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UBC Theses and Dissertations

Properties of eigenvalues of singular second order elliptic operators Welsh, K. Wayne

Abstract

This thesis investigates the properties of the L₂-eigenvalues of singular, elliptic, second order operators, primarily the operator L defined by [formula omitted]. Here the "potential function", V(x), is such that [formula omitted] is a norm on [formula omitted] being the usual norm in the Sobolev space W¹̕²(G) and [formula omitted] is the completion of [formula omitted] in the metric from this norm, identified with a subset L₂(G) ; Δ is the Laplacian and G is an arbitrary open domain of E[superscript n] . Several sufficient conditions are given on V and on G in order that L have spectrum satisfying [formula omitted] , for some real number [formula omitted] denote the spectrum and point spectrum of L , respectively). The properties of these lower eigenvalues are investigated by examining the eigenvalues of a coercive bilinear form corresponding to the operator. Such a form B , having domain [symbol omitted] , say, is defined to have eigenvalueλє¢ with corresponding eigenfunction [symbols omitted] if B[u,f] = λ (u,f) for all f є [symbol omitted] . Variational properties are discussed in detail; In particular, a condition is given which ensures that the numbers sup inf B[u,u] (the sup and inf being over appropriate sets involving [symbol omitted] and n ) are eigenvalues of B . These properties are applied to L to generalize the well-known classical property (G bounded) of monotonic dependence of the eigenvalues on the underlying domain G : G [symbol omitted] G* implies [formula omitted] for corresponding eigenvalues, with strict inclusion implying strict inequality. A few miscellaneous properties of the eigenvalues and eigenfunctions then follow from this dependence.

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