Exactly Solvable Schrödinger Operators

Abstract

Abstract. We systematically describe and classify one-dimensional Schrödinger equations that can be solved in terms of hypergeometric type functions. Beside the well-known families, we explicitly describe two new classes of exactly solvable Schrödinger equations that can be reduced to the Hermite equation.

Pseudo-hermitian andPT-symmetric quantum systems with energy-dependent potentials: Bound-state solutions and energy spectra

“…The reasoning is very similar to the argumentation that we presented above for the case of our Wronskian (51), such that we omit to show it here. It follows that the integral in (11) exists in the present case, such that we can proceed to calculate its value numerically. We find for the three applicable cases N (Ψ 0 ) = 1.875 N (Ψ 1 ) = 1.86566 N (Ψ 2 ) = 1.49046.…”

“…If N (Ψ n ) < ∞ is real-valued for all admissible values of n, then the normalized solutions of our boundary-value problem (1), (2) are given by Ψ n /N (Ψ n ). Finally let us note that (11) and (12) do not constitute a norm in the mathematical sense because they can become negative or take complex values.…”

“…Since there are only three of such functions due to n < 3, we proceed in a way that is similar to the previous application. In the first step we establish existence of the integral in (11). To this end, we evaluate Ψ * n (x) η Ψ n (…”

Renormalization of the Strongly Attractive Inverse Square Potential: Taming the Singularity

“…Thus, by standard results (cf., e.g., Theorem 8.1 on page 156 in [35]), it is possible to transform this differential equation into the hypergeometric equation. One can refer to [36,37] for the details of such calculation that leads to the following expression for the wavefunction in the near region 0 ≤ x < x 0 which is a linear combination of two independent solutions and could be written as…”

Bound State Solution of Dirac Equation for Generalized Pöschl-Teller plus Trigomometric Pöschl-Teller Non- Central Potential Using SUSY Quantum Mechanics

“…In this paper we will attempt to solve the Dirac equation for a charged particle moving in a field governed by a generalized Pöschl-Teller potential, which is discussed in Derezinski and Wrochna [24] and, with simultaneous presence of a trigonometric Pöschl-Teller non-central potential, in Flugge [25], using super symmetric quantum mechanics (SUSY QM) with the idea of shape invariance in the case of exact spin symmetry. SUSY QM was developed based on Witten's proposal [26], while the idea of shape invariant potentials was proposed by Gendenshtein [27].…”