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We study discrete spectrum in spectral gaps of an elliptic periodic second order differential operator in L 2 (ℝd ) perturbed by a decaying potential. It is assumed that a perturbation is nonnegative and has a power-like behavior at infinity. We find asymptotics in the large coupling constant limit for the number of eigenvalues of the perturbed operator that have crossed a given point inside the gap or the edge of the gap. The corresponding asymptotics is power-like and depends on the observation point.

This paper deals with the derivation of two-sided estimates for the trace of the difference of two semigroups generated by two Schrodinger operators in L2(ℝ3) with trace class difference of the resolvents. Use is made of a purely operator-theoretic technique. The results are stated in a rather general abstract form. The sharpness of our estimates is confirmed by the fact that they imply the asymptotic behavior of the trace of the difference of the semigroups as t → +0. Our considerations are substantially based on the Krein-Lifshits formula and on the Birman-Solomyak representation for the spectral shift function.

For a periodic matrix elliptic operator \( A_\varepsilon \) with (xɛ-dependent) rapidly oscillating coefficients, a certain analog of the limit absorption principle is proved. It is shown that the bordered resolvent 〈x〉−1/2−·(\( A_\varepsilon \) − (η ± iɛσ)I)−1〈x〉−1/2−· has a limit in the operator norm in L2 as ɛ → 0 provided that η > 0, · > 0, and 0 < σ < 1/2.

The homogenization problem in the small period limit for the stationary periodic Maxwell system in ℝ3 is considered. It is assumed that the permittivity ηe(x)=η(e−x), e > 0, is a rapidly oscillating positive matrix function and the permeability µ0 is a constant positive matrix. For all four physical fields (the electric and magnetic field intensities, the electric displacement field, and the magnetic flux density), we obtain uniform approximations in the L2(ℝ3)-norm with order-sharp remainder estimates.

The homogenization procedure for a multidimensional periodic Schrodinger operator near the edge of an internal gap is discussed. We obtain an approximation for the resolvent in the small period limit with respect to the operator norm in L2(ℝd). This approximation contains oscillations but in a simpler form than the resolvent of the initial operator. Bibliography: 8 titles.

Let Ω ⊂R d be an unbounded domain, periodic along a chosen direction (a waveguide-type domain),P be a self-adjoint elliptic second order operator inL 2(Ω) periodic along the same direction, andV be a real-valued decaying potential. We suppose that the bottom of the spectrum ofP is λ=0 and study the asymptotic behaviour of the number of negative eigenvalues of the opeatorP−aV as the parameter α tends to +∞. We show that typically the Weyl asymptotic law for this quantity is violated and find a substitute for this law.

We consider the periodic Dirac operatorD inL 2(ℝ d ). The magnetic potentialA and the electric potentialV are periodic. Ford=2 the absolute continuity ofD is established forA,V∈L r, loc ,r>2; the proof is based on the estimates, obtained by the authors earlier [BSu2] for the periodic magnetic Schrodinger operatorM. Ford≥3 our considerations are based on the estimates forM, obtained in [So] forA∈C 2d+3 . Under the same condition onA, forV∈C, the absolute continuity ofD, d≥3, is proved. ForA=0 the arguments of the paper give a new (and much simpler) proof of the main result of [D].

Several formula representations for the I. M. Lifshits — M. G. Krein spectral shift function (SSF) are discussed and intercompared. It is pointed out that the equivalence of these representations is not apparent, and different properties of the SSF are revealed by different formulas. The presentation is informal and contains no proofs.

We study the eigenvalue behaviour of a class of ODE acting on the semiaxis. Using interpolation of quasinormed spaces as one of our main tools, we obtain sharp estimates of the counting function of eigenvalues n(λ). In particular, some of our results become reversible if a matrix-valued “weight” -function is nonnegative. The asymptotic behaviour of n(λ) is also discussed.

The following problems are considered: the problem of the electromagnetic oscillations of a hollow resonator with an ideally conductive boundary; the steady-state Stokes equations under conditions of slippery walls; the problem of elasticity theory under the boundary condition of a rigid contact. Formally, in polyhedral domains these three problems coincide. It is shown that in a meaningful sense such a coincidence is realized only in convex polyhedra. Related effects are discussed. Bibliography: 10 titles.

