Exactly solvable nonlinear eigenvalue problems

TL;DR

This paper introduces the first exactly solvable nonlinear eigenvalue problem related to semi-transcendental differential equations, establishing a link to linear eigenvalue problems and solving a quantum harmonic oscillator case.

Contribution

It demonstrates the equivalence of certain nonlinear eigenvalue problems to linear ones and provides the first exact solutions for these nonlinear problems.

Findings

Nonlinear eigenvalue problems can be transformed into linear eigenvalue problems.

The nonlinear eigenvalue problem for the quantum harmonic oscillator is solved exactly.

Extended nonlinear equations are also analyzed for solvability.

Abstract

The nonlinear eigenvalue problem of a class of second order semi-transcendental differential equations is studied. A nonlinear eigenvalue is defined as the initial condition which gives rise a separatrix solution. A semi-transcendental equation can be integrated once to a first order nonlinear equation, e.g., the Ricatti equation. It is shown that the nonlinear eigenvalue problems of these semi-transcendental equations are equivalent to linear eigenvalue problems. They share the exactly same eigenvalues. The eigensolutions in the two problems are closely related. The nonlinear eigenvalue problem equivalent to the (half) harmonic oscillator in quantum mechanics is solved exactly. This is the first solvable nonlinear eigenvalue problem. The nonlinear eigenvalue problems of some extended equations are also studied.