Generalized Gibbs Ensemble of the Ablowitz–Ladik Lattice, Circular $$\beta $$-Ensemble and Double Confluent Heun Equation

Ince’s limits for confluent and double-confluent Heun equations

We find pairs of solutions to a differential equation which is obtained as a special limit of a generalized spheroidal wave equation (this is also known as confluent Heun equation). One solution in each pair is given by a series of hypergeometric functions and converges for any finite value of the independent variable z, while the other is given by a series of modified Bessel functions and conv...

An example of double confluent Heun equation: Schrödinger equation with supersingular plus Coulomb potential

A recently proposed algorithm to obtain global solutions of the double confluent Heun equation is applied to solve the quantum mechanical problem of finding the energies and wave functions of a particle bound in a potential sum of a repulsive supersingular term, A r, plus an attractive Coulombian one, −Z r. The existence of exact algebraic solutions for certain values of A is discussed. Supersi...

Gibbs Ensemble Techniques

Integral relations and new solutions to the double-confluent Heun equation

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 convergent in different regions of the complex plane. Integral relations are also established between solutions given by series of Coulomb wave functions. The...

Integral relations for solutions of the confluent Heun equation

Abstract: Firstly, we construct kernels for integral relations among solutions of the confluent Heun equation (CHE). Additional kernels are systematically generated by applying substitutions of variables. Secondly, we establish integral relations between known solutions of the CHE that are power series and solutions that are series of special functions. Thirdly, by using one of the integral rel...