Krein's Spectral Shift Function Research Articles

A relative trace formula for obstacle scattering

We consider the case of scattering by several obstacles in Rd for d≥2. In this setting, the absolutely continuous part of the Laplace operator Δ with Dirichlet boundary conditions and the free Laplace operator Δ0 are unitarily equivalent. For suitable functions that decay sufficiently fast, we have that the difference g(Δ)−g(Δ0) is a trace-class operator and its trace is described by the Krein spectral shift function. In this article, we study the contribution to the trace (and hence the Krein spectral shift function) that arises from assembling several obstacles relative to a setting where the obstacles are completely separated. In the case of two obstacles, we consider the Laplace operators Δ1 and Δ2 obtained by imposing Dirichlet boundary conditions only on one of the objects. Our main result in this case states that then g(Δ)−g(Δ1)−g(Δ2)+g(Δ0) is a trace-class operator for a much larger class of functions (including functions of polynomial growth) and that this trace may still be computed by a modification of the Birman–Krein formula. In case g(x)=x1 2, the relative trace has a physical meaning as the vacuum energy of the massless scalar field and is expressible as an integral involving boundary layer operators. Such integrals have been derived in the physics literature using nonrigorous path integral derivations and our formula provides both a rigorous justification as well as a generalization.

$$L_p-$$Bounds for the Krein Spectral Shift Function: $$0<p<\infty$$

We extend the inequalities originally obtained by D. Hundertmark and B. Simon for $$L_p-$$ bounds, $$1\leq p\leq \infty$$ , for the Krein spectral shift function to the setting of general semifinite von Neumann algebras. We also complete these results by showing that in the quasi-normed setting, for example, for $$L^p$$ -spaces with $$0<p\le 1$$ , a converse inequality holds.

Threshold Singularities of the Spectral Shift Function for Geometric Perturbations of Magnetic Hamiltonians

We consider the 3D Schr\"odinger operator $H_0$ with constant magnetic field $B$ of scalar intensity $b>0$, and its perturbations $H_+$ (resp., $H_-$) obtained by imposing Dirichlet (resp., Neumann) conditions on the boundary of the bounded domain $\Omega_{\rm in} \subset {\mathbb R}^3$. We introduce the Krein spectral shift functions $\xi(E;H_\pm,H_0)$, $E \geq 0$, for the operator pairs $(H_\pm,H_0)$, and study their singularities at the Landau levels $\Lambda_q : = b(2q+1)$, $q \in {\mathbb Z}_+$, which play the role of thresholds in the spectrum of $H_0$. We show that $\xi(E;H_+,H_0)$ remains bounded as $E \uparrow \Lambda_q$, $q \in {\mathbb Z}_+$ being fixed, and obtain three asymptotic terms of $\xi(E;H_-,H_0)$ as $E \uparrow \Lambda_q$, and of $\xi(E;H_\pm,H_0)$ as $E \downarrow \Lambda_q$. The first two terms are independent of the perturbation while the third one involves the {\em logarithmic capacity} of the projection of $\Omega_{\rm in}$ onto the plane perpendicular to $B$.

Trace formulas for a class of non-Fredholm operators: A review

Take a one-parameter family of self-adjoint Fredholm operators [Formula: see text] on a Hilbert space [Formula: see text], joining endpoints [Formula: see text]. There is a long history of work on the question of whether the spectral flow along this path is given by the index of the operator [Formula: see text] acting in [Formula: see text], where [Formula: see text] denotes the multiplication operator [Formula: see text] for [Formula: see text]. Most results are about the case where the operators [Formula: see text] have compact resolvent. In this article, we review what is known when these operators have some essential spectrum and describe some new results. Using the operators [Formula: see text], [Formula: see text], an abstract trace formula for Fredholm operators with essential spectrum was proved in [23], extending a result of Pushnitski [35], although, still under strong hypotheses on [Formula: see text]: [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text]. Associated to the pairs [Formula: see text] and [Formula: see text] are Krein spectral shift functions [Formula: see text] and [Formula: see text], respectively. From the trace formula, it was shown that there is a second, Pushnitski-type, formula: [Formula: see text] This can be employed to establish the desired equality, [Formula: see text] This equality was generalized to non-Fredholm operators in [14] in the form [Formula: see text] replacing the Fredholm index on the left-hand side by the Witten index of [Formula: see text] and [Formula: see text] on the right-hand side by an appropriate arithmetic mean (assuming [Formula: see text] is a right and left Lebesgue point for [Formula: see text] denoted by [Formula: see text] and [Formula: see text], respectively). But this applies only under the restrictive assumption that the endpoint [Formula: see text] is a relatively trace class perturbation of [Formula: see text] (ruling out general differential operators). In addition to reviewing this previous work, we describe in this article some extensions using a [Formula: see text]-dimensional setup, where [Formula: see text] are non-Fredholm differential operators. By a careful analysis we prove, for a class of examples, that the preceding trace formula still holds in this more general situation. Then we prove that the Pushnitski-type formula for spectral shift functions also holds and this then gives the equality of spectral shift functions in the form [Formula: see text] for the [Formula: see text]-dimensional model operator at hand. This shows that neither the relatively trace class perturbation assumption nor the Fredholm assumption are required if one works with spectral shift functions. The results support the view that the spectral shift function should be a replacement for the spectral flow in certain non-Fredholm situations and also point the way to the study of higher-dimensional cases. We discuss the connection with summability questions in Fredholm modules in an appendix.

