Expansion in generalized eigenfunctions for Laplacians on graphs and metric measure spaces

Abstract

Abstract. We consider an arbitrary selfadjoint operator in a separable Hilbert space. To this operator we construct an expansion in generalized eigenfunctions, in which the original Hilbert space is decomposed as a direct integral of Hilbert spaces consisting of general eigenfunctions. This automatically gives a Plancherel type formula. For suitable operators on metric measure spaces we discuss some growth restrictions on the generalized eigenfunctions. For Laplacians on locally finite graphs the generalized eigenfunctions are exactly the solutions of the corresponding difference equation. Contents

Shnol’s theorem and the spectrum of long range operators

“…So far there have been various generalizations of the Shnol's theorem (see e.g. [17,18,4,7,6,19]). In this note, we generalize this result to the long range case.…”

Growth of Eigenfunctions and R-limits on Graphs

“…The proof of Theorem 2.2 depends on the existence of a generalized eigenfunction for each point in the spectrum admitting a suitable growth rate. The following statement provides a sufficient condition to get such generalized eigenfunction which can be found in [28,Theorem 3] in a more general setting. Theorem 4.14 (reverse Shnol's Theorem, [28,Theorem 3]).…”

“…The following statement provides a sufficient condition to get such generalized eigenfunction which can be found in [28,Theorem 3] in a more general setting. Theorem 4.14 (reverse Shnol's Theorem, [28,Theorem 3]). Let (G, v 0 ) be a connected, infinite, rooted graph and H be a Schrödinger operator on 2 (G) of the form (1.1) with spectral measure µ.…”

“…[3,4,6] and references therein. In the literature also the converse question is addressed [7,17,28]. To be more precise, one seeks to find for µ-almost every element in the spectrum, a generalized eigenfunction that has at most sub-exponential growth, where µ is the spectral measure of the operator H. Such a converse theorem is used in the proof of Theorem 2.2.…”

Agmon estimates for Schrödinger operators on graphs

“…Moreover, when talking about generalized eigenfunction one should mention the relation to Shnol' type theorems [ Š57, BdMLS09, CFKS87, BD19] which were recently studied in the context of graphs [BP20,HK11]. Furthermore, a general perspective is taken in [LT16] on studying generalized eigenfunctions on graphs.…”