Spectral Shift Function Research Articles

Two-sided estimates for the trace of the difference of two semigroups

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.

Operator Theoretic Methods for the Eigenvalue Counting Function in Spectral Gaps

Using the notion of spectral flow, we suggest a simple approach to various asymptotic problems involving eigenvalues in the gaps of the essential spectrum of self-adjoint operators. Our approach uses some elements of the spectral shift function theory. Using this approach, we provide generalisations and streamlined proofs of two results in this area already existing in the literature. We also give a new proof of the generalised Birman-Schwinger principle.

Operator Integrals, Spectral Shift, and Spectral Flow

Abstract. We present a new and simple approach to the theory of multiple operator integrals that applies to unbounded operators affiliated with general von Neumann algebras. For semifinite von Neumann algebras we give applications to the Fréchet differentiation of operator functions that sharpen existing results, and establish the Birman–Solomyak representation of the spectral shift function of M.G. Krein in terms of an average of spectral measures in the type II setting. We also exhibit a surprising connection between the spectral shift function and spectral flow.

Resonances and spectral shift function for a magnetic Schrödinger operator

We consider the three-dimensional Schrödinger operator H0 with a constant magnetic field and subject to an electric potential v0 depending only on the variable along the magnetic field x3. The operator H0 has infinitely many eigenvalues of infinite multiplicity embedded in its continuous spectrum. We perturb H0 by smooth scalar potentials V=O(⟨(x1,x2)⟩−δ⊥⟨x3⟩−δ∥), δ⊥>2, δ∥>1. We assume also that V and v0 have an analytic continuation, in the magnetic field direction, in a complex sector outside a compact set. We define the resonances of H=H0+V as the eigenvalues of the nonself-adjoint operator obtained from H by analytic distortions of Rx3. We study their distribution near any fixed real eigenvalue of H0, 2bq+λ for q∊N. In a ring centered at 2bq+λ with radii (r,2r), we establish an upper bound, as r tends to 0, of the number of resonances. This upper bound depends on the decay of V at infinity only in the directions (x1,x2). Finally, we deduce a representation of the derivative of the spectral shift function for the operator pair (H0,H) in terms of resonances. This representation justifies the Breit–Wigner approximation and implies a local trace formula.

Higher order spectral shift

We construct higher order spectral shift functions, extending the perturbation theory results of M.G. Krein [M.G. Krein, On a trace formula in perturbation theory, Mat. Sb. 33 (1953) 597–626 (in Russian)] and L.S. Koplienko [L.S. Koplienko, Trace formula for perturbations of nonnuclear type, Sibirsk. Mat. Zh. 25 (1984) 62–71 (in Russian); translation in: Trace formula for nontrace-class perturbations, Siberian Math. J. 25 (1984) 735–743] on representations for the remainders of the first and second order Taylor-type approximations of operator functions. The higher order spectral shift functions represent the remainders of higher order Taylor-type approximations; they can be expressed recursively via the lower order (in particular, Krein's and Koplienko's) ones. We also obtain higher order spectral averaging formulas generalizing the Birman–Solomyak spectral averaging formula. The results are obtained in the semi-finite von Neumann algebra setting, with the perturbation taken in the Hilbert–Schmidt class of the algebra.

Trace inequalities and spectral shift

We derive monotonicity and convexity inequalities for traces of operator functions defined on self-adjoint elements of a semi-finite von Neumann algebra. Among tools involved in the proofs are a generalized Birman-Solomyak spectral averaging formula (obtained in the paper), a generalized Birman-Schwinger principle, and Koplienko’s spectral shift function, a new, more straightforward, approach to which is developed in the paper. Mathematics subject classification (2000): Primary 26A48, 26A51, 47A56; Secondary 47A55, 47C15.

On Properties of the ξ-Function in Semi-finite von Neumann Algebras

Versions of the Birman-Schwinger principle for (relative) trace class perturbation problems of dissipative operators in a semi-finite von Neumann algebra and self-adjoint operators affiliated with the algebra are obtained and applied in the study of the spectral shift function.

