Integrable quantum Stäckel systems

The Stäckel separability of a Hamiltonian system is well known to ensure existence of a complete set of Poisson commuting integrals of motion quadratic in the momenta. We consider a class of Stäckel separable systems where the entries of the Stäckel matrix are monomials in the separation variables. We show that the only systems in this class for which the integrals of motion arising from the Stäckel construction keep commuting after quantization are, up to natural equivalence transformations, the so-called Benenti systems. Moreover, it turns out that the latter are the only quantum separable systems in the class under study.     

Flat minimal quantizations of Stäckel systems and quantum separability

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, natural Hamiltonian systems from a given Riemann space are expressed by some 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 remains separable after any of admissible flat minimal quantizations.     

Separable quantizations of Stäckel systems

In this article we prove that many Hamiltonian systems that cannot be separably quantized in the classical approach of Robertson and Eisenhart 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 (defined on 2n$2n$ dimensional Poisson manifold) for which Stäckel matrix consists of monomials in position coordinates there exist infinitely many quantizations–parametrized by n$n$ arbitrary functions–that turn this system into a quantum separable Stäckel system.     

Non-Hamiltonian systems separable by Hamilton–Jacobi method

We show that with every separable classical Stäckel system of Benenti type on a Riemannian space one can associate, by a proper deformation of the metric tensor, a multi-parameter family of non-Hamiltonian systems on the same space, sharing the same trajectories and related to the seed system by appropriate reciprocal transformations. These systems are known as bi-cofactor systems and are integrable in quadratures as the seed Hamiltonian system is. We show that with each class of bi-cofactor systems a pair of separation curves can be related. We also investigate the conditions under which a given flat bi-cofactor system can be deformed to a family of geodesically equivalent flat bi-cofactor systems.     

Integration of the dirac equation,which does not presume complete separation of variables,in Stäckel spaces

The method of noncommutative integration of linear differential equations is used to construct an exact solution of the Dirac equation, which does not presume complete separation of variables, in Stäckel spaces. The Dirac equation in an external electromagnetic field is integrated by this method, using one example. The Stäckel space under consideration does not enable one to solve this equation exactly within the framework of the theory of separation of variables.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 31–37, January, 1996.     

Classification of null-Stäckel electrovac metrics with cosmological constant

The complete classification of the null Stäckel electrovac spacetimes is realized. For these spacetimes it is possible to integrate the geodetic equations by the complete separation of variables in the Hamilton-Jacobi equation.     

Stäckel electrovacuum spaces that admit full separation of variables in a diagonalized Dirac equation. Type (2.1)

We list all electrovacuum Stäckel spaces of the type (2.1) that admit diagonalization and full separation of variables in the Dirac equation.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 116–120, February, 1992.     

Separation of Variables for Bi-Hamiltonian Systems

We address the problem of the separation of variables for the Hamilton–Jacobi equation within the theoretical scheme of bi-Hamiltonian geometry. We use the properties of a special class of bi-Hamiltonian manifolds, called N manifolds, to give intrisic tests of separability (and Stäckel separability) for Hamiltonian systems. The separation variables are naturally associated with the geometrical structures of the N manifold itself. We apply these results to bi-Hamiltonian systems of the Gel'fand–Zakharevich type and we give explicit procedures to find the separated coordinates and the separation relations.     

Exact solutions of vacuum Brans-Dicke equations

Vacuum Stäckel type (2.1) spaces are classified in Brans-Dicke theory for the case when the scalar field depends on all the variables.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 86–90, January, 1992.     

Stäckel electrovacuum spaces of type (2.1)

Classification of Stäckel electrovacuum spaces of type (2.1) is carried out. All metrics and potentials associated with the given type of spaces are obtained.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 54–56, February, 1989.     

Integrability Versus Separability for the Multi-Centre Metrics

The multi-centre metrics are a family of euclidean solutions of the empty space Einstein equations with self-dual curvature. For this full class, we determine which metrics do exhibit an extra conserved quantity quadratic in the momenta, induced by a Killing-Stäckel tensor. Our systematic approach brings to light a subclass of metrics which correspond to new classically integrable dynamical systems. Within this subclass we analyze on the one hand the separation of coordinates in the Hamilton-Jacobi equation and on the other hand the construction of some new Killing-Yano tensors.     

