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Integrability of some quantum problems related to the root system \(B_2\). (English. Russian original) Zbl 0952.35111

Mosc. Univ. Math. Bull. 54, No. 2, 5-8 (1999); translation from Vestn. Mosk. Univ., Ser. I 1999, No. 2, 6-9 (1999).
The author proves the algebraic integrability of the two-dimensional analogue of the Lamé operator related to the system of roots \(B_2\) \[ L=-\Delta + 2(\wp(x) +\wp(y))+4(\wp(x+y) +\wp(x-y)), \] where \(\wp\) is the elliptic Weierstrass function. Besides, the integrals are determined for the operator related to the deformed system of roots \(B_2(l,m)\).

MSC:

35Q40 PDEs in connection with quantum mechanics
37K20 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions