Second order periodic differential operators. Threshold properties and homogenization

Authors:
M. Sh. Birman and T. A. Suslina

Translated by:
T. A. Suslina

Original publication:
Algebra i Analiz, tom **15** (2003), nomer 5.

Journal:
St. Petersburg Math. J. **15** (2004), 639-714

MSC (2000):
Primary 35P99, 35Q99

Published electronically:
August 2, 2004

MathSciNet review:
2068790

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Abstract: The vector periodic differential operators (DO's) admitting a factorization , where is a first order homogeneous DO, are considered in . Many operators of mathematical physics have this form. The effects that depend only on a rough behavior of the spectral expansion of in a small neighborhood of zero are called *threshold effects* at the point . An example of a threshold effect is the behavior of a DO in the small period limit (the homogenization effect). Another example is related to the negative discrete spectrum of the operator , , where and as . ``Effective characteristics'', such as the homogenized medium, effective mass, effective Hamiltonian, etc., arise in these problems. The general approach to these problems proposed in this paper is based on the spectral perturbation theory for operator-valued functions admitting analytic factorization. Most of the arguments are carried out in abstract terms. As to applications, the main attention is paid to homogenization of DO's.

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Additional Information

**M. Sh. Birman**

Affiliation:
Department of Physics, St. Petersburg State University, Ul’yanovskaya 1, Petrodvorets, St. Petersburg 198504, Russia

**T. A. Suslina**

Affiliation:
Department of Physics, St. Petersburg State University, Ul’yanovskaya 1, Petrodvorets, St. Petersburg 198504, Russia

Email:
tanya@petrov.stoic.spb.su

DOI:
https://doi.org/10.1090/S1061-0022-04-00827-1

Keywords:
Periodic operators,
threshold effect,
homogenization

Received by editor(s):
June 25, 2003

Published electronically:
August 2, 2004

Additional Notes:
Supported by RFBR (grant no. 02-01-00798).

Article copyright:
© Copyright 2004
American Mathematical Society