 3 and 4. For the third order operator we have considered the generic case with constant coefficients in the leading term, and found the classical three term Calogero-Moser system (in terms of Weierstrass functions) and two PQI (three term) systems, one of which was first reported in [9]. These systems also have a fourth order commuting operator, but only for the PQI cases is the algebraic relation between the three operators nontrivial. It is also possible to have integrable four term potentials if the commuting operator is of fourth order. Our analysis was here restricted to 1/x2 -type potential terms and we have identified two PQI examples. Whether these potentials are also algebraically integrable is an open question: in principle one should find for them another, still higher order invariant, but this seems to be a formidable task. With these results we have only scratched the surface of pure quantum integrability. What is missing in particular is a geometric characterization of the phenomena and comprehensive extensions to higher orders and dimensions. The classification based on Lie algebra root systems [11], that works so well in the classical case, does not include these new PQI systems. Acknowledgments I would like to thank A.P. Veselov for calling my attention to this problem, and F. van Dienen and L. Floria for pointing out further references. References [1] V.I. Arnold, Mathematical Methods of Classical Mechanics, (Springer, 1978) p. 271. [2] J. von Neumann, Mathematical Foundations of Quantum Mechanics, (Princeton UP, 1955) p. 173. 14 [3] J. Hietarinta, Direct methods for the search of the second invariant, Physics Reports 147, 87-154 (1987). [4] J. Hietarinta, Classical versus quantum integrability, J. Math. Phys. 25, 1833 (1984). J. Hietarinta and B. Grammaticos, On the h̄2 -correction terms in quantum mechanics, J. Phys. A: Math. Gen. 22, 1315-1322 (1989). [5] T. Oshima, Completely integrable systems with a symmetry in coordinates, preprint TUMS 94-6 (1994); H. Ochiai and T. Oshima, Commuting differential operators of type B2 , preprint UTMS 94-65 (1994); H. Ochiai, T. Oshima and H. Sekiguchi Commuting families of symmetric differential operators, Proc. Japan Acad. 70 A, 62-66 (1994); T. Oshima and H. Sekiguchi, Commuting families of differential operators invariant under the action of a Weyl group, J. Math. Sci. Univ. Tokyo 2, 1-75 (1995); H. Ochiai, Commuting differential operators of rank two, preprint. [6] J. Hietarinta, Solvability in quantum mechanics and classically superfluous invariants, J. Phys. A: Math. Gen. 22, L143-L147 (1989). [7] J.L. Burchnall and T.W. Chaundy, Commutative Ordinary Differential Operators, Proc. Roy. Soc. (London) 118, 557-583 (1928); ibid 134, 471-485 (1932). [8] O.A. Chalykh and A.P. Veselov, Commutative Rings of Partial Differential Operators and Lie Algebras, Comm. Math. Phys. 126, 597-611 (1990). [9] A.P. Veselov, M.V. Feigin and O.A. Chalykh, New integrable deformations of the Calogero-Moser quantum problem, Russ. Math. Surveys 51, 573-574 (1996). [10] A.P. Veselov, Huygen’s principle and integrability, unpublished (1996). [11] M.A. Olshanetsky and A.M. Perelomov, Classically integrable finite dimensional systems related to Lie algebras, Phys. Rep. 71, 313-400 (1981); M.A. Olshanetsky and A.M. Perelomov, Quantum integrable systems related to Lie algebras, Phys. Rep. 94, 313-404 (1981). 15