Low Energy Asymptotics of the Spectral Shift Function for Pauli Operators with Nonconstant Magnetic Fields

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].

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.

A new method is proposed to study supersymmetric quantum mechanics. A basic relation is derived between Krein's spectral shift function and the Witten index as a powerful tool for explicit model investigations. The topological invariance of relevant quantities like the index, the anomaly, the spectral asymmetry and the spectral shift function is proved. As an illustration, some model calculations are presented, in particular the two-dimensional magnetic field problem, without assuming the magnetic flux to be quantized.

The energy-dependent trace-class time-delay operator associated with the transit time of a scattering system through a finite space region Σ⊆R2 is used to define a local (Σ-dependent) version of the Krein spectral shift function. If the region Σ is a disk of radius r, it is proved that as r→∞ the local spectral shift function converges, for almost all energies, to the original spectral shift function of Krein. This result continues to be valid for systems exhibiting zero-energy resonance behavior.

A functional analytic proof of the existence of Krein's spectral shift function and the associated trace formula is given for a pair of unitary operators, the difference of which is trace class.

Spectral distributions for long range perturbations

In [Spectral asymmetry and Riemannian geometry. III, Math. Proc. Cambridge Philos. Soc. 79 (1976) 71–99] Atiyah, Patodi and Singer introduced spectral flow for elliptic operators on odd-dimensional compact manifolds. They argued that it could be computed from the Fredholm index of an elliptic operator on a manifold of one higher dimension. A general proof of this fact was produced by Robbin–Salamon [The spectral flow and the Maslov index, Bull. London Math. Soc. 27(1) (1995) 1–33, MR 1331677]. In [F. Gesztesy, Y. Latushkin, K. Makarov, F. Sukochev and Y. Tomilov, The index formula and the spectral shift function for relatively trace class perturbations, Adv. Math. 227(1) (2011) 319–420, MR 2782197], a start was made on extending these ideas to operators with some essential spectrum as occurs on non-compact manifolds. The new ingredient introduced there was to exploit scattering theory following the fundamental paper [A. Pushnitski, The spectral flow, the Fredholm index, and the spectral shift function, in Spectral Theory of Differential Operators, American Mathematical Society Translations: Series 2, Vol. 225 (American Mathematical Society, Providence, RI, 2008), pp. 141–155, MR 2509781]. These results do not apply to differential operators directly, only to pseudo-differential operators on manifolds, due to the restrictive assumption that spectral flow is considered between an operator and its perturbation by a relatively trace-class operator. In this paper, we extend the main results of these earlier papers to spectral flow between an operator and a perturbation satisfying a higher pth Schatten class condition for [Formula: see text], thus allowing differential operators on manifolds of any dimension [Formula: see text]. In fact our main result does not assume any ellipticity or Fredholm properties at all and proves an operator theoretic trace formula motivated by [M.-T. Benameur, A. Carey, J. Phillips, A. Rennie, F. Sukochev and K. Wojciechowski, An analytic approach to spectral flow in von Neumann algebras, in Analysis, Geometry and Topology of Elliptic Operators (World Scientific Publisher, Hackensack, NJ, 2006), pp. 297–352, MR 2246773; A. Carey, H. Grosse and J. Kaad, On a spectral flow formula for the homological index, Adv. Math. 289 (2016) 1106–1156, MR 3439708]. We illustrate our results using Dirac type operators on [Formula: see text] for arbitrary [Formula: see text] (see Sec. 8). In this setting Theorem 6.4 substantially extends Theorem 3.5 of [A. Carey, F. Gesztesy, H. Grosse, G. Levitina, D. Potapov, F. Sukochev and D. Zanin, Trace formulas for a class of non-Fredholm operators: a review, Rev. Math. Phys. 28(10) (2016) 1630002, MR 3572626], where the case d = 1 was treated.

RésuméNous considérons les perturbations H := H0 + V et D := D0 + V des Hamiltoniens libres H0 de Pauli et D0 de Dirac en dimension 3 avec champ magnétique non constant, V étant un potentiel électrique qui décroıt super-exponentiellement dans la direction du champ magnétique. Nous montrons que dans des espaces de Banach appropriés, les résolvantes de H et D définies sur le demi-plan supérieur admettent des prolongements méromorphes. Nous définissons les résonances de H et D comme étant les pôles de ces extensions méromorphes. D’une part, nous étudions la répartition des résonances de H prés de l’origine 0 et d’autre part, celle des résonances de D près de ±m o ùm est la masse d’une particule. Dans les deux cas, nous obtenons d’abord des majorations du nombre de résonances dans de petits domaines au voisinage de 0 et ±m. Sous des hypothèses supplémentaires, nous obtenons des développements asymptotiques du nombre de résonances qui entraınent leur accumulation près des seuils 0 et ±m. En particulier, pour une perturbation V de signe défini, nous obtenons des informations sur la répartition des valeurs propres de H et D près de 0 et ±m respectivement.

