Euler integral symmetries for a deformed Heun equation and symmetries of the Painlevé PVI equation

Euler integral transformations relate solutions of ordinary linear differential equations and generate integral representations of the solutions in a number of cases or relations between solutions of constrained equations (Euler symmetries) in some other cases. These relations lead to the corresponding symmetries of the monodromy matrices. We discuss Euler symmetries in the case of the simplest Fuchsian system that is equivalent to a deformed Heun equation, which is in turn related to the Painlevé PVI equation. The existence of integral symmetries of the deformed Heun equation leads to the corresponding symmetries of the PVI equation. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 155, No. 2, pp. 252–264, May, 2008.     

Isomonodromic deformations and “antiquantization” for the simplest ordinary differential equations

We consider three different models of linear differential equations and their isomonodromic deformations. We show that each of the models has its own specificity, although all of them lead to the same final result. It turns out that isomonodromic deformations are closely related to the Hamiltonian structure of both classical mechanics and quantum mechanics. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 150, No. 1, pp. 143–151, January, 2007.     

Integrable dynamical systems generated by quantum models with an adiabatic parameter

Several models solvable in terms of special functions of the Heun class are widely used in quantum mechanics. They are all characterized by the presence of a parameter that can be regarded as an adiabatic variable. An antiquantization procedure applied to such a model generates a dynamical model with properties of the Painlevé equations. The mentioned parameter plays the role of time. We consider examples of such models.     

Confluence of fuchsian second-order differential equations

Ordinary linear homogeneous second-order differential equations with polynomial coefficients including one in front of the second derivative are studied. Fundamental definitions for these equations: of s-rank of the singularity (different from Poincaré rank), of s-multisymbol of the equation, and of s-homotopic transformations are proposed. The generalization of Fuchs' theorem for confluent Fuchsian equations is proved. The tree structure of types of equations is shown, and the generalized confluence theorem is proved.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 104, No. 2, pp. 233–247, August, 1995.     

Isomonodromic deformations of Heun and Painlevé equations

Continuing the study of the relationship between the Heun and the Painlevé classes of equations reported in two previous papers, we formulate and prove the main theorem expressing this relationship. We give a Hamiltonian interpretation of the isomonodromic deformation condition and propose an alternative classification of the Painlevé equations, which includes ten equations. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 123, No. 3, pp. 395–406, June, 2000.     

Recurrent calculations of multipole matrix elements

We propose a new method for calculating multipole matrix elements between wave eigenfunctions of the one-dimensional Schrödinger equation. The method is based on the transition to the auxiliary third- and fourth-order equations, to which an analogue of the Laplace transform is then applied. The resulting recursive procedure allows us to evaluate matrix elements starting with a number of eigenvalues that are assumed to be known and several basis matrix elements. As an example, we consider the multipole matrix elements between the wave functions of the harmonic and nonharmonic oscillators.     

Antiquantization of deformed Heun-class equations

We consider deformed Heun-class equations, i.e., equations of the Heun class with an added apparent singularity. We prove that each deformed Heun-class equation under antiquantization realizes a transfer from the Heun-class equation to the corresponding Painlevé equation, and we completely list such transfers.     

Relations Between Linear Equations and Painlevé’s Equations

Painlevé equations are studied on the basis of linear equations, which are generic for them. Different possible approaches are compared to each other. Formulas binding these approaches are derived. Symmetries demonstrated in the equations are also a subject of discussion.