Singular Bilinear Integrals in Quantum Physics

Bilinear integrals of operator-valued functions with respect to spectral measures and integrals of scalar functions with respect to the product of two spectral measures arise in many problems in scattering theory and spectral analysis. Unfortunately, the theory of bilinear integration with respect to a vector measure originating from the work of Bartle cannot be applied due to the singular variational properties of spectral measures. In this work, it is shown how ``decoupled'' bilinear integration may be used to find solutions \(X\) of operator equations \(AX-XB=Y\) with respect to the spectral measure of \(A\) and to apply such representations to the spectral decomposition of block operator matrices. A new proof is given of Peller's characterisation of the space \(L^1((P\otimes Q)_{\mathcal L(\mathcal H)})\) of double operator integrable functions for spectral measures \(P\), \(Q\) acting in a Hilbert space \(\mathcal H\) and applied to the representation of the trace of \(\int_{\Lambda\times\Lambda}\varphi\,d(PTP)\) for a trace class operator \(T\). The method of double operator integrals due to Birman and Solomyak is used to obtain an elementary proof of the existence of Krein's spectral shift function.

The spectral shift function for planar obstacle scattering at low energy

AbstractLet H signify the free non‐negative Laplacian on \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbb {R}^2$\end{document} and HY the non‐negative Dirichlet Laplacian on the complement Y of a nonpolar compact subset K in the plane. We derive the low‐energy expansion for the Krein spectral shift function (scattering phase) for the obstacle scattering system \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\left\lbrace \,H_Y\hspace*{-3.0pt},\,H\right\rbrace$\end{document} including detailed expressions for the first three coefficients. We use this to investigate the large time behaviour of the expected volume of the pinned Wiener sausage associated to K.

Spectral shift functions in the fixed energy inverse scattering

In this paper the notion of the Krein spectral shift function is extended to the radial Schrodinger operator with fixed energy. Then we analyze the connections between the tail of the potential and the decay rate of the fixed-energy phase shifts. Finally we extend former results on the uniqueness of the fixed-energy inverse scattering problem to a general class of potentials.

Large time volume of the pinned Wiener sausage to second order

AbstractWe investigate the large time behaviour of the expected volume of the pinned Wiener sausage associated to a compact subset K in \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$ {\mathbb R}^d $\end{document} for d ⩾ 3. The structure of the asymptotic series is known: it depends strongly on whether the dimension d is odd or even; and the leading coefficient is given by the Newtonian capacity of K. In this article, we obtain detailed expressions for the second order coefficients. It is noteworthy that these coefficients feature novel potential‐theoretic terms that do not appear in the unpinned case. The proof exploits a trace formula of Krein. In the course of the argument, we derive a low‐energy expansion to second order for the Krein spectral shift function for an obstacle scattering system. © 2011 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim

Relative oscillation theory for Dirac operators

We develop relative oscillation theory for one-dimensional Dirac 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 number of eigenvalues in essential spectral gaps. Furthermore, the connection with Krein's spectral shift function is established. As an application we extend a result by K.M. Schmidt on the finiteness/infiniteness of the number of eigenvalues in essential spectral gaps of perturbed periodic Dirac operators.

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.

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.

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].

