相关论文: Finite-gap potential, Heun's differential equation…
We construct several expansions of the solutions of the confluent Heun equation in terms of the incomplete Beta functions and the Appell generalized hypergeometric functions of two variables of the fist kind. The coefficients of different…
The paper describes known and new results about finite difference calculus on configuration spaces. We describe finite difference geometry on configuration spaces, connect finite difference operators with cannonical commutation relations,…
We obtain several new results for the complex generalized associated Lame potential V(x)= a(a+1)m sn^2(y,m)+ b(b+1)m sn^2(y+K(m),m) + f(f+1)m sn^2(y+K(m)+iK'(m),m)+ g(g+1)m sn^2(y+iK'(m),m), where y = x-K(m)/2-iK'(m)/2, sn(y,m) is a Jacobi…
We present in total fifteen potentials for which the stationary Klein-Gordon equation is solvable in terms of the confluent Heun functions. Because of the symmetry of the confluent Heun equation with respect to the transposition of its…
This work provides a spacetime interpretation of the confluent Heun functions within black hole perturbation theory (BHPT) and explores their relationship to the hyperboloidal framework. In BHPT, the confluent Heun functions are solutions…
We investigate the quantum geometry of the Seiberg-Witten curve for $\mathcal{N}=2$, $\mathrm{SU(2)}^n$ linear quiver gauge theories. By applying the Weyl quantization prescription to the algebraic curve, we derive the corresponding…
We describe a connection between finite--dimensional representations of quantum affine algebras and affine Hecke algebras.
The Darboux transformation applied recurrently on a Schroedinger operator generates what is called a {\em dressing chain}, or from a different point of view, a set of supersymmetric shape invariant potentials. The finite-gap potential…
We find a new class of the Fuchsian equations, which have an algebraic geometric solutions with the parameter belonging to a hyperelliptic curve. Methods of calculating the algebraic genus of the curve, and its branching points, are…
Integral relations and transformation rules are used to obtain, out of an asymptotic solution, a new group of four pairs of solutions to the double-confluent Heun equation. Each pair presents the same series coefficients but has solutions…
In this paper we study AKNS hierarchy. We find explicit necessary conditions for functions $p$ and $q$ to be solution of some equation of AKNS hierarchy. Then we construct finite-gap Schrodinger potential using functions $p$ and $q$.
The $q$-Heun equation and its variants arise as degenerations of Ruijsenaars-van Diejen operators with one particle. We investigate local properties of these equations. In particular we characterize the variants of the $q$-Heun equation by…
We construct integrable modifications of 2d lattice gauge theories with finite gauge groups.
It is shown that Schr\"odinger equation with combination of three potentials U = - {\alpha} r^{-1} + {\beta} r + kr^{2}, Coulomb, linear and harmonic, the potential often used to describe quarkonium, is reduced to a bi-confluent Heun…
We introduce a nine-parameter Heun-type differential equation and obtain three classes of its solutions as series of square integrable functions written in terms of the Jacobi polynomial. The expansion coefficients of the series satisfy…
We reexamine and extend a group of solutions in series of Bessel functions for a limiting case of the confluent Heun equation and, then, apply such solutions to the one-dimensional Schr\"odinger equation with an inverted quasi-exactly…
Taking into account results of WKB-approximation, we derive exact quantum energies and wave functions of even and odd states in the one-dimensional Coulomb potential
The present article discusses the two point connection problem for Heun's differential equation. We employ a contour integral method to derive connection matrices for a sub-class of the Heun equation containing 3 free parameters. Explicit…
In this work we explain the relevance of the Differential Galois Theory in the semiclassical (or WKB) quantification of some two degree of freedom potentials. The key point is that the semiclassical path integral quantification around a…
This paper examines some solutions for confluent and double-confluent Heun equations. In the first place, we review two Leaver's solutions in series of regular and irregular confluent hypergeometric functions for the confluent equation and…