this paper is a proxy for what deserves a book or at least a very long review article. In attempting to overview such a vast area in a few pages, I have had to focus on the high points. No proofs are given and I have settled for usually quoting the initial or especially significant papers. I have no doubt that I have left out some important papers, and if so, I ask the forgiveness of the reader (and their authors!).

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2. Eigenvalues and eigenfunctions 516 2.1 Spectral asymptotics 516 2.2 The isoperimetric problem 518

. This paper is the continuation of [17]. We investigate mapping and spectral properties of pseudodifferential operators of type \Psi 1;fl with 2 IR and 0 fl 1 in the weighted function spaces B s p;q (IR n ; w(x)) and F s p;q (IR n ; w(x)) treated in [17]. Furthermore we study the distribution of eigenvalues and the behaviour of corresponding root spaces for degenerate pseudodifferential operators preferably of type b2 (x)b(x; D)b1(x), where b1 (x) and b2(x) are appropriate functions and b(x; D) 2 \Psi 1;fl . Finally, on the basis of the Birman-Schwinger principle, we deal with the "negative spectrum" (bound states) of related symmetric operators in L2 . Math. Subject Classification: 46E35, 47G30, 35S05 1. Introduction The spaces B s p;q and F s p;q with s 2 IR; 0 ! p 1 (p ! 1 for the F - spaces) and 0 ! q 1 on IR n cover many well-known classical spaces such as (fractional) Sobolev spaces, H older-Zygmund spaces, Besov spaces and (inhomogeneous) Hardy spaces. In [1...

Abstract. We explore the connections between Krein’s spectral shift function ξ(λ, H0, H) associated with the pair of self-adjoint operators (H0, H), H = H0+V in the Hilbert space H and the recently introduced concept of a spectral shift operator Ξ(J+K ∗ (H0−λ−i0) −1 K) associated with the Herglotz operator J + K ∗ (H0 − z) −1 K, Im(z)> 0 in H, where V = KJK ∗ and J = sgn(V). Our principal results include a new representation for ξ(λ, H0, H) in terms of an averaged index for the Fredholm pair of self-adjoint spectral projections (E J+A(λ)+tB(λ)((−∞, 0)), EJ((−∞, 0))), t ∈ R, where A(λ) = Re(K ∗ (H0 − λ −i0) −1 K), B(λ) = Im(K ∗ (H0 −λ−i0) −1 K) a.e. Moreover, introducing the new concept of a trindex for a pair of operators (A, P) in H, where A is bounded and P is an orthogonal projection, we prove that ξ(λ, H0, H) coincides with the trindex associated with the pair (Ξ(J + K ∗ (H0 − λ − i0) −1 K),Ξ(J)). We also provide an abstract version of the Birman-Krein formula relating ξ(λ, H0, H) and the scattering operator for the pair (H0, H), and discuss an operator-valued approach to spectral averaging, extending ideas of Birman and Solomyak. 1.

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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 variable metric. In Appendix we study the periodic elliptic operators of the second order and discuss the behavior of the first band function near its minimum point.

Two results are proved for nul PA, the dimension of the kernel of the Pauli 1 operator PA = σ · i ∇ +)} 2 ⃗A in [L2(R3)] 2: (i) for | ⃗B | ∈ L3/2 (R3), where ⃗B = curl ⃗ A is the magnetic field, nul PtA = { 0 except for a finite number of values of t in any compact subset of (0, ∞); (ii) ⃗B: nul PA = 0, | ⃗ B | ∈ L3/2 (R3}

. Let -- i (H l (V )) denote the negative eigenvalues of the operator H l (V )u := (\Gamma\Delta) l u \Gamma V (x)u; V 0; x 2 R d on L 2 (R d ): We prove the two-sided estimate ~ L(d; l) Z R d V (x)dx X k j-- k (H l (V ))j 1\Gamma L(d; l; 1 \Gamma ) Z R d V (x)dx; = d=2l ! 1: We discuss bounds on the Riesz means P k j-- k (H l (V ))j ¯ if 0 ! ¯ ! 1 \Gamma : 1. Introduction 1.1. We consider the quadratic form h l (V )[u; u] := Z R d jr l uj 2 dx \Gamma Z R d V juj 2 dx; 0 V 2 L loc 1 (R d ); l 2 N + ; defined on functions u 2 C 1 0 (R d ): If the function V vanishes properly at infinity, this form can be closed. Its closure generates the self-adjoint operator H l (V ) := (\Gamma\Delta) l \Gamma V (x) (1) on L 2 (R d ); the negative spectrum of which is discrete and bounded from below. Let f-- k (H l (V ))g stand for the non-decreasing, finite or infinite sequence of the negative eigenvalues of the operator H l (V ): Estimates on the...