Eigenvalues of the fractional Laplace operator in the unit ball

We describe a highly efficient numerical scheme for finding two-sided bounds for the eigenvalues of the fractional Laplace operator (-Delta)^{alpha/2} in the unit ball D in R^d, with a Dirichlet condition in the complement of D. The standard Rayleigh-Ritz variational method is used for the upper bou...

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Bibliographic Details
Published in:arXiv.org 2016-08
Main Authors: Dyda, Bartłomiej, Kuznetsov, Alexey, Kwaśnicki, Mateusz
Format: Article
Language:English
Subjects:
Dirichlet problem
Eigenvalues
Laplace transforms
Lower bounds
Mathematical analysis
Nonlinear programming
Operators (mathematics)
Polynomials
Upper bounds
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Summary:We describe a highly efficient numerical scheme for finding two-sided bounds for the eigenvalues of the fractional Laplace operator (-Delta)^{alpha/2} in the unit ball D in R^d, with a Dirichlet condition in the complement of D. The standard Rayleigh-Ritz variational method is used for the upper bounds, while the lower bounds involve the less-known Aronszajn method of intermediate problems. Both require explicit expressions for the fractional Laplace operator applied to a linearly dense set of functions in L^2(D). We use appropriate Jacobi-type orthogonal polynomials, which were studied in a companion paper "Fractional Laplace operator and Meijer G-function". Our numerical scheme can be applied analytically when polynomials of degree two are involved. This is used to partially resolve the conjecture of T. Kulczycki, which claims that the second smallest eigenvalue corresponds to an antisymmetric function: we prove that this is the case when either d
ISSN:2331-8422
DOI:10.48550/arxiv.1509.08533