Approximation of isolated eigenvalues of general singular ordinary differential operators |
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Authors: | Günter Stolz Joachim Weidmann |
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Institution: | 1. Department of Mathematics, CH 452, University of Alabama, Birmingham, AL, 35294-1170, USA 2. Fachbereich Mathematik, Johann Wolfgang Goethe-UniversitΞt, D-60054, Frankfurt, Germany
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Abstract: | Let A be a self-adjoint operator defined by a general singular ordinary differential expression τ on an interval (a, b), ? ∞ ≤ a < b ≤ ∞. We show that isolated eigenvalues in any gap of the essential spectrum of A are exactly the limits of eigenvalues of suitably chosen self-adjoint realizations An of τ on subintervals (an, bn) of (a, b) with an → a, bn → b. This means that eigenvalues of singular ordinary differential operators can be approximated by eigenvalues of regular operators. In the course of the proof we extend a result, which is well known for quasiregular differential expressions, to the general case: If the spectrum of A is not the whole real line, then the boundary conditions needed to define A can be given using solutions of (τ ? λ)u = 0, where λ is contained in the regularity domain of the minimal operator corresponding to τ. |
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