Under mild assumption, integral representations of the form (*) $$f(A_1 ) \cdot \mathfrak{J} - \mathfrak{J} \cdot f(A_1 ) = \int {\int {\frac{{f(\mu ) - f(\lambda )}}{{\mu - \lambda }}} } dE_1 (\mu )(A_1 \mathfrak{J} - \mathfrak{J}A_0 )dE_0 (\mu ),$$ are justified. Here Ak, k=0, 1, is a self-adjoint operator in a Hilbert space Hk, is an operator from H0 H1; in general, all the operators are unbounded; Ek is the spectral measure of the operator Ak. On the basis of the representation (*), estimates of the s-numbers of the operator $$f(A_1 ) \cdot \mathfrak{J} - \mathfrak{J} \cdot f(A_0 )$$ in terms of the s-numbers of the operator $$A_1 \mathfrak{J} - \mathfrak{J}A_0$$ are given. Analogous results are obtained for commutators and antocommutators.

Let A0, A1 be positive selfadjoint operators in a Hilbert space, let |·| be the norm in some symmetrically normed ideal of operators, satisjying the majorization condition. In the paper one obtains estimate and also some of its generalizations. Examples, with A0, A1 differential operators, are considered.

The perturbation of a Schrodinger operator H0 with an arbitrary bounded potential function q, decreasing sufficiently fast at infinity, is considered. With the aid of results of the nuclear theory, for the corresponding pair of Hamiltonians H0, H=H0+q, one establishes the existence and the completeness of the wave operators. Generalizations are given to a wider class of unperturbed operators H0, and also to perturbations by firstorder differential operators. In addition, perturbations by integral operators of Fourier type are investigated.

The problem of the construction in a bounded domain Ω ⊂ ℝm with a Lipschitz boundary of a function Φ ∃ H2(Ω), for which the conormal derivative on ∂Ω coincides with the normal component of a given vector field u ∃ H1(Ω,C3), is discussed. The solution of this problem is given for piecewise smooth boundaries in the case m=3.

We calculate the tensor of effective masses for the two-dimensional periodic Pauli operator. The explicit representation for this tensor is given in terms of the magnetic field. It is proved that the tensor of effective masses is circular symmetric and always proportional to the unit matrix. We also consider the generalized Pauli operator with a variable metric. In the appendix we study the periodic elliptic operators of the second order and discuss the behaviour of the first band function near its minimum point.

1. We start from the traditional problem on the negative spectrum of the Schrodinger operator in R d , d ≥ 3. Let A = -Δ, $$ A(\alpha ) = - \Delta \, - \alpha V,\,\,V(x)\, \geqslant \,0,\,\,\alpha > 0, $$ (1) and ∈ ≥ 0. By N A (α,∈) we denote the number of eigenvalues of the operator (1) on the left-hand side of the point ∈. Then for potentials \(V\, \in \,L_{d/2} (R^d )\) we have the well-known asymptotics $$ N_A (\alpha ,\,\lambda ) \sim \,(2\pi )^{ - d} \omega _d \alpha ^{d/2} \,\,\int {V^{d/2} dx,\,} \,\alpha \to \infty , $$ (2) with ω d the volume of the unit ball in R d . We call potentials \(V\, \in \,L_{d/2} (R^d )\) “regular” perturbations of the operator A (cf. [BS1]). The asymptotic (2) do not depend on ∈ ≥ 0, it’s character is determined by the behavior of the symbol |ξ|2 - αV(x) for large |ξ| only. In [BS1] the asymptotics of N A (α,∈) are discussed precisely for potentials violating the assumption \(V\, \in \,L_{d/2} (R^d )\) because of a slow decrease as |x| → ∞ (“non-regular perturbations”).

