The string of variable density: further results

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.

54 pages, 11 figures, 7 tables

References in corpus (4)

• Variational collocation on finite intervals
• Spectroscopy of drums and quantum billiards: perturbative and non-perturbative results
• The string of variable density: perturbative and non-perturbative results
• Can one hear the density of a drum? Weyl's law for inhomogeneous media

Cited by in corpus (6)

• Exact sum rules for inhomogeneous strings
• A perturbative approach to the spectral zeta functions of strings, drums and quantum billiards
• Isospectral flows for the inhomogeneous string density problem
• Weakly bound states in heterogeneous waveguides
• $\mathcal{P}\mathcal{T}$-symmetric strings
• Heterogeneous systems in $d$ dimensions: lower spectrum