Asymptotic behaviour of the positive spectrum of a family of periodic Sturm-Liouville problems under continuous passage from a definite problem to an indefinite one

摘要

We consider the problem of the spectrum of a parameter-dependent family of periodic Sturm-Liouville problems for the equation U '' + lambda(2)(g(x) - a)u = 0, where a is an element of R is the parameter of the family and A is the spectral parameter. It is assumed that g: R - R is a sufficiently smooth 2 pi-periodic function with one simple maximum g(x(max)) = a(1) > 0 and one simple minimum g(x(min)) = a(2) > 0 over a period, and that the functions g(x - x(min)) and g(x - x(max)) are even. Under these assumptions, the first two asymptotic terms are calculated explicitly for the positive eigenvalues on the whole interval 0 <= a <= a(1), including the neighbourhoods of the points a = a(1) and a = a(2). For lambda 1, it is shown that the spectrum consists of two branches lambda = lambda(+/-) (a, p), indexed by the signs and by an integer p is an element of Z(+), p 1. A. unified interpolation formula is derived to describe the asymptotic behaviour of the spectrum branches in the passage from the definite (classical) problem with a < a(2) to the indefinite problem with a > a(2).