Enclosure Theorems for Eigenvalues of Elliptic Operators
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چکیده
are to be considered when the coefficients a,-,-, b, and c are continuous real-valued functions with &=^0, c>0 in En. The ellipticity of L implies that the symmetric matrix (a,,) is everywhere positive definite. A "solution" u of Lu = 0 is supposed to be of class C1 and all derivatives involved in (1.1) are supposed to exist, be continuous, and satisfy Lu = 0 at every point. The eigenvalue problem for L on En will be called the basic problem. The only assumption to be made is that there exists at least one eigenvalue X for this problem whose associated eigenfunctions are "L-strongly asymptotic to zero" as x—>=° (definition in §2). Our purpose is to obtain variational formulae for the eigenvalues and eigenfunctions of L when E" is perturbed to an w-disk of large radius a, and the null boundary condition is adjoined on the bounding (« —1)sphere. If the eigenspace of X is w-dimensional, our first theorem shows in particular that at least m eigenvalues of the perturbed problem converge to X as a—> °°. Our other results are refinements of this which lead to asymptotic estimates for eigenfunctions. The method of estimation used here is due to H. F. Bohnenblust. The problem at hand of estimating eigenvalues and eigenfunctions for large domains has its physical origin in certain models of enclosed quantum mechanical systems, considered by a number of authors including de Groot and ten Seldam [2], [12], Dingle [3], Hull and Julius [6], Sommerfeld and Hartman [7]. In the case that the Schrodinger operator (a special case of (1.1)) is separable, the problem reduces to a domain-perturbation problem for a singular second-
منابع مشابه
On the Spectral Properties of Degenerate Non-selfadjoint Elliptic systems of Differential Operators
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تاریخ انتشار 2010