Relative Oscillation Theory, Weighted Zeros of the Wronskian, and the Spectral Shift Function

We develop an analog of classical oscillation theory for Sturm-Liouville operators which, rather than measuring the spectrum of one single operator, measures the difference between the spectra of two different operators. This is done by replacing zeros of solutions of one operator by weighted zeros of Wronskians of solutions of two different operators. In particular, we show that a Sturm-type comparison theorem still holds in this situation and demonstrate how this can be used to investigate the finiteness of eigenvalues in essential spectral gaps. Furthermore, the connection with Krein's spectral shift function is established.

Scattering matrices and Weyl functions

For a scattering system {AΘ, A0} consisting of self-adjoint extensions AΘ and A0 of a symmetric operator A with finite deficiency indices, the scattering matrix {SΘ(λ)} and a spectral shift function ξΘ are calculated in terms of the Weyl function associated with a boundary triplet for A*, and a simple proof of the Krein–Birman formula is given. The results are applied to singular Sturm–Liouville operators with scalar and matrix potentials, to Dirac operators and to Schrödinger operators with point interactions.

Spectral averaging for trace compatible operators

In this note the notions of trace compatible operators and infin- itesimal spectral flow are introduced. We define the spectral shift function as the integral of infinitesimal spectral flow. It is proved that the spectral shift function thus defined is absolutely continuous and Krein's formula is es- tablished. Some examples of trace compatible affine spaces ofoperators are given.

Relative oscillation theory for Sturm–Liouville operators extended

We extend relative oscillation theory to the case of Sturm–Liouville operators H u = r −1 ( − ( p u ′ ) ′ + q u ) with different p's. We show that the weighted number of zeros of Wronskians of certain solutions equals the value of Krein's spectral shift function inside essential spectral gaps.

RESONANCES AND SPECTRAL SHIFT FUNCTION FOR THE SEMI-CLASSICAL DIRAC OPERATOR

We consider the selfadjoint operator H = H0+ V, where H0is the free semi-classical Dirac operator on ℝ3. We suppose that the smooth matrix-valued potential V = O(〈x〉-δ), δ > 0, has an analytic continuation in a complex sector outside a compact. We define the resonances as the eigenvalues of the non-selfadjoint operator obtained from the Dirac operator H by complex distortions of ℝ3. We establish an upper bound O(h-3) for the number of resonances in any compact domain. For δ > 3, a representation of the derivative of the spectral shift function ξ(λ,h) related to the semi-classical resonances of H and a local trace formula are obtained. In particular, if V is an electro-magnetic potential, we deduce a Weyl-type asymptotics of the spectral shift function. As a by-product, we obtain an upper bound O(h-2) for the number of resonances close to non-critical energy levels in domains of width h and a Breit–Wigner approximation formula for the derivative of the spectral shift function.

The Spectral Shift Function and Spectral Flow

At the 1974 International Congress, I. M. Singer proposed that eta invariants and hence spectral flow should be thought of as the integral of a one form. In the intervening years this idea has lead to many interesting developments in the study of both eta invariants and spectral flow. Using ideas of [24] Singer’s proposal was brought to an advanced level in [16] where a very general formula for spectral flow as the integral of a one form was produced in the framework of noncommutative geometry. This formula can be used for computing spectral flow in a general semifinite von Neumann algebra as described and reviewed in [5]. In the present paper we take the analytic approach to spectral flow much further by giving a large family of formulae for spectral flow between a pair of unbounded self-adjoint operators D and D + V with D having compact resolvent belonging to a general semifinite von Neumann algebra $${\mathcal{N}}$$ and the perturbation $$V \in {\mathcal{N}}$$ . In noncommutative geometry terms we remove summability hypotheses. This level of generality is made possible by introducing a new idea from [3]. There it was observed that M. G. Krein’s spectral shift function (in certain restricted cases with V trace class) computes spectral flow. The present paper extends Krein’s theory to the setting of semifinite spectral triples where D has compact resolvent belonging to $${\mathcal{N}}$$ and V is any bounded self-adjoint operator in $${\mathcal{N}}$$ . We give a definition of the spectral shift function under these hypotheses and show that it computes spectral flow. This is made possible by the understanding discovered in the present paper of the interplay between spectral shift function theory and the analytic theory of spectral flow. It is this interplay that enables us to take Singer’s idea much further to create a large class of one forms whose integrals calculate spectral flow. These advances depend critically on a new approach to the calculus of functions of non-commuting operators discovered in [3] which generalizes the double operator integral formalism of [8–10]. One surprising conclusion that follows from our results is that the Krein spectral shift function is computed, in certain circumstances, by the Atiyah-Patodi-Singer index theorem [2].