Cluster properties of one particle Schrödinger operators are proved, where the potential is the sum of two terms and the center of one is moved to infinity. These cluster properties hold for the on-shell S-matrix elements, the spectral shift functions (and thus the total phase shift) as well as their Laplace transforms, i.e. the traces of the differences of appropriate heat kernels.

We consider the 3D-Schrödinger operator H_{0} with constant magnetic field B of scalar intensity b>0 , and its perturbations H= H_{0}+V by a potential V of not necessarily fixed sign. Then we study the singularities at the Landau levels of the Kreĭn spectral shift functions \xi(\cdot;H,H_{0}) for the pair (H,H_{0}) . We consider power-like decaying potentials and cylindrically supported potentials (with axis B ).

In this paper the notion of the Krein spectral shift function is extended to the radial Schrödinger 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.

We study the resonances of the two-dimensional Schrodinger operator P1(B;β)=(Dx−By)2+Dy2+βx+V(x,y), B > 0, β > 0, with constant magnetic and electric fields. We define the resonances of P 1 (B; β) and the spectral shift function ξ(λ) related to P 1 (B; β) and P 0 (B; β) = P 1 (B; β) − V(x, y) without any restriction on B and β. For strong magnetic fields (B → ∞) we obtain a representation of the derivative of ξ(λ), a trace formula for tr(f(P1(B;β))−f(P0(B;β))) and an upper bound for the number of the resonances lying in {z∈3/8:|ℜz−(2n−1)B|≤αB, Imz≥μImθ}, 0 0 is a constant independent of B and n ∈ ℕ* = ℕ {0}.

For the pair $$-\Delta,-\Delta-\alpha\delta_\mathcal{C}$$ of self-adjoint Schröodinger operators in $$L^2(\mathbb{R}^n)$$ a spectral shift function is determined in an explicit form with the help of (energy parameter dependent) Dirichlet-to-Neumann maps. Here δC denotes a singular δ-potential which is supported on a smooth compact hypersurface $$ { C}\subset(\mathbb{R}^n)$$ and δ is a real-valued function on C.

The goal of the workshop was to bring together specialists working in various branches of spectral theory with applications to solid state physics, superconductivity, quantum mechanics etc. The meeting was attended by more than 45 participants from Europe, Japan, Russia, South America and US. During the five days 26 talks were delivered. A special care was taken that apart from the recognized experts in the field, young participants also had an opportunity to speak about their results. The Wednesday morning session, preceding the traditional afternoon hike, consisted of talks of survey nature, which was appreciated by all. There were several major themes in the workshop. One was the study of discrete spectra, including Lieb-Thirring estimates, properties of resonances. A substantial number of talks was concerned with the connection between the continuous and discrete spectra. These include, in particular, the study of the so-called trace formulas. The investigation of various characteristics of the continuous spectra (e.g. density of states, spectral shift function) was featured in a number of talks in the context of random or magnetic operators. A variety of new results were also reported on the theory of periodic operators. They concerned the Liouville Theorem, the problem of absolute continuity, and the classical problem of homogenization. A relatively low number of talks gave the participants an opportunity for discussions in small groups outside the scheduled lecture time. It is hoped that these contacts will result in further collaboration. It is our pleasure to thank the administration and staff of the \textsl{Mathematisches Forschungsinstitut Oberwolfach} for creating comfortable and genuinely inspiring atmosphere, which facilitated the work of the organisers and contributed to the success of the workshop.

In this note we study the behaviour of functions of maximal dissipative operators under relatively bounded and relatively trace class perturbations. We introduce the class of analytic relatively operator Lipschitz functions.We obtain a formula for the derivative in the strong operator topology in the parameter of functions of one-parametric families of dissipative operators.We also establish a trace formula for the difference of a function of a perturbed operator and the function of the initial operator. It turns out that the corresponding spectral shift function is integrable with weight (1+|x|). Moreover, the maximal class of functions, for which the trace formula holds for all pairs of maximal dissipative operators under relatively trace class perturbations coincides with the class of analytic relatively operator Lipschitz functions.