Results 1  10
of
29
THE Ξ OPERATOR AND ITS RELATION TO KREIN’S SPECTRAL SHIFT FUNCTION
, 1999
"... We explore the connections between Krein’s spectral shift function ξ(λ, H0, H) associated with the pair of selfadjoint 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 ..."
Abstract

Cited by 21 (3 self)
 Add to MetaCart
We explore the connections between Krein’s spectral shift function ξ(λ, H0, H) associated with the pair of selfadjoint 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 selfadjoint 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 BirmanKrein formula relating ξ(λ, H0, H) and the scattering operator for the pair (H0, H), and discuss an operatorvalued approach to spectral averaging, extending ideas of Birman and Solomyak.
Entropy numbers in weighted function spaces and eigenvalue distributions of some degenerate pseudodifferential operators II
, 1994
"... . 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 distri ..."
Abstract

Cited by 12 (5 self)
 Add to MetaCart
. 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 BirmanSchwinger 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 wellknown classical spaces such as (fractional) Sobolev spaces, H olderZygmund spaces, Besov spaces and (inhomogeneous) Hardy spaces. In [1...
L p spectral theory of higherorder elliptic differential operators
 Bull. London Math. Soc
, 1997
"... 2. Eigenvalues and eigenfunctions 516 2.1 Spectral asymptotics 516 2.2 The isoperimetric problem 518 ..."
Abstract

Cited by 12 (1 self)
 Add to MetaCart
2. Eigenvalues and eigenfunctions 516 2.1 Spectral asymptotics 516 2.2 The isoperimetric problem 518
Spectral shift function of the Schrödinger operator in the large coupling constant limit
, 1998
"... ..."
On the zero modes of Pauli operators
 J. Funct. Analysis
"... 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 ..."
Abstract

Cited by 9 (0 self)
 Add to MetaCart
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}
TwoDimensional Periodic Pauli OPERATOR. THE EFFECTIVE MASSES AT THE LOWER EDGE OF THE SPECTRUM
, 1998
"... We calculate the tensor of effective masses for the twodimensional 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 als ..."
Abstract

Cited by 7 (3 self)
 Add to MetaCart
We calculate the tensor of effective masses for the twodimensional 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.
The Discrete Spectrum of Selfadjoint Operators under Perturbations of Variable Sign
 COMM. PARTIAL DIFFERENTIAL EQUATIONS
, 2000
"... Given two selfadjoint operators A and V = V+ V , we study the motion of the eigenvalues of the operator A(t) = A tV as t increases. Let > 0 and let be a regular point for A. We consider the quantity N(; A; W+ ; W ; ) defined as the difference between the number of the eigenvalues of A(t) that pas ..."
Abstract

Cited by 5 (1 self)
 Add to MetaCart
Given two selfadjoint operators A and V = V+ V , we study the motion of the eigenvalues of the operator A(t) = A tV as t increases. Let > 0 and let be a regular point for A. We consider the quantity N(; A; W+ ; W ; ) defined as the difference between the number of the eigenvalues of A(t) that pass the point from right to left and the number of the eigenvalues passing from left to right as t increases from 0 to : We study the asymptotic behavior of N(; A; W+ ; W ; ) as !1: Applications to Schrödinger and Dirac operators are given.