Weak asymptotics of the spectral shift function

AbstractWe consider the three‐dimensional Schrödinger operator with constant magnetic field of strength b > 0, and with smooth electric potential. The weak asymptotics of the spectral shift function with respect to b ↗ +∞ is studied. First, we fix the distance to the Landau levels, then the distance to Landau levels tends to infinity as b ↗ +∞. In particular we give explicitly the leading terms in the asymptotics and in some case we obtain full asymptotics expansions. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

The Schrödinger operator: Perturbation determinants, the spectral shift function, trace identities, and all that

We discuss applications of the M. G. Krein theory of the spectral shift function to the multidimensional Schrodinger operator. Specific properties of this function, for example, its high-energy asymptotics are studied. Trace identities are derived.

Resonances and Spectral Shift Function near the Landau levels

On etudie l'operateur de Schrodinger magnetique en dimension 3, H = H o + V ou Ho (-iV - A) 2 -b, A est un potentiel magnetique generant un champ magnetique constant de force b > 0 fixee et V est un potentiel electrique qui decroit super-exponentiellernent; dans la direction du champ magnetique. On montre que la resolvante de H admet un prolongement meromorphe du plan superieur une certaine surface de Riemann M et on definit les resonances de H comme les poles de cette extension ineromorphe. On etudie leur repartition pres d'un niveau de Landau fixe 2bq, q ∈ N. On obtient d'abord des majorations du nombre de resonances dans des petits domaines proches de 2bq. Sous des hypothses supplementaires, on prouve des minorations du nombre de resonances qui implique la presence d'une infinite de resonances ou bien l'absence de resonances dans certains secteurs de sommet 2bq. Finalement, on montre que la fonction de decalage spectral (FDS) associee a la paire (H, H 0 ) est la somme de mesures harmoniques associees aux resonances et. de la. partie imaginaire d'une fonction holomorphe. Cette formule justifie l'approximation de Breit-Wigner, implique une formule de trace a la Sjostrand et donne des informations sur les singularites de la FDS aux niveaux de Landau.

Transmission through quantum dots: Focus on phase lapses

Measurements of the transmission phase in transport through a quantum dot embedded in an Aharonov-Bohm interferometer show systematic sequences of phase lapses separated by Coulomb peaks. Using a two-level quantum dot as an example we show that this phenomenon can be accounted for by the combined effect of asymmetric dot-lead coupling and interaction-induced ``population switching'' of the levels, rendering this behavior generic. In addition, we use the notion of spectral shift function to analyze the relationship between transmission phase lapses and the Friedel sum rule.

Vibrational spectrum of disordered linear chain with light isotopic impurities. Region of the initial band

The vibrational spectrum of disordered linear chains with light impurities of substitution is calculated within the region of the unperturbed band. The overall structure of the spectrum and the spectral shift function are determined. The results of a numerical calculation are compared with the known analytical expressions. The character of the statistical distribution of vibrational frequencies over different parts of the spectrum is investigated, and it is shown that at a sufficient impurity concentration in not-too-wide spectral intervals a completely random distribution of eigenfrequencies is realized. It is found that, with increasing concentration, sequences of minima and maxima arise with increasing clarity in the high-frequency half of the band, their positions being independent of the concentration and clumped towards the outer edge of the band. An expression giving the values of these special frequencies is found.

The Krein Spectral Shift Function in Semifinite von Neumann Algebras

We show the existence of a spectral shift function in the sense of Krein for bounded trace class perturbations of a self-adjoint operator affiliated with a semifinite von Neumann algebra

Spectral Shift Function in the Large Coupling Constant Limit

In the large-coupling constant limit we obtain an asymptotic expansion in powers of \( \mu ^{{ - \frac{1} {\delta }}} \) of the derivative of the spectral shift function corresponding to \( {\left( { - \Delta + \mu W(x), - \Delta } \right)} \). Here the potential W(x) is positive and \( W(x) \sim \omega _{0} (\frac{x} {{|x|}})|x|^{{ - \delta }} \) near infinity for some δ> n and ω0 ∈C∞ (Sn-1).