On complex oscillation theory, quasi-exact solvability and Fredholm Integral Equations

TL;DR

This paper develops a complex oscillation theory for the Biconfluent Heun equation, establishing conditions for explicit eigen-solutions, their orthogonality, related integral equations, and connections to quasi-exact solvability in quantum mechanics.

Contribution

It introduces a theory of periodic BHE, constructs eigen-solutions with orthogonality properties, and links these to Fredholm integral equations and quasi-exact solvability.

Findings

Established explicit eigen-solution construction for PBHE

Derived orthogonality relations for solutions

Connected results to quasi-exact solvable Schrödinger equations

Abstract

Biconfluent Heun equation (BHE) is a confluent case of the general Heun equation which has one more regular singular points than the Gauss hypergeometric equation on the Riemann sphere $\hat{\mathbb{C}}$. Motivated by a Nevanlinna theory (complex oscillation theory) approach, we have established a theory of \textit{periodic} BHE (PBHE) in parallel with the Lam\'e equation verses the Heun equation, and the Mathieu equation verses the confluent Heun equation. We have established condition that lead to explicit construction of eigen-solutions of PBHE, and their single and double orthogonality, and a related first-order Fredholm-type integral equation for which the corresponding eigen-solutions must satisfy. We have also established a Bessel polynomials analogue at the BHE level which is based on the observation that both the Bessel equation and the BHE have a regular singular point at the origin and an irregular singular point at infinity on the Riemann sphere $\hat{\mathbb{C}}$, and that the former equation has orthogonal polynomial solutions with respect to a complex weight. Finally, we relate our results to an equation considered by Turbiner, Bender and Dunne, etc concerning a quasi-exact solvable Schr\"odinger equation generated by first order operators such that the second order operators possess a finite-dimensional invariant subspace in a Lie algebra of $SL_2(\mathbb{C})$