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Kato Class Potentials for Higher Order Elliptic Operators
Author(s) -
Davies E. B.,
Hinz A. M.
Publication year - 1998
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610798006565
Subject(s) - semigroup , mathematics , sobolev space , standard probability space , order (exchange) , operator (biology) , elliptic operator , closure (psychology) , generator (circuit theory) , domain (mathematical analysis) , pure mathematics , measure (data warehouse) , quadratic form (statistics) , discrete mathematics , combinatorics , mathematical analysis , physics , quantum mechanics , power (physics) , finance , repressor , database , computer science , transcription factor , economics , gene , biochemistry , chemistry , market economy
Our goal in this paper is to determine conditions on a potential V which ensure that an operator such as 1 H : = ( − Δ ) m + Vacting on L 2 ( R N ) defines a semigroup in L p ( R N ) for various values of p including p =1. The operator is defined as a quadratic form sum. That is, we putQ 0 ( f ) : = ∫ (( ‐ Δ ) m f ) f ¯for f ∈ C c ∞(all integrals are on R N and are with respect to Lebesgue measure), and note that the closure of the form is non‐negative and has domain equal to the Sobolev space W m ,2 . We then assume that the potential has quadratic form bound less than 1 with respect to Q 0 , and define Q ( f ) : = Q 0 ( f ) + ∫ V| f | 2This form is closed and is associated with a semibounded self‐adjoint operator H in L 2 (see [ 17 , p. 348; 5 , Theorem 4.23]). One can then ask whether the semigroup e − Ht defined on L 2 for t ⩾0 is extendable to a strongly continuous one‐parameter semigroup on L p for other values of p , and if so whether one can describe the domain and spectrum of its generator.