Journal article
Trotter's limit formula for the Schrodinger equation with singular potential
Journal of mathematical physics, Vol.58(12), p.122101
12/01/2017
DOI: 10.1063/1.5013243
Abstract
We discuss the Schrodinger equation with singular potentials. Our focus is non-relativistic Schrodinger operators H with scalar potentials V defined on R-d, hence covering such quantum systems as atoms, molecules, and subatomic particles whether free, bound, or localized. By a "singular potential" V, we refer to the case when the corresponding Schrodinger operators H, with their natural minimal domain in L-2(R-d), are not essentially self-adjoint. Since V is assumed real valued, the corresponding Hermitian symmetric operator H commutes with the conjugation in L-2(R-d), and so (by von Neumann's theorem), H has deficiency indices (n, n). The case of singular potentials V refers to when n > 0. Hence, by von Neumann's theory, we know the full variety of all the self-adjoint extensions. Since the Trotter formula is restricted to the case when n = 0, and here n > 0, two questions arise: (i) existence of the Trotter limit and (ii) the nature of this limit. We answer (i) affirmatively. Our answer to (ii) is that when n > 0, the Trotter limit is a strongly continuous contraction semigroup; so it is not time-reversible. Published by AIP Publishing.
Details
- Title: Subtitle
- Trotter's limit formula for the Schrodinger equation with singular potential
- Creators
- Ekaterina S Nathanson - Georgia Gwinnett Coll, Sch Sci & Technol, 1000 Univ Ctr Lane, Lawrenceville, GA 30043 USAPalle E. T Jorgensen - Univ Iowa, Dept Math, Iowa City, IA 52242 USA
- Resource Type
- Journal article
- Publication Details
- Journal of mathematical physics, Vol.58(12), p.122101
- Publisher
- AMER INST PHYSICS
- DOI
- 10.1063/1.5013243
- ISSN
- 0022-2488
- eISSN
- 1089-7658
- Number of pages
- 13
- Language
- English
- Date published
- 12/01/2017
- Academic Unit
- Mathematics
- Record Identifier
- 9984240864602771
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