Alexander Strohmaier

• Professor
• Professor (Full) at Leibniz Universität Hannover

We prove a local version of the index theorem for Lorentzian Dirac-type operators on globally hyperbolic Lorentzian manifolds with Cauchy boundary. In case the Cauchy hypersurface is compact we do not assume self-adjointness of the Dirac operator on the spacetime or the associated elliptic Dirac operator on the boundary. In this case integration of...

This short letter considers the case of acoustic scattering by several obstacles in Rd+r\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^{d+r}$$\end{document...

The generator of time-translations on the solution space of the wave equation on stationary spacetimes specialises to the square root of the Laplacian on Riemannian manifolds when the spacetime is ultrastatic. Its spectral analysis therefore constitutes a generalization of classical spectral geometry. If the spacetime is spatially compact the spect...

Computing the Casimir force and energy between objects is a classical problem of quantum theory going back to the 1940s. Several different approaches have been developed in the literature often based on different physical principles. Most notably a representation of the Casimir energy in terms of determinants of boundary layer operators makes it ac...

We consider scattering theory of the Laplace Beltrami operator on differential forms on a Riemannian manifold that is Euclidean near infinity. Allowing for compact boundaries of low regularity we prove a Birman-Krein formula on the space of co-closed differential forms. In the case of dimension three this reduces to a Birman-Krein formula in Maxwel...

There recently has been some interest in the space of functions on an interval satisfying the heat equation for positive time in the interior of this interval. Such functions were characterised as being analytic on a square with the original interval as its diagonal. In this short note we provide a direct argument that the analogue of this result h...

This paper establishes trace-formulae for a class of operators defined in terms of the functional calculus for the Laplace operator on divergence-free vector fields with relative and absolute boundary conditions on Lipschitz domains in $\mathbb{R}^3$. Spectral and scattering theory of the absolute and relative Laplacian is equivalent to the spectra...

We consider the case of scattering by several obstacles in $${\mathbb {R}}^d$$ R d , $$d \ge 2$$ d ≥ 2 for the Laplace operator $$\Delta $$ Δ with Dirichlet boundary conditions imposed on the obstacles. In the case of two obstacles, we have the Laplace operators $$\Delta _1$$ Δ 1 and $$\Delta _2$$ Δ 2 obtained by imposing Dirichlet boundary conditi...

Starting from the construction of the free quantum scalar field of mass m≥0, we give mathematically precise and rigorous versions of three different approaches to computing the Casimir forces between compact obstacles. We then prove that they are equivalent.

In this article I give a rigorous construction of the classical and quantum photon field on non-compact manifolds with boundary and in possibly inhomogeneous media. Such a construction is complicated by zero-modes that appear in the presence of non-trivial topology of the manifold or the boundary. An important special case is R3\documentclass[12pt]...

We consider scattering theory of the Laplace Beltrami operator on differential forms on a Riemannian manifold that is Euclidean near infinity. Allowing for compact boundaries of low regularity we prove a Birman-Krein formula on the space of co-closed differential forms. In the case of dimension three this reduces to a Birman-Krein formula in Maxwel...

Starting from the construction of the free quantum scalar field of mass $m\geq 0$ we give mathematically precise and rigorous versions of three different approaches to computing the Casimir forces between compact obstacles. We then prove that they are equivalent.

We consider the case of scattering of several obstacles in $\mathbb{R}^d$ for $d \geq 2$ for the Laplace operator $\Delta$ with Dirichlet boundary conditions imposed on the obstacles. In the case of two obstacles, we have the Laplace operators $\Delta_1$ and $\Delta_2$ obtained by imposing Dirichlet boundary conditions only on one of the objects. T...

This article reviews the microlocal construction of Feynman propagators for normally hyperbolic operators acting on vector bundles over globally hyperbolic spacetimes and its consequences. It is shown that for normally hyperbolic operators that are selfadjoint with respect to a hermitian bundle metric, the Feynman propagators can be constructed to...