1. Let us consider the periodic operator $$ {\text{A = - div(g(x)grad) + p(x)}} $$ (1) acting in L 2(R d) Assume that \( g,p \in {L_\infty }({R^d}) \) are real functions obeying $$ g(x + n) = g(x),\quad p(x + n) = p(x),\quad n \in {{\rm Z}^d} $$ and that the matrix function g is positive definite. If the matrix g is constant, then the operator A with the lattice periods Z d is equivalent to the Schrodinger operator with arbitrary lattice periods. Further, together with A the perturbed operator $$ A(\alpha ) = A + \alpha V,\quad V(x) \geqslant 0,\quad \alpha > 0, $$ (2) is considered where \( V \in {L_{\infty ,loc}} \) behaves like $$ V(x) \sim {\left| x \right|^{ - 2\sigma }}f(\vartheta ),\quad \vartheta = \frac{x}{{\left| x \right|}},\quad \sigma > 0 $$ (3) as |x| → ∈ The exact definition of the operators (1),(2) is given by quadratic forms. Let the interval \( ({\lambda _ - },{\lambda _ + }) \) be a gap of the spectrum of the operator A and let A be a fixed number in the closed gap, i.e. \( {\lambda _ - } \leqslant \lambda \leqslant {\lambda _ + } \). By N(α, λ) we denote the number of eigenvalues of the operator A(t) which go through the point ∈ if the coupling constant t increases from 0 to α.

Estimates for the number of negative eigenvalues of the Schrodinger operator and its generalizations by M. Sh. Birman and M. Z. Solomyak Discrete spectrum in the gaps of a continuous one for perturbations with large coupling constant by M. Sh. Birman Discrete spectrum in the gaps for perturbations of the magnetic Schrodinger operator by M. Sh. Birman and G. D. Raikov Boundedness conditions and spectrum estimates for the operators $b(X)a(D)$ and their analogs by M. Sh. Birman, G. E. Karadzhov, and M. Z. Solomyak Reflection operators and their applications to asymptotic investigations of semiclassical integral equations by A. M. Budylin and V. S. Buslaev Weyl asymptotics for the discrete spectrum of the perturbed Hill operator by A. V. Sobolev On solutions of the Schrodinger equation with radiation conditions at infinity by D. R. Yafaev.

ContentsIntroduction § 1. Operator spaces and function spaces § 2. Estimates of singular numbers based on the method of piecewise-polynomial approximation § 3. Interpolation methods § 4. General results on estimates (statements) § 5. Proof of the main results on estimates § 6. Operators with a difference kernel § 7. Tests for nuclearity § 8. Multipliers in classes of kernels of integral operators § 9. Analytical results for the multiplier problem § 10. Operators with a homogeneous difference kernel § 11. Some special casesAppendix. M. Z. Solomyak, The Interpolation of bilinear maps from function spaces into operator spacesReferences

A class of pseudodifferential operators with a small parameter in the amplitude in considered in . The quasiclassical asymptotics of the spectral characteristics is investigated. The results are applicable to statistical estimation theory.

CONTENTSIntroduction § 1. Classes of functions. Boundary conditions § 2. Definition of the Maxwell operator. Inclusion in the elliptic theory § 3. Bounded domains with Lipschitz boundary § 4. Domains with piecewise-smooth boundary § 5. Asymptotic properties of the discrete spectrum of the operator § 6. Comments and bibliographic notes References

One computes the principal term of the asymptotics of the spectrum of variational problems of the form B[u]/A[u] considered on the solutions of elliptic systems Lu=0 on a compact manifold with boundary. Previously, the corresponding result has been obtained by the authors under the additional condition of the existence of an elliptic boundary problem for the operator L. Now this restriction is removed.

In the problem of the oscillations of a hollow resonator with ideally conducting walls, one describes the domain of definition of the corresponding self-adjoint Maxwell operator. One discusses the case when the boundary has inward edges.

In the present paper we investigate operators in separable Hilbert space H given by integrals of the type $$ Q = \int {\int \varphi } \left( {\lambda ,\mu } \right)d{F_\mu }Td{E_\lambda }. $$ (1.1)

Let K0, K denote two separable Hilbert spaces. The set of bounded linear operators mapping K° into K we denote by R(K0, K). The sets R(K, K0), R(K0), R(K) are defined similarly. When no confusion is possible, we will write simply R. We denote similarly by S1 and Soo the sets of nuclear and completely continuous operators. I0, I denote the identity operators in K0 and K respectively. The symbol s-lim denotes the strong operator limit. If A is a densely defined operator, then D(A) denotes its domain and A* the adjoint operator. We denote by Z the real (spectral) axis.