Series solutions of confluent Heun equations in terms of incomplete Gamma-functions

We present a simple systematic algorithm for construction of expansions of the solutions of ordinary differential equations with rational coefficients in terms of mathematical functions having indefinite integral representation. The approach employs an auxiliary equation involving only the derivatives of a solution of the equation under consideration. Using power-series expansions of the solutions of this auxiliary equation, we construct several expansions of the four confluent Heun equations'' solutions in terms of the incomplete Gamma-functions. In the cases of single- and double-confluent Heun equations the coefficients of the expansions obey four-term recurrence relations, while for the bi- and tri-confluent Heun equations the recurrence relations in general involve five terms. Other expansions for which the expansion coefficients obey recurrence relations involving more terms are also possible. The particular cases when these relations reduce to ones involving less number of terms are identified. The conditions for deriving closed-form finite-sum solutions via right-hand side termination of the constructed series are discussed.     

Above-barrier reflection of cold atoms by resonant laser light within the Gross-Pitaevskii approximation

Above-barrier reflection of cold alkali atoms by resonant laser light was considered analytically within the Gross-Pitaevskii approximation. Correction for the reflection coefficient because of a weak nonlinearity of the stationary Schrödinger equation has been derived using multiscale analysis as a form of perturbation theory. The nonlinearity adds spatial harmonics to linear incident and reflecting waves. It was shown that the role of nonlinearity increases when the kinetic energy of an atom is nearly to the height of the potential barrier. Results are compared to the known numerical derivations for wave functions of the Gross-Pitaevskii equation with the step potential.     

Maslov index for power-law potentials

The Maslov index in the semiclassical Bohr–Sommerfeld quantization rule is calculated for one-dimensional power-law potentials V (x) = ?V ₀/x ^s, x > 0, 0 < s < 2 The result for the potential V(x)=-V ₀/x ^1/2 is compared with the recently reported exact solution. The case of a spherically symmetric power-law potential is also considered.     

A Schrödinger Potential Conditionally Integrable in Terms of the Hermite Functions

We introduce a biconfluent Heun potential well for the one-dimensional stationary Schrödinger equation which is composed of a confining fraction-power term and a repulsive centrifugal-barrier core. This is a conditionally integrable potential in that the strength of the centrifugal barrier is fixed to a constant. The potential supports a countable infinite number of bound states. We present the general solution of the Schrödinger equation, deduce the exact equation for the energy spectrumand derive a highly accurate approximation for energy levels. The bound state wave functions are written as irreducible linear combinations with constant coefficients of two Hermite functions of a scaled and shifted argument.

     

Accurate Treatment of Tunnel Ionization,Induced by a Constant Electric Field,From a Power-Law Singular Potential

The tunnel ionization, induced by a weak electrostatic field, of a particle bound in an attractive one-dimensional power-law singular potential is analysed. An accurate approximation applicable for the general case of non-fractional power exponent is suggested.