Limiting absorption principle and virtual levels of operators in Banach spaces

Abstract

We review the concept of the limiting absorption principle and its connection to virtual levels of operators in Banach spaces. R ÉSUM É. Nous passons en revue le principe d'absorption limite et sa relation avec les niveaux virtuels pour des opérateurs dans les espaces de Banach. Keywords limiting absorption principle • nonselfadjoint operators • threshold resonances • virtual levels • virtual states PACS 02.30.Tb • 02.30.Jr Mathematics Subject Classification (2010) 35P05 • 47Axx • 47B01 To Alexander Shnirelman on the occasion of his 75th birthday 1 Limiting absorption principleThe idea of introducing a small absorption into the wave equation for selecting particular solutions goes back to Ignatowsky [Ign05] and is closely related to the Sommerfeld radiation condition [Som12]. We start with the Helmholtz equation−∆ − zI) −1 represented by the convolution with e −|x| √ −z /(4π|x|), Re √ −z > 0.