Abstract
In L 2 \(\left( {{\mathbb{R}^d}} \right), \) we consider vector periodic DO A admitting a factorization A = X*X, where X is a homogeneous DO of first order. Many operators of mathematical physics have this form. The effects that depend only on a rough behavior of the spectral decomposition of A in a small neighborhood of zero are called threshold effects at λ = O. An example of a threshold effect is the behavior of a DO in the small period limit Another example is related to the negative discrete spectrum of the operator A- α V, α> 0, where V(x) ≥ 0 and V(x) → 0 as |x|→ ∞. The “effective characteristics”, namely, the homogenized medium, the effective mass, the effective Hamiltonian, etc. arise in these problems. We propose a general approach to these problems based on the spectral perturbation theory for operator-valued functions admitting analytic factorization. A great deal of considerations is done in abstract terms.
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References
G. Allaire and C. Conca, Bloch wave homogenization and spectral asymptotic analysis, J. Math. Pures Appl., 77 (1998), 153–208.
N. S. Bakhvalov and G. P. Panasenko, Homogenization: averaging processes in periodic media. Mathematical problems in the mechanics of composite materials, (Russian), Nauka, Moscow, 1984, 352 pp; English transi.: Mathematics and its Applications (Soviet Series), 36, Kluwer Academic Publishers Group, Dordrecht, 1989, 366 pp.
A. Bensoussan, J. L. Lions and G. Papanicolaou, Asymptotic analysis for periodic structures, North Holland Publishing Company, Amsterdam, 1978, 700 pp.
M. Sh. Birman, The discrete spectrum of the periodic Schrödinger operator perturbed by a decreasing potential, (Russian) Algebra i Analiz, 8 (1996), no. 1, 3–20; English transi.: St. Petersburg Math. J., 8 (1997), no. 1, 1–14.
M. Sh. Birman, Discrete spectrum in the gaps of the perturbed periodic Schrödinger operator. II. Non-regular perturbations, (Russian) Algebra i Analiz, 9 (1997), no. 6, 62–89; English transl.: St. Petersburg Math. J., 9 (1998), no. 6, 1073–1095.
M. Sh. Birman, A. Laptev and T. A. Suslina, Discrete spectrum of the two-dimensional periodic elliptic second order operator perturbed by a decaying potential.I. Semibounded gap, (Russian), Algebra i Analiz, 12 (2000), no. 4, 36–78; English transl.: St. Petersburg Math. J., 12 (2001), no. 4.
M. Sh. Birman and T. A. Suslina, Absolute continuity of the two-dimensional periodic magnetic Hamiltonian with discontinuous vector-valued potential, (Russian) Algebra i Analiz, 10 (1998), no. 4, 1–36; English transi.: St. Petersburg Math. J., 10 (1999), no. 4, 579–601.
M. Sh. Birman and T. A. Suslina, Two-dimensional periodic Pauli operator. The effective masses at the lower edge of the spectrum, Oper. Theory Adv. Appl., 108 (1999), Birkhäuser, Basel, 13–31.
C. Conca and M. Vanninathan, Homogenization of periodic structures via Bloch decomposition, SIAM J. Appl. Math., 57 (1997), no. 6, 1639–1659.
V. Ivrii, Accurate spectral asymptotics for periodic operators, Journées “Equations aux derivées partielles”, Saint-Jean-de-Monts (1999), 1–11.
V. V. Jikov, S. M. Kozlov and O. A. Oleinik, Homogenization of differential operators and integral functionals, (Russian), Fizmatgiz, Moscow, 1993, 464 pp.; English transl.: Springer-Verlag, Berlin, 1994, 570 pp.
W. Kirsh and B. Simon, Comparison theorems for the gap of Schrödinger operators,J. Funct. Anal., 75 (1987), no. 2, 396–410.
E. V. Sevost’yanova, Asymptotic expansion of the solution of a second-order elliptic equation with periodic rapidly oscillating coefficients, (Russian) Mat. Sbornik, 115 (1981), 204–222; English transl.: Math. USSR-Sb., 43 (1982), no. 2, 181–198.
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Birman, M., Suslina, T. (2001). Threshold Effects near the Lower Edge of the Spectrum for Periodic Differential Operators of Mathematical Physics. In: Borichev, A.A., Nikolski, N.K. (eds) Systems, Approximation, Singular Integral Operators, and Related Topics. Operator Theory: Advances and Applications, vol 129. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8362-7_4
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DOI: https://doi.org/10.1007/978-3-0348-8362-7_4
Publisher Name: Birkhäuser, Basel
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