Eigenvalues of the fractional Laplace operator in the unit ball

Abstract

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 <= 2 and 0 < alpha <= 2, or d <= 9 and alpha = 1, and we provide strong numerical evidence for d <= 9 and general alpha in (0,2].Comment: 23 pages, 4 tables, 3 figure

Ultrarelativistic bound states in the spherical well

“…Interestingly, the d = 3 investigation of the spherical well analog of the d = 1 fractional (and thus also Cauchy) infinite well problem has been initiated only recently, [18][19][20]. The existence of solutions to the eigenvalue problem has been demonstrated, together with that of a non-decreasing unbounded sequence of eigenvalues, the lowest eigenvalue being positive and simple, [19].…”

“…The existence of solutions to the eigenvalue problem has been demonstrated, together with that of a non-decreasing unbounded sequence of eigenvalues, the lowest eigenvalue being positive and simple, [19]. An analysis has been focused on finding two-sided bounds for the eigenvalues of the fractional Laplace operator in the unit ball.…”

Fractional Laplace Operator and Meijer G-function

“…The following proposition will play an important role in the study of eigenvalues of the fractional Laplace operator in [9]. For the special case of this result when d = α = 2, see [11].…”

“…Our goal in this paper is to extend the above list: We want to find more functions f for which (−∆) α/2 f can be computed explicitly. Our main motivation for doing this comes from the study of spectral properties of the fractional Laplace operator acting on functions supported in the unit ball -we consider this problem in a companion paper [9].…”

“…As one important application of this result, we produce a complete system of eigenfunctions of the operator (1 − |x| 2 ) α/2 + (−∆) α/2 with the Dirichlet boundary conditions outside of the unit ball. The latter result will be used to estimate the eigenvalues of the fractional Laplace operator in the unit ball in a companion paper [9]. …”