We prove a local version of the index theorem for Lorentzian Dirac-type operators on globally hyperbolic Lorentzian manifolds with Cauchy boundary. In case the Cauchy hypersurface is compact we do not assume self-adjointness of the Dirac operator on the spacetime or the associated elliptic Dirac operator on the boundary. In this case integration of...

In this article I give a rigorous construction of the classical and quantum photon field on non-compact manifolds with boundary and in possibly inhomogeneous media. Such a construction is complicated by the presence of zero-modes that may appear in the presence of non-trivial topology of the manifold or the boundary. An important special case is $\...

We give a relativistic generalization of the Gutzwiller-Duistermaat-Guillemin trace formula for the wave group of a compact Riemannian manifold to globally hyperbolic stationary space-times with compact Cauchy hypersurfaces. We introduce several (essentially equivalent) notions of trace of self-adjoint operators on the null-space ker⁡□ of the wave...

We study the spectrum of a timelike Killing vector field Z acting as a differential operator on the solution space Hm:={u∣(□g+m2)u=0} of the Klein–Gordon equation on a globally hyperbolic stationary spacetime (M,g) with compact Cauchy hypersurface Σ. We endow Hm with a natural inner product, so that DZ=1i∇Z is a self-adjoint operator on Hm with dis...

For the heat equation on a bounded Lipschitz domain $\Omega \subset \mathbb{R}^d$ we show that at positive time all solutions to the heat equation are analytically extendable to a geometrically determined subdomain $\mathcal{E}(\Omega)$ of $\mathbb{C}^d$ containing $\Omega$. This result is sharp in the sense that there is no larger domain for which...

We consider scattering theory of the Laplace Beltrami operator on differential forms on a Riemannian manifold that is Euclidean at infinity. The manifold may have several boundary components caused by obstacles at which relative boundary conditions are imposed. Scattering takes place because of the presence of these obstacles and possible non-trivi...

We review our recent relativistic generalization of the Gutzwiller–Duistermaat–Guillemin trace formula and Weyl law on globally hyperbolic stationary space-times with compact Cauchy hypersurfaces. We also discuss anticipated generalizations to non-compact Cauchy hypersurface cases.

We consider the case of scattering of several obstacles in $\mathbb{R}^d$ for $d \geq 2$. Then the absolutely continuous part of the Laplace operator $\Delta$ with Dirichlet boundary conditions and the free Laplace operator $\Delta_0$ are unitarily equivalent. For suitable functions that decay sufficiently fast we have that the difference $g(\Delta...

We study the spectrum $\{\lambda_j(m)\}_{j=1}^{\infty}$ of a timeline Killing vector field $Z$ acting as a differential operator $D_Z$ on the Hilbert space of solutions of the massive Klein-Gordon equation $(\Box_g + m^2) u = 0$ on a globally hyperbolic stationary spacetime $(M, g)$ with compact Cauchy hypersurface. The inverse mass $m^{-1}$ is for...

In this paper we describe a simple method that allows for a fast direct computation of the scattering matrix for a surface with hyperbolic cusps from the Neumann-to-Dirichlet map on the compact manifold with boundary obtained by removing the cusps. We illustrate that even if the Neumann-to-Dirichlet map is obtained by a finite element method (FEM)...

We show that the Dirac operator on a compact globally hyperbolic Lorentzian spacetime with spacelike Cauchy boundary is a Fredholm operator if appropriate boundary conditions are imposed. We prove that the index of this operator is given by the same expression as in the index formula of Atiyah-Patodi-Singer for Riemannian manifolds with boundary. T...

We consider scattering theory of the Laplace Beltrami operator on differential forms on a Riemannian manifold that is Euclidean at infinity. The manifold may have several boundary components caused by obstacles at which relative boundary conditions are imposed. Scattering takes place because of the presence of these obstacles and possible non-trivi...

