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Matrix Sturm-Liouville operators with boundary conditions dependent on the spectral parameter. (English) Zbl 1327.34049

Summary: Let \(L\) denote the operator generated in \(L_2(\mathbb{R}_+,E)\) by the differential expression \[ l(y)=- y''+ Q(x)y,\quad x\in\mathbb{R}_+, \] and the boundary condition \((A_0+ A_1\lambda)Y'(0,\lambda)- (B_0+ B_1\lambda)Y(0,\lambda)= 0\) where \(Q\) is a matrix-valued function and \(A_0\), \(A_1\), \(B_0\), \(B_1\) are non-singular matrices, with \(A_0B_1- A_1B_0\neq 0\). Using the uniqueness theorems of analytic functions, we investigate the eigenvalues and the spectral singularities of \(L\).
In particular, we obtain the conditions on \(q\) under which the operator \(L\) has a finite number of eigenvalues and spectral singularities.

MSC:

34B24 Sturm-Liouville theory
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
34L05 General spectral theory of ordinary differential operators
34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators
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