Spectral Shift Function in the Large Coupling Constant Limit

Given two selfadjoint operators H0 and V=V+−V−, we study the spectrum of the operator H(α)=H0+αV, α>0. We consider the quantity ξ(λ, H(α), H0), λ∈R, which coincides with Krein's spectral shift function of the pair H(α), H0 if V is of trace class and study its asymptotic behavior as α→∞. Applications to differential operators are given.

The Ξ operator and its relation to Krein's spectral shift function

We explore connections between Krein's spectral shift function ζ(λ,H 0, H) associated with the pair of self-adjoint operators (H 0, H),H=H 0+V, in a Hilbert spaceH and the recently introduced concept of a spectral shift operator Ξ(J+K *(H 0−λ−i0)−1 K) associated with the operator-valued Herglotz functionJ+K *(H 0−z)−1 K, Im(z)>0 inH, whereV=KJK * andJ=sgn(V). Our principal results include a new representation for ζ(λ,H 0,H) in terms of an averaged index for the Fredholm pair of self-adjoint spectral projections (E J+A(λ)+tB(λ)(−∞, 0)),E J((−∞, 0))), ℝ, whereA(λ)=Re(K *(H 0−λ−i0−1 K),B(λ)=Im(K *(H 0−λ-i0)−1 K) a.e. Moreover, introducing the new concept of a trindex for a pair of operators (A, P) inH, whereA is bounded andP is an orthogonal projection, we prove that ζ(λ,H 0, H) coincides with the trindex associated with the pair (Ξ(J+K *(H 0−λ−i0)K), Ξ(J)). In addition, we discuss a variant of the Birman-Krein formula relating the trindex of a pair of Ξ operators and the Fredholm determinant of the abstract scattering matrix. We also provide a generalization of the classical Birman—Schwinger principle, replacing the traditional eigenvalue counting functions by appropriate spectral shift functions.

THE DENSITY OF STATES AND THE SPECTRAL SHIFT DENSITY OF RANDOM SCHRÖDINGER OPERATORS

In this article we continue our analysis of Schrödinger operators with a random potential using scattering theory. In particular the theory of Krein's spectral shift function leads to an alternative construction of the density of states in arbitrary dimensions. For arbitrary dimension we show existence of the spectral shift density, which is defined as the bulk limit of the spectral shift function per unit interaction volume. This density equals the difference of the density of states for the free and the interaction theory. This extends the results previously obtained by the authors in one dimension. Also we consider the case where the interaction is concentrated near a hyperplane.

The Spectral Shift Function for Certain Block Operator Matrices

Let L0 be a 2 × 2 diagonal self-adjoint block operator matrix with entries A and D. If operators B and B* are added to the off diagonal zeros, certain parts of the spectrum of L0 move to the right and other parts move to the left. In this paper it is shown that, correspondingly, if B is a trace class operator M. G. Krein's spectral shift function is of constant sign on certain intervals.

Large Time Volume of the Pinned Wiener Sausage

We calculate the form of the large time asymptotic expansion of the expected volume of the pinned Wiener sausage associated to a compact set K in Rd in dimensions d⩾3. In each case the leading coefficient is given by the Newtonian capacity of K. If K is a ball of radius a>0 the first three coefficients are calculated explicitly. As a byproduct of our results we obtain the low energy behaviour of the Krein spectral shift function for obstacle scattering systems.

Estimates for the spectral shift function of the polyharmonic operator

The Lifshits–Krein spectral shift function is considered for the pair of operators H0=(−Δ)l, l&amp;gt;0 and H=H0+V in L2(Rd), 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&amp;gt;0 and the integral norms of V. These estimates are in a good agreement with the ones predicted by the classical phase space volume considerations.

SCATTERING THEORY APPROACH TO RANDOM SCHRÖDINGER OPERATORS IN ONE DIMENSION

Methods from scattering theory are introduced to analyze random Schrödinger operators in one dimension by applying a volume cutoff to the potential. The key ingredient is the Lifshitz–Krein spectral shift function, which is related to the scattering phase by the theorem of Birman and Krein. The spectral shift density is defined as the "thermodynamic limit" of the spectral shift function per unit length of the interaction region. This density is shown to be equal to the difference of the densities of states for the free and the interacting Hamiltonians. Based on this construction, we give a new proof of the Thouless formula. We provide a prescription how to obtain the Lyapunov exponent from the scattering matrix, which suggest a way how to extend this notion to the higher dimensional case. This prescription also allows a characterization of those energies which have vanishing Lyapunov exponent.

Spectral shift function, amazing and multifaceted

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.