# Extended weak coupling limit for Friedrichs Hamiltonians

## Abstract

We study a class of self-adjoint operators defined on the direct sum of two Hilbert spaces: a finite dimensional one called sometimes a 'small subsystem' and an infinite dimensional one called a 'reservoir'. The operator, which we call a 'Friedrichs Hamiltonian', has a small coupling constant in front of its off-diagonal term. It is well known that under some conditions in the weak coupling limit the appropriately rescaled evolution in the interaction picture converges to a contractive semigroup when restricted to the subsystem. We show that in this model, the properly renormalized and rescaled evolution converges on the whole space to a new unitary evolution, which is a dilation of the above mentioned semigroup. Similar results have been studied before ( 1990) in more complicated models under the name of 'stochastic limit'.

- Authors:

- Department of Mathematical Methods in Physics, Warsaw University, Hoza 74, 00-682 Warsaw (Poland)
- (Belgium)

- Publication Date:

- OSTI Identifier:
- 20929606

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Mathematical Physics; Journal Volume: 48; Journal Issue: 1; Other Information: DOI: 10.1063/1.2405402; (c) 2007 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; COUPLING CONSTANTS; HAMILTONIANS; HILBERT SPACE; INTERACTIONS; MATHEMATICAL EVOLUTION; RENORMALIZATION

### Citation Formats

```
Derezinski, Jan, Roeck, Wojciech de, and Instituut voor Theoretische Fysica, K.U. Leuven.
```*Extended weak coupling limit for Friedrichs Hamiltonians*. United States: N. p., 2007.
Web. doi:10.1063/1.2405402.

```
Derezinski, Jan, Roeck, Wojciech de, & Instituut voor Theoretische Fysica, K.U. Leuven.
```*Extended weak coupling limit for Friedrichs Hamiltonians*. United States. doi:10.1063/1.2405402.

```
Derezinski, Jan, Roeck, Wojciech de, and Instituut voor Theoretische Fysica, K.U. Leuven. Mon .
"Extended weak coupling limit for Friedrichs Hamiltonians". United States.
doi:10.1063/1.2405402.
```

```
@article{osti_20929606,
```

title = {Extended weak coupling limit for Friedrichs Hamiltonians},

author = {Derezinski, Jan and Roeck, Wojciech de and Instituut voor Theoretische Fysica, K.U. Leuven},

abstractNote = {We study a class of self-adjoint operators defined on the direct sum of two Hilbert spaces: a finite dimensional one called sometimes a 'small subsystem' and an infinite dimensional one called a 'reservoir'. The operator, which we call a 'Friedrichs Hamiltonian', has a small coupling constant in front of its off-diagonal term. It is well known that under some conditions in the weak coupling limit the appropriately rescaled evolution in the interaction picture converges to a contractive semigroup when restricted to the subsystem. We show that in this model, the properly renormalized and rescaled evolution converges on the whole space to a new unitary evolution, which is a dilation of the above mentioned semigroup. Similar results have been studied before ( 1990) in more complicated models under the name of 'stochastic limit'.},

doi = {10.1063/1.2405402},

journal = {Journal of Mathematical Physics},

number = 1,

volume = 48,

place = {United States},

year = {Mon Jan 15 00:00:00 EST 2007},

month = {Mon Jan 15 00:00:00 EST 2007}

}