Two-term Kummer function solutions of the 1D Schrödinger equation

We explore the application of fundamental solutions, constructed as irreducible linear combinations with variable coefficients of two Kummer confluent hypergeometric functions, to solve the one-dimensional stationary Schrödinger equation. This approach offers a new technique for tackling this equation, particularly well-suited for singular potentials exhibiting non-monotonic behavior, a challenge for traditional methods. We focus on a specific family of four-parameter potential wells defined on the positive semi-axis using the Lambert-W function. For these wells, the potential energy increases toward the origin but tends to zero at infinity. We present solutions in the form of linear combinations of Kummer functions and their derivatives, along with power functions and exponents. We then explore two specific well types within this family, analyzing the energy spectrum of the quantum system for each case.