By exploiting a recently developed connection between Heun's differential equation and the generalized associated Lamé equation, we not only recover the well known periodic solutions, but also obtain a large class of new, quasi-periodic solutions of Heun's equation. Each of the quasi-periodic solutions is doubly degenerate.
A new bi-parametric $su(1,1)$ algebraization of the Heun class of equations is explored. This yields additional quasi-polynomial solutions of the form $\{z^αP_N(z): \ α\in \mathbb{C}, \ N \in \mathbb{N}_0\}$ to the General Heun eqaution and its confluent versions. Explicit conditions leading to these quasi-polynomials have been provided for the individual equations to allow direct use. For the Confluent and the Doubly-confluent Heun equations, specific parametric situations leading to (i) an infinite number of quasi-polynomials and (ii) non-algebraizability of the equation have been identified…
We construct new solutions in series of confluent hypergeometric functions for the confluent Heun equation (CHE). Some of these solutions are applied to the one-dimensional stationary Schrödinger equation with hyperbolic and trigonometric quasiexactly solvable potentials.
It is shown that the Confluent Heun Equation (CHEq) reduces for certain conditions of the parameters to a particular class of Quasi-Exactly Solvable models, associated with the Lie algebra $sl (2,{\mathbb R})$. As a consequence it is possible to find a set of polynomial solutions of this quasi-exactly solvable version of the CHEq. These finite solutions encompass previously known polynomial solutions of the Generalized Spheroidal Equation, Razavy Eq., Whittaker-Hill Eq., etc. The analysis is applied to obtain and describe special eigen-functions of the quantum Hamiltonian of two fixed Coulombi…
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 using analysis of regular singularities. We also consider the quasi-exact solvability of the $q$-Heun equation and its variants. Namely we investigate finite-dimensional subspaces which are invariant under the action of the $q$-Heun operator or variants of the $q$-Heun operator.
Connection formulas relating Frobenius solutions of linear ODEs at different Fuchsian singular points can be expressed in terms of the large order asymptotics of the corresponding power series. We demonstrate that for the usual, confluent and reduced confluent Heun equation, the series expansion of the relevant asymptotic amplitude in a suitable parameter can be systematically computed to arbitrary order. This allows to check a recent conjecture of Bonelli-Iossa-Panea Lichtig-Tanzini expressing the Heun connection matrix in terms of quasiclassical Virasoro conformal blocks.
We study polynomial-type solutions of the $q$-Heun equation, which is related with quasi-exact solvability. The condition that the $q$-Heun equation has a non-zero polynomial-type solution is described by the roots of the spectral polynomial, whose variable is the accessory parameter $E$. We obtain sufficient conditions that the roots of the spectral polynomial are all real and distinct. We consider the ultradiscrete limit to clarify the roots of the spectral polynomial and the zeros of the polynomial-type solution of the $q$-Heun equation.
We apply solutions of Heun's general equation to the stationary Schrödinger equation with two quasi-exactly solvable elliptic potentials which depend on a real parameter $\ell$. We get finite-series solutions from power series expansions for Heun's equation if $\ell$ is an integer, except if $\ell=-1,-2,-3,-4$. If $\ell\neq-5/2$ is half an odd integer, we obtain finite series in terms of hypergeometric functions. The quasi-exact solvability is expressed by the finite series solutions. However, for any value of $\ell$, we find infinite-series eigenfunctions which are convergent and bounded for …
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 Schrödinger equation with an inverted quasi-exactly solvable potential as well as to the angular equation for an electron in the field of a point electric dipole. For the first problem we find finite- and infinite-series solutions which are convergent and bounded for any value of the independent variable. For the angular equation, we also find expansions in series of Jacobi polynomials.
