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On the existence of eigenvalues of a divergence-form operator \(A+\lambda B\) in a gap of \(\sigma(A)\). (English) Zbl 0806.47042
Summary: We consider uniformly elliptic divergence type operators \(A= - \sum\partial_ j a_{ij}(x)\partial_ i\) with bounded, Lipschitz continuous coefficients, acting in the Hilbert space \(L_ 2(\mathbb{R}^ \nu)\). It is easy to see that such an operator cannot have discrete eigenvalues below the infimum of the essential spectrum. In order to produce eigenvalues with exponentially decaying eigenfunctions we study the family of operators \[ A+ \lambda B= -\sum \partial_ j(a_{ij}(x)+ \lambda b_{ij}(x)) \partial_ i,\quad \lambda\geq 0, \] where \(A\) is supposed to have a spectral gap, while \((b_{ij})\geq 0\) and \(b_{ij}(x)\to 0\), as \(x\to \infty\). One of our main results assures that discrete eigenvalues of \(A+ \lambda B\) move into the gap, as \(\lambda\) increases, if the support of the matrix function \((b_{ij})\) is large enough. In addition, we analyze the connection between decay properties of the coefficient matrix \((b_{ij})\) and the asymptotics of the associated eigenvalue counting function; these results are modeled on our earlier work in the Schrödinger case.

47F05 General theory of partial differential operators