The string of variable density: further results

TL;DR

This paper develops a perturbative and iterative analytical framework to compute eigenvalues and eigenfunctions of a string with arbitrary density, covering both high and low energy regimes, improving understanding of spectral properties.

Contribution

It introduces a novel perturbative scheme based on WKB functions and an iterative method for analyzing strings with variable density, including rapidly oscillating cases.

Findings

Explicit expressions for eigenvalues and eigenfunctions at high energy

Analytical formulas for low energy behavior of eigenvalues

Recovery of known results with simplified methods

Abstract

We analyze the problem of calculating the solutions and the spectrum of a string with arbitrary density and fixed ends. We build a perturbative scheme which uses a basis of WKB-type functions and obtain explicit expressions for the eigenvalues and eigenfunctions of the string. Using this approach we show that it is possible to derive the asymptotic (high energy) behavior of the string, obtaining explicit expressions for the first three coefficients (the first two can also be obtained with the WKB method). Finally using an iterative approach we also obtain analytical expressions for the low energy behavior of the eigenvalues and eigenfunctions of a string with rapidly oscillating density, recovering (in a simpler way) results in the literature.