Integrable quantum Stäckel systems

Annals of Physics, 2016

In this article we prove that many Hamiltonian systems that can not be separably quantized in the classical approach of Robertson and Eisenhardt can be separably quantized if we extend the class of admissible quantizations through a suitable choice of Riemann space adapted to the Poisson geometry of the system. Actually, in this article we prove that for every quadratic in momenta Stäckel system (de…ned on 2n dimensional Poisson manifold) for which Stäckel matrix consists of monomials in position coordinates there exist in…nitely many quantizations-parametrized by n arbitrary functions-that turn this system into a quantum separable Stäckel system.

Annals of Physics, 2014

In this paper, we consider the problem of quantization of classical Stäckel systems and the problem of separability of related quantum Hamiltonians. First, using the concept of Stäckel transform, all considered systems are expressed by flat coordinates of related Euclidean configuration space. Then, the so-called flat minimal quantization procedure is applied in order to construct an appropriate Hermitian operator in the respective Hilbert space. Finally, we distinguish a class of Stäckel systems which remain separable after any of admissible flat minimal quantizations.

Physics of Atomic Nuclei, 2008

The two-dimensional quantum superintegrable systems with quadratic integrals of motion on a manifold are classified by using the quadratic associative algebra of the integrals of motion. There are six general fundamental classes of quantum superintegrable systems corresponding to the classical ones. Analytic formulas for the involved integrals are calculated in all the cases. All the known quantum superintegrable systems with quadratic integrals are classified as special cases of these six general classes. The coefficients of the quadratic associative algebra of integrals are calculated and they are compared to the coefficients of the corresponding coefficients of the Poisson quadratic algebra of the classical systems. The quantum coefficients are similar to the classical ones multiplied by a quantum coefficient − 2 plus a quantum deformation of order 4 and 6. The systems inside the classes are transformed using St¨ackel transforms in the quantum case as in the classical case. The general form of the St¨ackel transform between superintegrable systems is discussed.

Symmetry, Integrability and Geometry: Methods and Applications, 2017

In this paper we discuss maximal superintegrability of both classical and quantum Stäckel systems. We prove a sufficient condition for a flat or constant curvature Stäckel system to be maximally superintegrable. Further, we prove a sufficient condition for a Stäckel transform to preserve maximal superintegrability and we apply this condition to our class of Stäckel systems, which yields new maximally superintegrable systems as conformal deformations of the original systems. Further, we demonstrate how to perform the procedure of minimal quantization to considered systems in order to produce quantum superintegrable and quantum separable systems.

In this article we prove that many Hamiltonian systems that can not be separably quantized in the classical approach of Robertson and Eisenhardt can be separably quantized if we extend the class of admissible quantizations through a suitable choice of Riemann space adapted to the Poisson geometry of the system. Actually, in this article we prove that for every quadratic in momenta St\"ackel system (defined on n dimensional Poisson manifold) for which St\"ackel matrix consists of monomials in position coordinates there exist infinitely many quantizations - parametrized by n arbitrary functions - that turn this system into a quantum separable St\"ackel system.

Physics Reports, 1983

Introduction 316 14. Systems with v(q) = I~I 367 1. Examples. Systems with one degree of freedom 318 15. Systems with v(q) = 8(q). Bethe Ansatz 369 2. General description 327 16*. Miscellanea 372 3*~Abstract quantum systems, related to root systems 331 16.1. Factorization of the ground-state wave function 372 4. The proof of complete integrability of the systems 335 16.2. Green's functions on symmetric spaces 376 5~.Complete integrability in the abstract case 337 Appendices 6*. Wave functions 340 A. Groups generated by reflections and their root systems 378 7*~Systems of type I (v(q) = q 2) 344 B. Symmetric spaces 38'7 8*. Systems of type II (v(q) = sinh2 q) 350 C. Laplace operators and spherical functions 394 9* Systems of type III (v(q) = sin2 q) 352 D. Connection between Hamiltonians and Laplace operators 31 0*. Systems of type IV (v(q) =~(q)) 354 E. Proof of Propositions of section 5 401 11*. Systems of type V (v(q) = q2+ w2q2) 355 References 402 12*. Systems of type VI (v(q)=expq) 363 13*. Systems of type VI' (Generalized periodic Toda lattices) 366

Lecture Notes in Physics, 2004

The key concept discussed in these lectures is the relation between the Hamiltonians of a quantum integrable system and the Casimir elements in the underlying hidden symmetry algebra. (In typical applications the latter is either the universal enveloping algebra of an affine Lie algebra, or its q-deformation.) A similar relation also holds in the classical case. We discuss different guises of this very important relation and its implication for the description of the spectrum and the eigenfunctions of the quantum system. Parallels between the classical and the quantum cases are thoroughly discussed.

Astérisque, 1994

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Journal of Mathematical Physics, 2006

We propose a general scheme of constructing of soliton hierarchies from finite dimensional Stäckel systems and related separation relations. In particular, we concentrate on the simplest class of separation relations, called Benenti class, i.e. certain Stäckel systems with quadratic in momenta integrals of motion.

Physics Letters A, 1998

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 toh 2 , 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.