138 research outputs found

### Exact solution of the Schr\"odinger equation for the inverse square root potential $V_0/{\sqrt{x}}$

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

### 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

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