In this paper we describe a simple method that allows for a fast direct computation of the scattering matrix for a surface with hyperbolic cusps from the Neumann-to-Dirichlet map on the compact manifold with boundary obtained by removing the cusps. We illustrate that even if the Neumann-to-Dirichlet map is obtained by a Finite Element Method (FEM)...

We give a relativistic generalization of the Gutzwiller (-Duistermaat-Guillemin) trace formula for the wave group of a compact Riemannian manifold to globally hyperbolic stationary space-times with compact Cauchy hypersurfaces. We introduce several (essentially equivalent) notions of trace of self-adjoint operators on the null-space $\ker \Box$ of...

We give a relativistic generalization of the Gutzwiller-Duistermaat-Guillemin trace formula for the wave group of a compact Riemannian manifold to globally hyperbolic stationary space-times with compact Cauchy hypersurfaces. We introduce several (essentially equivalent) notions of trace of self-adjoint operators on the null-space ker□ of the wave o...

The example in section 7.3 page 854 titled “The surface with symmetry group Z5×Z2” was given incorrect Fenchel–Nielsen coordinates.

In Section 5.1 in [1] it is incorrectly claimed that condition (A) is equivalent to the vanishing of the operator B in the expansion. © 2017 Springer International Publishing AG, part of Springer Nature

These are lecture notes from a series of three lectures given at the summer school “Geometric and Computational Spectral Theory” in Montreal in June 2015. The aim of the lecture was to explain the mathematical theory behind computations of eigenvalues and spectral determinants in geometrically non-trivial contexts.

We discuss the chiral anomaly for a Weyl field in a curved background and show that a novel index theorem for the Lorentzian Dirac operator can be applied to describe the gravitational chiral anomaly. A formula for the total charge generated by the gravitational and gauge field background is derived directly in Lorentzian signature and in a mathema...

These are lecture notes from a series of three lectures given at the summer school "Geometric and Computational Spectral Theory" in Montreal in June 2015. The aim of the lecture was to explain the mathematical theory behind computations of eigenvalues and spectral determinants in geometrically non-trivial contexts.

We show that not feeling the boundary estimates for heat kernels hold for any non-negative self-adjoint extension of the Laplace operator acting on vector-valued compactly supported functions on a domain in $\mathbb{R}^d$. They are therefore valid for any choice of boundary condition and we show that the implied constants can be chosen independent...

We show that not feeling the boundary estimates for heat kernels hold for any non-negative self-adjoint extension of the Laplace operator acting on vector-valued compactly supported functions on a domain in $\mathbb{R}^d$. They are therefore valid for any choice of boundary condition and we show that the implied constants can be chosen independent...

Let $P$ be a non-negative self-adjoint Laplace type operator acting on sections of a hermitian vector bundle over a closed Riemannian manifold. In this paper we review the close relations between various $P$-related coefficients such as the mollified spectral counting coefficients, the heat trace coefficients, the resolvent trace coefficients, the...

Let $P$ be a non-negative self-adjoint Laplace type operator acting on sections of a hermitian vector bundle over a closed Riemannian manifold. In this paper we review the close relations between various $P$-related coefficients such as the mollified spectral counting coefficients, the heat trace coefficients, the resolvent trace coefficients, the...

Let $(M, {g})$ be a compact, $d$-dimensional Riemannian manifold without boundary. Suppose further that $(M,g)$ is either two dimensional and has no conjugate points or $(M,g)$ has non-positive sectional curvature. The goal of this note is to show that the long time parametrix obtained for such manifolds by B\'erard can be used to prove a logarithm...

We give a complete framework for the Gupta-Bleuler quantization of the free electromagnetic field on globally hyperbolic space-times. We describe one-particle structures that give rise to states satisfying the microlocal spectrum condition. The field algebra in the so-called Gupta-Bleuler representations satisfies the time-slice axiom, and the corr...

