Vol. 47, No. 2, 1973

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Hilbert transforms, and a problem in scattering theory

Michael John Westwater

Vol. 47 (1973), No. 2, 567–608

Let H be a separable infinite dimensional complex Hilbert space, B(H) the set of bounded linear operators on H. Consider a holomorphic map z K(z) from a complex neighborhood of some interval of the real axis to B(H), such that for z real K(z) is hermitian. These conditions are satisfied by K(z) = H z1, with H hermitian. In this special case K(z) has a bounded inverse S(z) (the resolvent of H), for z not on the real axis, and S(z) can be represented as the Hilbert transform of a measure whose values are bounded positive operators (the spectral measure of H); for z on the real axis K(z) has a (generally unbounded) inverse for z not in the point spectrum of H; closely related to the spectral representation of S(z) is an approximation theorem which asserts roughly that for most real values of z,[K(z)]1 can be approximated by operators of finite rank obtained by taking the orthogonal projector P onto a finite dimensional subspace D and inverting PK(z)P on D. The object of this paper is to give conditions on K(z) sufficient to imply the conlusions just noted in the special case K(z) = H z1.

The main theorems are Theorem 1 in §1, and Theorem 4 in §4; each has two parts—a representation for [K(z)]1 for z complex, and an approximation theorem for [K(z)]1 for z real. Theorem 4 is used in §5 to prove a convergence theorem (Theorem 5) for the Kohn variational method in quantum mechanical potential scattering (the relevant terms are defined in that section). This application motivated the writing of the paper.

Mathematical Subject Classification 2000
Primary: 47A40
Received: 25 May 1972
Revised: 24 August 1972
Published: 1 August 1973
Michael John Westwater