# On the analogues of Szegő's theorem for ergodic operators

## Abstract

Szegő's theorem on the asymptotic behaviour of the determinants of large Toeplitz matrices is generalized to the class of ergodic operators. The generalization is formulated in terms of a triple consisting of an ergodic operator and two functions, the symbol and the test function. It is shown that in the case of the one-dimensional discrete Schrödinger operator with random ergodic or quasiperiodic potential and various choices of the symbol and the test function this generalization leads to asymptotic formulae which have no analogues in the situation of Toeplitz operators. Bibliography: 22 titles.

- Authors:

- FernUniversität in Hagen, Hagen (Germany)
- B.Verkin Institute for Low Temperature Physics and Engineering, National Academy of Sciences of Ukraine, Khar'kov (Ukraine)

- Publication Date:

- OSTI Identifier:
- 22590462

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Sbornik. Mathematics; Journal Volume: 206; Journal Issue: 1; Other Information: Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICAL METHODS AND COMPUTING; ASYMPTOTIC SOLUTIONS; MATHEMATICAL OPERATORS; ONE-DIMENSIONAL CALCULATIONS; RANDOMNESS

### Citation Formats

```
Kirsch, W, and Pastur, L A.
```*On the analogues of Szegő's theorem for ergodic operators*. United States: N. p., 2015.
Web. doi:10.1070/SM2015V206N01ABEH004448.

```
Kirsch, W, & Pastur, L A.
```*On the analogues of Szegő's theorem for ergodic operators*. United States. doi:10.1070/SM2015V206N01ABEH004448.

```
Kirsch, W, and Pastur, L A. Sat .
"On the analogues of Szegő's theorem for ergodic operators". United States.
doi:10.1070/SM2015V206N01ABEH004448.
```

```
@article{osti_22590462,
```

title = {On the analogues of Szegő's theorem for ergodic operators},

author = {Kirsch, W and Pastur, L A},

abstractNote = {Szegő's theorem on the asymptotic behaviour of the determinants of large Toeplitz matrices is generalized to the class of ergodic operators. The generalization is formulated in terms of a triple consisting of an ergodic operator and two functions, the symbol and the test function. It is shown that in the case of the one-dimensional discrete Schrödinger operator with random ergodic or quasiperiodic potential and various choices of the symbol and the test function this generalization leads to asymptotic formulae which have no analogues in the situation of Toeplitz operators. Bibliography: 22 titles.},

doi = {10.1070/SM2015V206N01ABEH004448},

journal = {Sbornik. Mathematics},

number = 1,

volume = 206,

place = {United States},

year = {Sat Jan 31 00:00:00 EST 2015},

month = {Sat Jan 31 00:00:00 EST 2015}

}

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