Generalized eigenfunctions and real axis limits of the resolvent

Abstract

Let (H, (., *)) be a Hilbert space and A, E be a selfadjoint operator and corresponding spectral measure in { (A f fAE(dA)). It is known that for a suitable positive subspace H+ CH and measure p the generalized eigenfunctions E([A -, A + hl )f E_(A) k11 un = l imim b-0 PdX h, A + bl) A-A p(A) exist in H., the corresponding negative space, for p-almost every A and f EJH+. It is shown that for each A the OX.f form a pre-Hilbert space H using the natural inner product (/f, bg)x = lim_.X((E(A)f, g)/p(A)), and that 1101-: 0), the whole question of eliminating the unwanted ones in the case of infinite domains by specifying a boundary condition at infinity (radiation condition) has only been settled in certain cases [171. The approach which has succeeded in the case of partial differential operators starts with the spectral measure E() arising in the spectral representation of P. One then shows that E (J) f(x) =f [fJ O(X, Y A)r(dA)] /(y) dy where J is an interval, 0 is a hermitian kernel which satisfies the equation P0(x, , A) = A0(x, , A) and r is a positive measure on the real line [91. This kind of result has been generalized to an abstract setting using Hilbert spaces with negative norm by the Russian school. Their work is discussed in a now translated book by l3erezanskiY [11. The present paper draws heavily on the theory in BerezanskiY's book. 1. Spaces with negative norm [1, Chapter 11. Let ({ (,0 1 ) be a separable Hilbert space containing a dense linear manifold }which is also a This content downloaded from 157.55.39.38 on Thu, 28 Apr 2016 06:14:01 UTC All use subject to http://about.jstor.org/terms 19721 GENERALIZED EIGENFUNCTIONS AND REAL AXIS LIMITS 491 Hilbert space under the norm || * || +> || * l o. Then, it can be shown that there exists an operator T: f(o -+ such that (/, g)O = (T/, Tg)+. Furthermore, if we consider the linear functionals ?b: J(+ -* C, l?b(/)i < j/II+,, we find they form a Hilbert space which is the closure of R( with respect to the inner product (/, g)_ = (Ti, Tg)0. We call this Hilbert space J_ hence, the name "space with negative norm". The case of special interest to us occurs when T is a Hilbert-Schmidt operator in Jo; that is Em,n (Ten, em)1 2 < for one and therefore for all c.o.n.s. (complete orthonormal sets) le 100 in Ho. We shall denote the trace E (Se , en) of a nonnegative operator S, whenever it exists, by tr(S). 2. The Hilbert space of generalized eigenfunctions. Suppose E( ) is a spectral measure and T, considered as a mapping of Y( into itself, is a one-toohe Hilbert-Schmidt contraction. Then the measure p0, defined by pO(\ = tr(T*E(A) T), is a bounded nonnegative regular Borel measure. Given any two nonnegative complete regular Borel measures or, p on R which are bounded on compact sets, using a theorem of de Guzman [11, Theorem 2.4], one can prove that vr ([ h, A + hi) h-0+ p'(Ii h-+ hi exists p-a.e. A and yields essentially the Radon-Nikodym derivative (do/dp)(X). To simplify notation, we shall henceforth denote such limits by lim ar (A\) A-X) tA -\p (A)9 and assume our measures are complete unless stated otherwise. Also, wk lim.= i will mean lima-8 da(g) = 0/g) V gEJ{+, where ? b EJ{ Our first theorem is a slight generalization of a result which is essentially known. Theorem 1. Let p be a locally bounded positive Borel measure. Then w ..E (A)~ wk lim () A -X p (A) exists p-a.e. for all f EY(+ yielding a functional 0 /, EI(. Furthermore, A ox,f 11 -5 (dp oldp) (A) 1 l + p-a.e. and for each A where both dpo/dp and dE( )/f/dp0 exist we have This content downloaded from 157.55.39.38 on Thu, 28 Apr 2016 06:14:01 UTC All use subject to http://about.jstor.org/terms 492 N. A. DERZKO [December dE(.) f dpo dE(/) f dp dp dp0 If f, g E J+ then (E( )f, g) is absolutely continuous with respect to pO. Proof. We first prove the theorem for p = po. Let ,1I be a countable dense subset of J{+. Then if f, g E we have I(E(A) f/ g)| = I(E(A) TT-'f, TT-'g)| = I(T*E(A) TT-'f, T-1g)l < W E (A) TT-'f, T-'f) Y2TWE(A) TT-1g, T-1g)"h < po(A) Ilf 11 +jjgjj The last inequality is a consequence of the fact that