A new integrable system on the sphere and conformally equivariant quantization

Taking full advantage of two independent projectively equivalent metrics on the ellipsoid leading to Liouville integrability of the geodesic flow via the well-known Jacobi–Moser system, we disclose a novel integrable system on the sphere Sⁿ$S^n$, namely the dual Moser system. The latter falls, along with the Jacobi–Moser and Neumann–Uhlenbeck systems, into the category of (locally) Stäckel systems. Moreover, it is proved that quantum integrability of both Neumann–Uhlenbeck and dual Moser systems is ensured by means of the conformally equivariant quantization procedure.     

Stäckel electrovacuum spaces with isotropic complete sets. Integration of field equations for the metric generalizing a space of type (1.1)

A system of Einstein-Maxwell equations is integrated for the metric that generalizes the metric of a space with full set of type (1.1). This is necessary for carrying out the classification of the corresponding Stäckel electrovacuum spaces. An algebraic classification of the solutions obtained is carried out.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 12, pp. 17–20, December, 1987.     

Classification of the quantum two-dimensional superintegrable systems with quadratic integrals and the Stäckel transforms

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 Stäckel transforms in the quantum case as in the classical case. The general form of the Stäckel transform between superintegrable systems is discussed.     

Bihamiltonian structures and Stäckel separability

It is shown that a class of Stäckel separable systems is characterized in terms of a Gel’fand–Zakharevich bihamiltonian structure. This structure arises as an extension of a Poisson–Nijenhuis structure on phase space. It is also shown that the Casimir of the Gel’fand–Zakharevich bihamiltonian structure provides the family of commuting Killing tensors found by Benenti and that, because of Eisenhart’s theorem, characterize orthogonal separability. It is also shown that recently found properties of quasi-bihamiltonian systems are natural consequences of the geometry of the extension of the Poisson–Nijenhuis structure.     

A Physical Application of Kerr-Schild Groups

The present work deals with the search of useful physical applications of some generalized groups of metric transformations. We put forward different proposals and focus our attention on the implementation of one of them. Particularly, the results show how one can control very efficiently the kind of spacetimes related by a Generalized Kerr-Schild (GKS) Ansatz through Kerr-Schild groups. Finally a preliminary study regarding other generalized groups of metric transformations is undertaken which is aimed at giving some hints in new Ansätz to finding useful solutions to Einstein's equations.     

Nonlinear mean field Fokker-Planck equations. Application to the chemotaxis of biological populations

We study a general class of nonlinear mean field Fokker-Planck equations in relation with an effective generalized thermodynamical (E.G.T.) formalism. We show that these equations describe several physical systems such as: chemotaxis of bacterial populations, Bose-Einstein condensation in the canonical ensemble, porous media, generalized Cahn-Hilliard equations, Kuramoto model, BMF model, Burgers equation, Smoluchowski-Poisson system for self-gravitating Brownian particles, Debye-Hückel theory of electrolytes, two-dimensional turbulence... In particular, we show that nonlinear mean field Fokker-Planck equations can provide generalized Keller-Segel models for the chemotaxis of biological populations. As an example, we introduce a new model of chemotaxis incorporating both effects of anomalous diffusion and exclusion principle (volume filling). Therefore, the notion of generalized thermodynamics can have applications for concrete physical systems. We also consider nonlinear mean field Fokker-Planck equations in phase space and show the passage from the generalized Kramers equation to the generalized Smoluchowski equation in a strong friction limit. Our formalism is simple and illustrated by several explicit examples corresponding to Boltzmann, Tsallis, Fermi-Dirac and Bose-Einstein entropies among others.     

On some integrable systems in the extended lobachevsky space

Some classical and quantum-mechanical problems previously studied in Lobachevsky space are generalized to the extended Lobachevsky space (unification of the real, imaginary Lobachevsky spaces and absolute). Solutions of the Schrödinger equation with Coulomb potential in two coordinate systems of the imaginary Lobachevsky space are considered. The problem of motion of a charged particle in the homogeneous magnetic field in the imaginary Lobachevsky space is treated both classically and quantum mechanically. In the classical case, Hamilton-Jacoby equation is solved by separation of variables, and constraints for integrals of motion are derived. In the quantum case, solutions of Klein-Fock-Gordon equation are found.     

On holomorphic maps and Generalized Complex Geometry

We introduce a natural notion of holomorphic map between generalized complex manifolds and we prove some related results on Dirac structures and generalized Kähler manifolds.