Expansions of the solutions of the general Heun equation governed by two-term recurrence relations for coefficients

We examine the expansions of the solutions of the general Heun equation in terms of the Gauss hypergeometric functions. We present several expansions using functions, the forms of which differ from those applied before. In general, the coefficients of the expansions obey three-term recurrence relations. However, there exist certain choices of the parameters for which the recurrence relations become two-term. The coefficients of the expansions are then explicitly expressed in terms of the gamma functions. Discussing the termination of the presented series, we show that the finite-sum solutions of the general Heun equation in terms of generally irreducible hypergeometric functions have a representation through a single generalized hypergeometric function. Consequently, the power-series expansion of the Heun function for any such case is governed by a two-term recurrence relation

Exact solution of the Schr\"odinger equation for the inverse square root potential V0/xV_0/{\sqrt{x}}

We present the exact solution of the stationary Schr\"odinger equation equation for the potential $V=V_0/{\sqrt{x}}$. Each of the two fundamental solutions that compose the general solution of the problem is given by a combination with non-constant coefficients of two confluent hypergeometric functions of a shifted argument. Alternatively, the solution is written through the first derivative of a tri-confluent Heun function. Apart from the quasi-polynomial solutions provided by the energy specification $E_n=E_1{n^{-2/3}}$, we discuss the bound-state wave functions vanishing both at infinity and in the origin. The exact spectrum equation involves two Hermite functions of non-integer order which are not polynomials. An accurate approximation for the spectrum providing a relative error less than $10^{-3}$ is $E_n=E_1{(n-1/(2 \pi))^{-2/3}}$ . Each of the wave functions of bound states in general involves a combination with non-constant coefficients of two confluent hypergeometric and two non-integer order Hermite functions of a scaled and shifted coordinate

A new exactly integrable hypergeometric potential for the Schr\"odinger equation

We introduce a new exactly integrable potential for the Schr\"odinger equation for which the solution of the problem may be expressed in terms of the Gauss hypergeometric functions. This is a potential step with variable height and steepness. We present the general solution of the problem, discuss the transmission of a quantum particle above the barrier, and derive explicit expressions for the reflection and transmission coefficients