×

Semigroup growth bounds. (English) Zbl 1114.47040

Via the Legendre transform the relationship between the norms of a one-parameter semigroup and those of its resolvent operators is studied. In particular, information on the short time behaviour are obtained. The author succeed in obtaining lower bounds, not on the semigroup norm themselves, but on certain regularizations. It is further shown that it is not possible to obtain similar upper bounds from numerical information about the resolvent norms, however accurate this information may be. Both of these facts are completely invisible if one only looks at the spectrum of the resolvent operator. The paper contains also results concerning the long time behaviour. It is shown that the norm of the Schr”odinger semigroup grows exponentially, although the spectrum of the operator is purely imaginary. Further, the long time behaviour of the norms of diffusion semigroups with self-adjoint generators may be entirely different for the \(L^1\) and \(L^2\) norm, although the generator has t he same spectrum in the two spaces.

MSC:

47D06 One-parameter semigroups and linear evolution equations
35J10 Schrödinger operator, Schrödinger equation
47D08 Schrödinger and Feynman-Kac semigroups
65J10 Numerical solutions to equations with linear operators
47F05 General theory of partial differential operators
PDFBibTeX XMLCite
Full Text: arXiv