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

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 $a>a_2$.