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.

Weakly bound states in heterogeneous waveguides

“…(1), in one or more dimensions, have been studied by one of the present authors. The aspects studied earlier include the description of non-perturbative methods for the calculation of the lowest part of the spectrum [18,20,21], of a perturbation method for the calculation of the eigenvalues of two-dimensional domains obtained from a small deformation of a reference (solvable) domain [22], the derivation of spectral zeta functions associated to heterogeneous systems in one and two dimensions [23] and the sum rules of heterogeneous domains in one or more dimensions, for different boundary conditions [19,24,25].…”

Accurate calculation of the eigenvalues of non-uniform strings and membranes

“…This property has already been exploited in Ref. [21] to obtain a perturbative expansion around that basis set. It was pointed out in Ref.…”

“…[20]), have been recently discussed by Amore [21]. By means of the change of variable = σ ( )/σ (L) it is straightforward to verify that these…”

PT-symmetric strings

“…In the case of a string with Dirichlet boundary conditions at ±L Amore [4] showed that if the density satisfies the differential equation…”

“…In a series of papers Amore studied the spectral problems of inhomogeneous strings and drums [2][3][4][5][6][7]. In this paper we enlarge the class of such problems to include vibrating strings with complex densities Σ(x) that satisfy Σ(−x) * = Σ(x).…”