We derive explicit bounds for the remainder term in the local Weyl law for locally hyperbolic manifolds, we also give the estimates of the derivative of this remainder. We use these to obtain explicit bounds for the C^k-norms of the L^2-normalised eigenfunctions in the case spectrum of the Laplacian is discrete, e.g. for closed Riemannian manifolds...

We analyze the semiclassical limit of spectral theory on manifolds whose metrics have jump-like discontinuities. Such systems are quite different from manifolds with smooth Riemannian metrics because the semiclassical limit does not relate to a classical flow but rather to branching (ray-splitting) billiard dynamics. In order to describe this syste...

Compact hyperbolic surfaces are two dimensional oriented Riemannian manifolds of constant negative curvature -1. They can be realized as quotients Γ∖ℍ of the upper half plane ℍ={(x,y)∣y>0} by a discrete hyperbolic co-compact subgroup Γ⊂SL(2,ℝ).

We introduce a class of functions near zero on the logarithmic cover of the complex plane that have convergent expansions into generalized power series. The construction covers cases where non-integer powers of $z$ and also terms containing $\log z$ can appear. We show that under natural assumptions some important theorems from complex analysis car...

We present a rigorous scheme that makes it possible to compute eigenvalues of the Laplace operator on hyperbolic surfaces within a given precision. The method is based on an adaptation of the method of particular solutions to the case of locally symmetric spaces and on explicit estimates for the approximation of eigenfunctions on hyperbolic surface...

At high energies relativistic quantum systems describing scalar particles behave classically. This observation plays an important role in the investigation of eigenfunctions of the Laplace operator on manifolds for large energies and allows to establish relations to the dynamics of the corresponding classical system. Relativistic quantum systems de...

In this paper we consider scattering theory on manifolds with special cusp-like metric singularities of warped product type g=dx^2 + x^(-2a)h, where a>0. These metrics form a natural subset in the class of metrics with warped product singularities and they can be thought of as interpolating between hyperbolic and cylindrical metrics. We prove that...

The spectral theory of the p-form Laplacian on manifolds with hyperbolic cusps has been extensively studied due to its relationships to number theory. For manifolds with infinite cylindrical ends, the spectral theory of the p-form Laplacian is related to index theory of manifolds with boundary and also gives a model for the physics of waveguides. I...

A well known result on pseudodifferential operators states that the noncommutative residue (Wodzicki residue) of a pseudodifferential projection vanishes. This statement is non-local and implies the regularity of the eta invariant at zero of Dirac type operators. We prove that in a filtered algebra the value of a projection under any residual trace...

Microlocal Analysis deals with the singular behavior of distributions in phase space. Distributions appear in physics in various forms: as mass distributions of point particles, as Green’s functions, and as propagators in quantum field theory. The theory of distributions not only has applications such as these in physics; but is also widely and int...

Scattering theory for p-forms on manifolds with cylindrical ends has a direct interpretation in terms of cohomology. Using the Hodge isomorphism, the scattering matrix at low energy may be regarded as an operator on the cohomology of the boundary. Its value at zero describes the image of the absolute cohomology in the cohomology of the boundary. We...

On a compact K\"ahler manifold there is a canonical action of a Lie-superalgebra on the space of differential forms. It is generated by the differentials, the Lefschetz operator and the adjoints of these operators. We determine the asymptotic distribution of irreducible representations of this Lie-superalgebra on the eigenspaces of the Laplace-Belt...

We relate high-energy limits of Laplace-type and Dirac-type operators to frame flows on the corresponding manifolds, and show that the ergodicity of frame flows implies quantum ergodicity in an appropriate sense for those operators. Observables for the corresponding quantum systems are matrix-valued pseudodifferential operators and therefore the sy...

We relate high-energy limits of Laplace-type and Dirac-type operators to frame flows on the corresponding manifolds, and show that the ergodicity of frame flows implies quantum ergodicity in an appropriate sense for those operators. Observables for the corresponding quantum systems are matrix-valued pseudodifferential operators, and therefore the s...

