A Note on Random Perturbations of a Multiple Eigenvalue of a Hermitian Operator |
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Authors: | G Gaines K Kaphle F Ruymgaart |
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Institution: | 1. Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX, 79409, USA 2. Division of Natural and Behavioral Sciences, University of Maine at Fort Kent, Fort Kent, ME, 04743, USA
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Abstract: | The problem of a random Hermitian perturbation of a multiple isolated eigenvalue of a Hermitian operator is considered. It is shown that the combined multiplicities of the perturbed eigenvalues converge in probability to the multiplicity of the eigenvalue of the target operator. Also the asymptotic distribution of a certain average of these eigenvalues, centered at the target, is obtained. As a tool differentiation of analytic functions of operators is employed in conjunction with an ensuing “delta-method”. The result is of a probabilistic rather than statistical nature. |
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