Spectral asymptotics of periodic elliptic operators

We demonstrate that the structure of complex second-order strongly elliptic operators H on with coefficients invariant under translation by can be analyzed through decomposition in terms of versions , , of H with z-periodic boundary conditions acting on where . If the s emigroup S generated by H has a H?lder continuous integral kernel satisfying Gaussian bounds then the semigroups generated by the have kernels with similar properties and extends to a function on which is analytic with respect to the trace norm. The sequence of semigroups obtained by rescaling the coefficients of by converges in trace norm to the semigroup generated by the homogenization of . These convergence properties allow asymptotic analysis of the spectrum of H. Received September 1, 1998; in final form January 14, 1999     

Spectral asymptotics of elliptic strongly degenerate operators of arbitrary order

We compute the principal term of the spectral asymptotics for elliptic operators of an arbitrary order which is degenerate at the boundary of the domain. The degree of the degeneracy is such that the order of the decrease of the eigenvalues of the boundary-value problems is different from the classical one and the asymptotic coefficient depends on the form of the boundary conditions (strong degeneracy).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 59, pp. 25–30, 1976.In conclusion, the author expresses his deep gratitude to M. Z. Solomyak for his guidance in the preparation of this paper.     

Spectral asymptotics of differential and pseudodifferential operators. II

In the article we obtain asymptotic formulae with remainder estimates for the distribution function of the eigenvalues of degenerate elliptic differential operators and Schrödinger operators with singular potential.Translated from Trudy Seminara imeni I. G. Petrovskogo, Vol. 10, pp. 78–106, 1984.     

An L2 model for selfadjoint elliptic differential operators with constant coefficients on bounded domains

The selfadjoint realization of a second order elliptic differential expression with Dirichlet boundary conditions is shown to be unitarily equivalent to the maximal multiplication operator with the independent variable in an explicit L² model space. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Semiclassical analysis of general second order elliptic operators on bounded domains

In this work we ascertain the semiclassical behavior of the fundamental energy and the ground state of an arbitrary second order elliptic operator, not necessarily selfadjoint, on a bounded domain. Our analysis provides us with substantial improvements of many previous results found in the context of quantum mechanics for perturbations of the Laplacian.

     

Spectral asymptotics and trace formulas for differential operators with unbounded coefficients

In the space L ²[0, π], we consider the operators $$ L = L_0 + V, L_0 = - y'' + (\nu ^2 - 1/4)r^{ - 2} y (\nu \geqslant 1/2) $$ with the Dirichlet boundary conditions. The potential V is the operator of multiplication by a function (in general, complex-valued) in L ²[0, π] satisfying the condition $$ \int\limits_0^\pi {r^\varepsilon } (\pi - r)^\varepsilon |V(r)|dr < \infty , \varepsilon \in [0,1] $$ . We prove the trace formula Σ _n=1 ^∞ [µ _n ? λ _n ? Σ _k=1 ^m α _k ⁽ⁿ⁾ ] = 0.     

L p-Theory of elliptic differential operators on manifolds of bounded geometry

This paper is devoted to some of the properties of uniformly elliptic differential operators with bounded coefficients on manifolds of bounded geometry in L ^pspaces. We prove the coincidence of minimal and maximal extensions of an operator of a considered type with a positive principal symbol, the existence of holomorphic semigroup, generated by it, and the estimates of L ^p-norms of the operators of this semigroup. Some spectral properties of such operators in L ^pspaces are also studied.     

On homogenization for non-self-adjoint locally periodic elliptic operators

In this note we consider the homogenization problem for a matrix locally periodic elliptic operator on R ^d of the form A ^ε = ?divA(x, x/ε)?. The function A is assumed to be Hölder continuous with exponent s ∈ [0, 1] in the “slow” variable and bounded in the “fast” variable. We construct approximations for (A ^ε ? μ)^?1, including one with a corrector, and for (?Δ) ^s/2(A ^ε ? μ)^?1 in the operator norm on L ₂(R ^d ) ⁿ . For s ≠ 0, we also give estimates of the rates of approximation.     

Asymptotic behavior of the spectra of second-order non-self-adjoint systems of differential operators

Translated from Matematicheskie Zametki, Vol. 51, No. 4, pp. 8–16, April, 1992.