The spectral function of an elliptic operator

Asymptotic distribution of eigenvalues of the elliptic operator system

Since the theory of spectral properties of non-self-accession differential operators on Sobolev spaces is an important field in mathematics, therefore, different techniques are used to study them. In this paper, two types of non-self-accession differential operators on Sobolev spaces are considered and their spectral properties are investigated with two different and new techniques.

Estimates for the spectral shift function of the polyharmonic operator

The Lifshits–Krein spectral shift function is considered for the pair of operators H0 = (−4)l, l > 0 and H = H0 + V in L2(R), d ≥ 1; here V is a multiplication operator. The estimates for this spectral shift function ξ(λ;H,H0) are obtained in terms of the spectral parameter λ > 0 and the integral norms of V . These estimates are in a good agreement with the ones predicted by the classical phase...

Spectral approach to homogenization of an elliptic operator periodic in some directions

The operator Aε = D1g1(x1/ε, x2)D1 + D2g2(x1/ε, x2)D2 is considered in L2(R ), where gj(x1, x2), j = 1, 2, are periodic in x1 with period 1, bounded and positive definite. Let function Q(x1, x2) be bounded, positive definite and periodic in x1 with period 1. Let Q(x1, x2) = Q(x1/ε, x2). The behavior of the operator (Aε + Q ) as ε → 0 is studied. It is proved that the operator (Aε +Q ) tends to ...

Thermoacoustic Tomography with an Arbitrary Elliptic Operator

Thermoacoustic tomography is a term for the inverse problem of determining of one of initial conditions of a hyperbolic equation from boundary measurements. In the past publications both stability estimates and convergent numerical methods for this problem were obtained only under some restrictive conditions imposed on the principal part of the elliptic operator. In this paper logarithmic stabi...

On the Spectral Properties of Degenerate Non-selfadjoint Elliptic systems of Differential Operators