We introduce the notion of a pseudo-Riemannian spectral triple which generalizes the notion of spectral triple and allows for a treatment of pseudo-Riemannian manifolds within a noncommutative setting. It turns out that the relevant spaces in noncommutative pseudo-Riemannian geometry are not Hilbert spaces any more but Krein spaces, and Dirac opera...

We show that for compact orientable hyperbolic orbisurfaces, the Laplace spectrum determines the length spectrum as well as the number of singular points of a given order. The converse also holds, giving a full generalization of Huber's theorem to the setting of compact orientable hyperbolic orbisurfaces.

Let X=G/K be a symmetric space of noncompact type and let L be the Laplacian associated with a G-invariant metric on X. We show that the resolvent kernel of L admits a holomorphic extension to a Riemann surface depending on the rank of the symmetric space. This Riemann surface is a branched cover of the complex plane with a certain part of the real...

Let X=G/K be a symmetric space of noncompact type and let L be the Laplacian associated with a G-invariant metric on X. We show that the resolvent kernel of L admits a holomorphic extension to a Riemann surface depending on the rank of the symmetric space. This Riemann surface is a branched cover of the complex plane with a certain part of the real...

Using the theory of quantized equivariant vector bundles over compact coadjoint orbits we determine the Chern characters of all noncommutative line bundles over the fuzzy sphere with regard to its derivation-based differential calculus. The associated Chern numbers (topological charges) arise to be noninteger, in the commutative limit the well-know...

We show in this article that the Reeh-Schlieder property holds for states of quantum fields on real analytic spacetimes if they satisfy an analytic microlocal spectrum condition. This result holds in the setting of general quantum field theory, i.e. without assuming the quantum field to obey a specific equation of motion. Moreover, quasifree states...

We introduce the notion of a semi-Riemannian spectral triple which generalizes the notion of spectral triple and allows for a treatment of semi-Riemannian manifolds within a noncommutative setting. It turns out that the relevant spaces in noncommutative semi-Riemannian geometry are not Hilbert spaces any more but Krein spaces, and Dirac operators a...

Let M be a connected Riemannian manifold and let D be a Dirac type operator acting on smooth compactly supported sections in a Hermitian vector bundle over M. Suppose D has a self-adjoint extension A in the Hilbert space of square-integrable sections. We show that any $L^2$-section $\phi$ contained in a closed A-invariant subspace onto which the re...

Using the theory of quantized equivariant vector bundles over compact coadjoint orbits we determine the Chern characters of all noncommutative line bundles over the fuzzy sphere with regard to its derivation based differential calculus. The associated Chern numbers (topological charges) arise to be non-integer, in the commutative limit the well kno...

This paper investigates wave-equations on spacetimes with a metric which is locally analytic in the time. We use recent results in the theory of the non-characteristic Cauchy problem to show that a solution to a wave-equation vanishing in an open set vanishes in the ``envelope'' of this set, which may be considerably larger and in the case of timel...

We show that as soon as a linear quantum field on a stationary spacetime satisfies a certain type of hyperbolic equation, the (quasifree) ground- and KMS-states with respect to the canonical time flow have the Reeh-Schlieder property. We also obtain an analog of Borchers' timelike tube theorem. The class of fields we consider contains the Dirac fie...

We prove the Reeh-Schlieder property for the ground- and KMS-states states of the massive Dirac Quantum field on a static globally hyperbolic 4 dimensional spacetime.

We give a noncommutative version of the complex projective space 2 and show that scalar QFT on this space is free of UV divergencies. The tools necessary to investigate quantum fields on this fuzzy 2 are developed and several possibilities to introduce spinors and Dirac operators are discussed.

. We give a noncommutative version of the complex projective space CP 2 and show that scalar QFT on this space is free of UV divergencies. The tools necessary to investigate Quantum fields on this fuzzy CP 2 are developed and several possibilities to introduce spinors and Dirac operators are discussed. Keywords: Regularization, Noncommutative Geome...