A Sturm-Liouville problem with physical and spectral parameters in boundary conditions
The spectrum of the problem \[ -u''+qu=\lambda u,\quad u'(0)=0,\quad u'(\pi)=m\lambda u(\pi), \] is investigated, where \(\lambda\) and \(m\) are a spectral and a physical parameter, respectively. For \(m\neq 0\), the operator \(L\), \[ L{\mathbf u}=L\begin{pmatrix} u\\ u(\pi)\end{pmatrix}=\begin{pmatrix} u''+qu\\ u'(\pi)/m\end{pmatrix} \] with a suitable domain, on the space \(H=L_2(0,\pi)\times \mathbb C\) with the inner product \[ ({\mathbf u},{\mathbf u})=\int_0^\pi|u(x)|^2+m|a|^2,\;{\mathbf u}=\begin{pmatrix} u\\ a \end{pmatrix}, \] is considered. The operator \(L\) is selfadjoint, where \(H\) is a Hilbert space if \(m>0\) and a Pontryagin space \(\Pi_1\) if \(m<0\). The main topic of this note is the behaviour of the eigenvalues as \(m\nearrow 0\). In the left-definite case, i.e., when the quadratic form \[ Q(u)=\int_0^\pi |u'(x)|^2+q(x)|u(x)|^2 dx \] is positive definite, the operator \(L\) can be considered as a selfadjoint operator in a Hilbert space, and so it has only real spectrum. Theorem 1 deals with the asymptotics of the eigenvalues and eigenfunctions in this case. In particular, there is one negative eigenvalue, which satisfies \(\lambda_0=-1/m^2+O(1)\) as \(m\nearrow 0\). In Theorem 2 it is shown that if the eigenvalue problem with \(\lambda\)-independent boundary conditions \(u'(0)=u(\pi)=0\) has a nonpositive eigenvalue \(\mu_1\), then there is a number \(m_0>0\) such that all eigenvalues of \(L\) are real for \(m\in (-m_0,0)\). The smallest eigenvalue for \(m=-m_0\) has geometric multiplicity \(1\) and algebraic multiplicity \(2\). If \(\mu_1<0\), then for sufficiently small \(\delta>0\) the problem has exactly one pair of nonreal eigenvalues for \(m\in (-m_0-\delta,-m_0)\).
scientific article; zbMATH DE number 1429621