Related papers: The Witten index and the spectral shift function

We establish uncertainty principles on compact Riemannian manifolds without boundary by combining restriction estimates for orthonormal systems with spectral projection bounds for Laplace-Beltrami and Schr\"odinger operators. Our results…

Let $\Op_t(a)$, for $t\in \mathbf R$, be the pseudo-differential operator $$ f(x) \mapsto (2\pi)^{-n}\iint a((1-t)x+ty,\xi)f(y)e^{i\scal {x-y}\xi} dyd\xi $$ and let $\mathscr I_p$ be the set of Schatten-von Neumann operators of order $p\in…

The usual Weyl calculus is intimately associated with the choice of the standard symplectic structure on $\mathbb{R}^{n}\oplus\mathbb{R}^{n}$. In this paper we will show that the replacement of this structure by an arbitrary symplectic…

We define a class of discrete operators acting on infinite, finite or periodic sequences mimicking the standard properties of pseudo-differential operators. In particular we can define the notion of order and regularity, and we recover the…

The abstract theory of self-adjoint extensions of symmetric operators is used to construct self-adjoint realizations of a second-order elliptic operator on $\mathbb{R}^{n}$ with linear boundary conditions on (a relatively open part of) a…

For a finite rank projective bundle over a compact manifold, so associated to a torsion, Dixmier-Douady, 3-class, w, on the manifold, we define the ring of differential operators `acting on sections of the projective bundle' in a formal…

We revisit a construction principle of Fredholm operators using Hilbert complexes of densely defined, closed linear operators and apply this to particular choices of differential operators. The resulting index is then computed with the help…

This paper is a continuation of arXiv:0706.3511, where we obtained a local index formula for matrix elliptic operators with shifts. Here we establish a cohomological index formula of Atiyah-Singer type for elliptic differential operators…

For the pair $\{-\Delta, -\Delta-\alpha\delta_\mathcal{C}\}$ of self-adjoint Schr\"{o}dinger operators in $L^2(\mathbb{R}^n)$ a spectral shift function is determined in an explicit form with the help of (energy parameter dependent)…

We consider the Schr{\"o}dinger operator $-\Delta +V(x)$ in $L^2({\bf R}^3)$ with a real short-range (integrable) potential $V$. Using the associated Fredholm determinant, we present new trace formulas, in particular, the ones in terms of…

We study the $\eta$-invariant of a Dirac operator on a manifold with boundary subject to local boundary conditions with the help of heat kernel methods. In even dimensions, we relate this invariant to $\eta$-invariants of a boundary Dirac…

The aim of this paper is to show how zeta functions and excision in cyclic cohomology may be combined to obtain index theorems. In the first part, we obtain a local index formula for "abstract elliptic pseudodifferential operators"…

We investigate the index of the Neuberger's Dirac operator in abelian gauge theories on finite lattices by numerically analyzing the spectrum of the hermitian Wilson-Dirac operator for a continuous family of gauge fields connecting…

We establish Fredholm properties for a class of nonlocal differential operators. Using mild convergence and localization conditions on the nonlocal terms, we also show how to compute Fredholm indices via a generalized spectral flow, using…

A classical theorem of Mihlin yields Lp estimates for spectral multipliers Lp(R^d) -> Lp(R^d); g -> F^{-1}[f(| |^2) Fg] in terms of L^\infty bounds of the multiplier function f and its weighted derivatives up to an order > d/2. This…

We derive a limiting absorption principle on any compact interval in $\mathbb{R} \backslash \{0\}$ for the free massless Dirac operator, $H_0 = \alpha \cdot (-i \nabla)$ in $[L^2(\mathbb{R}^n)]^N$, $n \geq 2$, $N=2^{\lfloor(n+1)/2\rfloor}$,…

We consider families $A(t)$ of self-adjoint operators with symmetry that causes the spectral flow of the family to vanish. We study the secondary $\mathbb{Z}_2$-valued spectral flow of such families. We prove an analog of the…

Differential equations on spaces of operators are very little developed in Mathematics, being in general very challenging. Here, we study a novel system of such (non-linear) differential equations. We show it has a unique solution for all…

We examine two kinds of spectral theoretic situations: First, we recall the case of self-adjoint half-line Schr\"odinger operators on $[a,\infty)$, $a\in\mathbb{R}$, with a regular finite end point $a$ and the case of Schr\"odinger…

We formulate a noncommutative generalization of the Ricci flow theory in the framework of spectral action approach to noncommutative geometry. Grisha Perelman's functionals are generated as commutative versions of certain spectral…