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• W1965660852 abstract "The theory of singular local perturbations of translation invariant positivity preserving semigroups on L2(R, dx) is developed. A powerful approximation theorem is proved which allows the treatment of a very general class of singular perturbations. Estimates on the local singularities of the kernels of the semigroups, e~'H, are given. Eigenfunction expansions are derived. The local singularities of the eigenfunction and generalized eigenfunctions are discussed. Results are illustrated with examples involving singular perturbations of —A. I. Introduction. The sum of an operator, //0, which generates a positivity preserving translation invariant semigroup on L2(RN, dNx) and a potential V is the subject of the present work. In §11 the class of such H0's is discussed more fully. Here we only remark that the operators corresponding to nonrel- ativistic and relativistic energy in quantum mechanics are included. The potentials considered are, in general, too singular to be operators and are given as forms, so that H0 + V must be defined as a form sum. A detailed description of the potentials is given in §111 and IV. The success of the perturbation program for the investigation of operator sums is impressive (20). For form sums such an analysis is more difficult because functions of forms are generally undefined. One technique for analyzing functions,/, of H0 + V is to: (1) approximate Kby bounded functions V; (2) show f(H0 + approximates/(//0 + V); and (3) analyze f(H0 + V) by (2) and a direct analysis of f(H0 + V). Such a procedure for (1) and (2) was developed by Kato (20) and Faris (10), using/(x) = (x + X)-1. It employs monotone convergence arguments and so is applicable only when the potential V can be written as the sum of a rather general nonnegative function V+ and a nonpositive function V_ which is a small form perturbation of H0. One truncates V+ and K_ to obtain functions V+ n and V_ which are absolutely bounded by the integer n. Then, for all" @default.
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• W1965660852 date "1978-01-01" @default.
• W1965660852 modified "2023-09-23" @default.
• W1965660852 title "Perturbation of translation invariant positivity preserving semigroups on $Lsp{2}({bf R}sp{N})$" @default.
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• W1965660852 doi "https://doi.org/10.1090/s0002-9947-1978-0470750-2" @default.
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