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

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

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.

**[BaPa]**N. S. Bakhvalov and G. P. Panasenko,*Homogenization: averaging processes in periodic media. Mathematical problems in the mechanics of composite materials*, ``Nauka'', Moscow, 1984; English transl., Math. Appl. (Soviet Ser.), vol. 36, Kluwer Acad. Publ. Group, Dordrecht, 1989. MR**0797571 (86m:73049)****[BeLP]**A. Bensoussan, J.-L. Lions, and G. Papanicolaou,*Asymptotic analysis for periodic structures*, Stud. Math. Appl., vol. 5, North-Holland Publishing Co., Amsterdam-New York, 1978. MR**0503330 (82h:35001)****[B1]**M. Sh. Birman,*The discrete spectrum of the periodic Schrödinger operator perturbed by a decreasing potential*, Algebra i Analiz**8**(1996), no. 1, 3–20 (Russian, with Russian summary); English transl., St. Petersburg Math. J.**8**(1997), no. 1, 1–14. MR**1392011****[B2]**M. Sh. Birman,*The discrete spectrum in gaps of the perturbed periodic Schrödinger operator. II. Nonregular perturbations*, Algebra i Analiz**9**(1997), no. 6, 62–89 (Russian, with Russian summary); English transl., St. Petersburg Math. J.**9**(1998), no. 6, 1073–1095. MR**1610239****[B3]**-,*On homogenization procedure for periodic operators near the edge of an internal gap*, Algebra i Analiz**15**(2003), no. 4, 61-71; English transl., St. Petersburg Math. J.**15**(2004), no. 4, 507-513.**[BLaSu]**M. Sh. Birman, A. Laptev, and T. A. Suslina,*The discrete spectrum of a two-dimensional second-order periodic elliptic operator perturbed by a decreasing potential. I. A semi-infinite gap*, Algebra i Analiz**12**(2000), no. 4, 36–78 (Russian, with Russian summary); English transl., St. Petersburg Math. J.**12**(2001), no. 4, 535–567. MR**1793617****[BSu1]**M. Sh. Birman and T. A. Suslina,*Two-dimensional periodic Pauli operator. The effective masses at the lower edge of the spectrum*, Mathematical results in quantum mechanics (Prague, 1998) Oper. Theory Adv. Appl., vol. 108, Birkhäuser, Basel, 1999, pp. 13–31. MR**1708785****[BSu2]**Michael Birman and Tatyana Suslina,*Threshold effects near the lower edge of the spectrum for periodic differential operators of mathematical physics*, Systems, approximation, singular integral operators, and related topics (Bordeaux, 2000) Oper. Theory Adv. Appl., vol. 129, Birkhäuser, Basel, 2001, pp. 71–107. MR**1882692****[CV]**Carlos Conca and Muthusamy Vanninathan,*Homogenization of periodic structures via Bloch decomposition*, SIAM J. Appl. Math.**57**(1997), no. 6, 1639–1659. MR**1484944**, https://doi.org/10.1137/S0036139995294743**[Ka]**Tosio Kato,*Perturbation theory for linear operators*, 2nd ed., Springer-Verlag, Berlin-New York, 1976. Grundlehren der Mathematischen Wissenschaften, Band 132. MR**0407617****[KiSi]**W. Kirsh and B. Simon,*Comparison theorems for the gap of Schrödinger operators*, J. Funct. Anal.**75**(1987), no. 2, 396-410. MR**0916759 (89b:35127)****[Ku]**Peter Kuchment,*The mathematics of photonic crystals*, Mathematical modeling in optical science, Frontiers Appl. Math., vol. 22, SIAM, Philadelphia, PA, 2001, pp. 207–272. MR**1831334****[Zh1]**V. V. Zhikov,*Spectral approach to asymptotic diffusion problems*, Differentsial'nye Uravneniya**25**(1989), no. 1, 44-50; English transl., Differential Equations**25**(1989), no. 1, 33-39. MR**0986395 (90a:35107)****[Zh2]**-,*Private communication*, 2002. (Russian)**[ZhKO]**V. V. Jikov, S. M. Kozlov, and O. A. Oleĭnik,*Homogenization of differential operators and integral functionals*, Springer-Verlag, Berlin, 1994. Translated from the Russian by G. A. Yosifian [G. A. Iosif′yan]. MR**1329546****[Sa]**E. Sanchez-Palencia,*Nonhomogeneous media and vibration theory*, Lecture Notes in Phys., vol. 127, Springer-Verlag, Berlin-New York, 1980. MR**0578345 (82j:35010)****[Se]**E. V. Sevost'yanova,*Asymptotic expansion of the solution of a second-order elliptic equation with periodic rapidly oscillating coefficients*, Mat. Sb. (N.S.)**115**(1981), no. 2, 204-222; English transl., Math. USSR-Sb.**43**(1982), no. 2, 181-198. MR**0622145 (83d:35038)****[Su]**T. A. Suslina,*On homogenization of a periodic elliptic operator in a strip*, Algebra i Analiz**16**(2004), no. 1, 269-292; English transl. in St. Petersburg Math. J.**16**(2005), no. 1 (to appear).**[Sh]**R. G. Shterenberg,*Example of a periodic magnetic Schrödinger operator with degenerate lower spectral edge*, Algebra i Analiz (to appear). (Russian)

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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