Symmetrized exponential oscillator

TL;DR

This paper investigates the bound states of a symmetric exponential potential using high-precision methods, comparing two approaches based on Bessel functions to analyze wave functions and secular equations.

Contribution

It introduces and compares two Bessel-function-based methods for accurately determining bound states in the exponential potential.

Findings

Both methods reliably compute wave functions.

The secular equation approaches are effective for this potential.

High-precision calculations enhance the accuracy of bound state analysis.

Abstract

Several properties of bound states in potential $ V(x)= g^2\exp (|x|)$ are studied. Firstly, with the emphasis on the reliability of our arbitrary-precision construction, wave functions are considered in the two alternative (viz., asymptotically decreasing or regular) exact Bessel-function forms which obey the asymptotic or matching conditions, respectively. The merits of the resulting complementary transcendental secular equation approaches are compared and their applicability is discussed.