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 introduce an additional expansion in series of irregular confluent hypergeometric functions. Then, we find the conditions under which one of these solutions can be written as a linear combination of the others. In the second place, by means of limiting procedures we generate solutions for the double-confluent equation as well as for special limits of both th…
New solutions for the elliptic Darboux equation are obtained as particular cases of solutions constructed for Heun's general equation. We consider two groups of power series expansions and two new groups of expansions in series of Gauss hypergeometric functions. The convergence of one group in power series is determined by means of ratio tests for infinite series, while the other groups are designed to solve problems which admit finite-series solutions. Actually, we envisage periodic quasi-exactly solvable potentials for which the stationary one-dimensional Schrödinger equation is reduced to t…
An analysis of the generalized confluent Heun equation $(α_2r^2+α_1r)\,y''+(β_2r^2+β_1r+β_0)\,y'-(\varepsilon_1r+\varepsilon_0)\,y=0$ in $d$-dimensional space, where $\{α_i, β_i, \varepsilon_i\}$ are real parameters, is presented. With the aid of these general results, the quasi exact solvability of the Schrödinger eigenproblem generated by the softcore Coulomb potential $V(r)=-e^2Z/(r+b),\, b>0$, is explicitly resolved. Necessary and sufficient conditions for polynomial solvability are given. A three-term recurrence relation is provided to generate the coefficients of polynomial solutions exp…
The Leaver solutions in series of Coulomb wave functions for the confluent Heun equation (CHE) are given by two-sided infinite series, that is, by series where the summation index $n$ runs from minus to plus infinity [E. W. Leaver, J. Math. Phys. 27, 1238 (1986)]. First we show that, in contrast to the D'Alembert test, under certain conditions the Raabe test assures that the domains of convergence of these solutions include an additional singular point. Further, by using a limit proposed by Leaver, we obtain solutions for the double-confluent Heun equation (DCHE). In addition, we get solutions…
The present paper analyze the constraints on the confluent Heun type-equation, $(a_{3,1}r^2+a_{3,2}r)y"+(a_{2,0}r^2+a_{2,1}r+a_{2,2})y'-(τ_{1,0}r+τ_{1,1})y=0,$ where $|a_{3,1}|^2+|a_{3,2}|^2\neq 0, $ and $a_{i,j},i=3,2,1, j=0,1,2$ are real parameters, to admit polynomial solutions. The necessary and sufficient conditions for the existence of these polynomials are given. A three-term recurrence relation is provided to generate the polynomial solutions explicitly. We, then, prove that these polynomial solutions are a source of finite sequences of orthogonal polynomials. Several properties, such …
We study in detail the relationship between the Tavis-Cummings Hamiltonian of quantum optics and a family of quasi-exactly solvable Schrödinger equations. The connection between them is stablished through the biconfluent Heun equation. We found that each invariant $n$-dimensional subspace of Tavis-Cummings Hamiltonian corresponds either to $n$ potentials, each with one known solution, or to one potential with $n$-known solutions. Among these Schrödinger potentials appear the quarkonium and the sextic oscillator.
We give explicit expressions for the finite frequency greybody factor, quasinormal modes and Love numbers of Kerr black holes by computing the exact connection coefficients of the radial and angular parts of the Teukolsky equation. This is obtained by solving the connection problem of the confluent Heun equation in terms of the explicit expression of irregular Virasoro conformal blocks as sums over partitions via the AGT correspondence. In the relevant approximation limits our results are in agreement with existing literature. The method we use can be extended to solve the linearized Einstein …
Recently, it has been proved that a nonlinear quantum oscillator, generalization of the isotonic one, is exactly solvable for certain values of its parameters. Here we show that the Schroedinger equation for such an oscillator can be transformed into a confluent Heun equation. We give a very simple and efficient algorithm to solve it numerically, no matter what the values of the parameters are. Algebraic quasi-polynomial solutions, for particular values of the parameters, are found.
We quantify the coupling asymptotics for the Lyapunov-exponent of a one-frequency quasi-periodic Schrödinger operator with analytic potential sampling function. The result refines the well-known lower bound of the Lyapunov-exponent by Sorets and Spencer.
In this paper we give the \emph {quantization rules} to determine the normalized stationary solutions to the cubic nonlinear Schrödinger equation with quasi-periodic conditions on a given interval. \ Similarly to what happen in the Floquet's theory for linear periodic operators, also in this case some kind of band functions there exist.
We present a set of smooth infinite energy global solutions (without spatial symmetry) to the non-integrable, nonlinear Schrödinger equations on $\Bbb R$. These solutions are space-time quasi-periodic with two frequencies each. Previous results [B2,1], and their generalizations [W2-4], are quasi-periodic in time, but periodic in space. This paper generalizes Bourgain's semi-algebraic set method [B3] to analyze nonlinear PDEs, in the non-compact space quasi-periodic setting on $\Bbb R$.
We discover that the distribution of (frequency and phase) resonances plays a role in determining the spectral type of supercritical quasi-periodic Schrödinger operators. In particular, we disprove the second spectral transition line conjecture of Jitomirskaya in the early 1990s.
We construct space quasi-periodic standing wave solutions to the nonlinear Schrödinger equations on R^d for arbitrary d. This is a type of quasi-periodic nonlinear Bloch-Floquet waves.
We consider CMV matrices, both standard and extended, with analytic quasi-periodic Verblunsky coefficients and prove Anderson localization in the regime of positive Lyapunov exponents. This establishes the CMV analog of a result Bourgain and Goldstein proved for discrete one-dimensional Schrödinger operators. We also prove a similar result for quantum walks on the integer lattice with suitable analytic quasi-periodic coins.
We consider certain degenerations of trigonal curves and hyperelliptic curves, which we call one step degeneration. We compute the limits of corresponding quasi-periodic solutions using the Sato Grassmannian. The mixing of solitons and quasi-periodic solutions is clearly visible in the obtained solutions.
We consider the quasi-periodic Schrödinger operator with the non-degenerate Gevrey potential for the Diophantine frequency. We prove that if the coupling number of the potential is large, then the spectrum is homogeneous.
We study random perturbations of quasi-periodic uniformly discrete sets in the $d$-dimensional euclidean space. By means of Diffraction Theory, we find conditions under which a quasi-periodic set $X$ can be almost surely recovered from its random perturbations. This extends the recent periodic case result of Yakir from "Recovery the lattice from its random perturbations" IMRN 2020 (arXiv:200201508).
