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Separation of Eigenvalues of the Wave Equation for the Unit Ball in R N
Author(s) -
Graham Keith D.
Publication year - 1973
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1002/sapm1973524329
Subject(s) - mathematics , bessel function , eigenvalues and eigenvectors , mathematical analysis , wave equation , dirichlet boundary condition , boundary value problem , boundary (topology) , physics , quantum mechanics
It is shown that π is the infinium gap between the consecutive square roots of the eigenvalues of the wave equation in a hypespherical domain for both the Neumann (free) and the full range of mixed (elastic) homogeneous boundary conditions. Previous literature contains the same information apparently only for the Dirichlet (fixed) boundary condition. These square roots of the eigenvalues are the zeros of solutions of a differential equation in Bessel functions (first kind) and their first derivatives. The infinium gap is uniform for Bessel functions of orders x ≥ ½ (as well as for x = 0). The intervals between the roots are actually monotone decreasing in length. These results are obtained by interlacing zeros of Bessel and associated functions and comparing their relative displacements with oscillation theory. If W l denotes the l th positive root for some fixed order x , the minimum gap property assures that {exp(± iw l t | l = 1, 2,...} form a Riesz basis in L 2 (0, τ) for τ > 2. This has application to the problem of controlling solutions of the wave equation by controlling the boundary values.