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
Abstract:
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 $a\in\mathbb R$ is the parameter of the family and $\lambda$ is the
spectral parameter. It is assumed that $g\colon\mathbb R\to\mathbb R$ 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 $0\le a<a_1$, including the
neighbourhoods of the points $a=a_1$ and $a=a_2$. For $\lambda\gg1$, it is
shown that the spectrum consists of two branches $\lambda=\lambda_{\pm}(a,p)$,
indexed by the signs $\pm$ and by an integer $p\in\mathbb Z^+$, $p\gg1$.
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 $a>a_2$.
Keywords:definite and indefinite Sturm–Liouville problems, asymptotic behaviour of the spectrum, turning points.
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