Spectrum of the Static Potential Schrodinger Equation over E n

Abstract

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.

On the eigenvalues of a second order elliptic operator in an unbounded domain

“…In the case where a(x, D) is the Laplace operator, Brownell [3] has shown that the fundamental singularity has exponential decay in R n . Next, we need a Green's function G R (τ, z, x) for the bounded cut off domain E R = {xe E: \x | < R}.…”

A note on Kato’s uniqueness criterion for Schrödinger operator self-adjoint extensions

“…Now consider the following condition on V. As stated in Corollary T.4 immediately thereafter, this condition is implied by Condition I, as we see from (15) 2 We note here that ^ need not be dense in L 2 (R n ) if n<3, although H x will not be a very respectable operator from the Hubert space viewpoint if ^ is not dense, in particular not being symmetric. This theorem is the same as our earlier one ( [1]), Theorem 5.3, p. 572) except for change in the initial domain from S^ = ^/^Π ^ there to ^ = ^fΐΊ ĥ ere. Merely sketching the proof, we first see…”

“…follows for φ e ^ and u e L λ (R n ) Π L 2 (R n ), the proof being unchanged from the earlier one ( [1], Theorem 5.1, p. 568) for φ having continuous second partials and vanishing outside a bounded set. Taking φ e J^ = <Λ" Π Si in (24) and using the facts that G ω is bounded Hermitian and that Since (19) and (16) follow from Condition I for large ω by Theorem T.3, this theorem follows from our earlier one ( [1], Theorem 6.4, p. 579).…”

“…Using the YoungTitchmarsh theorem on Fourier transforms, we generalize Kato's argument to general dimension n > 1. We show the connection of the resulting criterion with our earlier construction [1] of a self-adjoint extension as the inverse of a modified Green function integral operator. We also give a variational characterization of the spectrum here.…”

“…This may be defined (see [1], p. 555) uniquely by the requirements that n KJr) be continuous over r > 0, that n KJ\ x\)e L x {R n ) over x, and that [ω 2 + | y …”