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Alexander Strohmaier

Publications and source records attributed to Alexander Strohmaier.

At least 37 records · Page 2Linked to original sources
arXiv2020

A Gutzwiller trace formula for stationary space-times

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 the wave operator and define $U(t)$ to be translation by the flow $e^{t Z}$ of the timelike Killing vector field $Z$ on $\Box$. The spectrum of $Z$ on $\ker \Box$ is discrete and the singularities of $\rm{Tr}\;e^{t Z} |_{\ker \Box}$ occur at periods of periodic orbits of $\exp t Z$ on the symplectic manifold of null geodesics. The trace formula gives a Weyl law for the eigenvalues of $Z$ on $\ker \Box$.

math.AP↗
arXiv2020

Semi-classical mass asymptotics on stationary spacetimes

We study the spectrum $\{λ_j(m)\}_{j=1}^{\infty}$ of a timelike 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 formally like the Planck constant in a Schrödinger equation, and we give Weyl asymptotics as $m \to \infty$ for the number $$N_{ν, C}(m)= \# \{j \mid \frac{λ_j(m)}{m} \in [ν- \frac{C}{m}, ν+ \frac{C}{m} ]\}$$ for a given $C > 0$. The semi-classical mass asymptotics are governed by the dynamics of the Killing flow $e^{tZ} $ on the hypersurface in the space of mass $1$ geodesics $γ$ where $\langle \dotγ, Z \rangle= ν$.

math-ph↗
arXiv2020

Gupta-Bleuler Quantization of the Maxwell Field in Globally Hyperbolic Space-Times

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 algebras in the so-called Gupta-Bleuler representations satisfy the time-slice axiom, and the corresponding vacuum states satisfy the microlocal spectrum condition. We also give an explicit construction of ground states on ultrastatic space-times. Unlike previous constructions, our method does not require a spectral gap or the absence of zero modes. The only requirement, the absence of zero-resonance states, is shown to be stable under compact perturbations of topology and metric. Usual deformation arguments based on the time-slice axiom then lead to a construction of Gupta-Bleuler representations on a large class of globally hyperbolic space-times. As usual, the field algebra is represented on an indefinite inner product space, in which the physical states form a positive semi-definite subspace. Gauge transformations are incorporated in such a way that the field can be coupled perturbatively to a Dirac field. Our approach does not require any topological restrictions on the underlying space-time.

math-ph↗
arXiv2020

Geometric and Obstacle Scattering at Low Energy

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-trivial topology and geometry. Unlike in the case of functions eigenvalues generally exist at the bottom of the continuous spectrum and the corresponding eigenforms represent cohomology classes. We show that these eigenforms appear in the expansion of the resolvent, the scattering matrix, and the spectral measure in terms of the spectral parameter $λ$ near zero, and we determine the first terms in this expansion explicitly. In dimension two an additional cohomology class appears as a resonant state in the presence of an obstacle. In even dimensions the expansion is in terms of $λ$ and $\log λ$. The theory of Hahn holomorphic functions is used to describe these expansions effectively. We also give a Birman-Krein formula in this context. The case of one forms with relative boundary conditions has direct applications in physics as it describes the scattering of electromagnetic waves.

math.AP↗
arXiv2019

An index theorem for Lorentzian manifolds with compact spacelike Cauchy boundary

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. The index is also shown to equal that of a certain operator constructed from the evolution operator and a spectral projection on the boundary. In case the metric is of product type near the boundary a Feynman parametrix is constructed.

math.DG↗
arXiv2019

Computations of eigenvalues and resonances on perturbed hyperbolic surfaces with cusps

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) one can achieve good accuracy for the scattering matrix. We give various interesting examples of how this can be used to investigate the behaviour of resonances under conformal perturbations or when moving in Teichmüller space. For example, based on numerical experiments we rediscover the four arithmetic surfaces of genus one with one cusp. This demonstrates that it is possible to identify arithmetic objects using FEM.

math.SP↗
arXiv2017

An Algorithm for the Computation of Eigenvalues, Spectral Zeta Functions and Zeta-Determinants on Hyperbolic Surfaces

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 surfaces by certain basis functions. It can be applied to check whether or not there is an eigenvalue in an ε-neighborhood of a given number λ>0. This makes it possible to find all the eigenvalues in a specified interval, up to a given precision with rigorous error estimates. The method converges exponentially fast with the number of basis functions used. Combining the knowledge of the eigenvalues with the Selberg trace formula we are able to compute values and derivatives of the spectral zeta function again with error bounds. As an example we calculate the spectral determinant and the Casimir energy of the Bolza surface and other surfaces.

math.SP↗
arXiv2017

Scattering at low energies on manifolds with cylindrical ends and stable systoles

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 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 show that the so-called scattering length, the Eisenbud-Wigner time delay at zero energy, has a cohomological interpretation as well. Namely, it relates the norm of a cohomology class on the boundary to the norm of its image under the connecting homomorphism in the long exact sequence in cohomology. An interesting consequence of this is that one can estimate the scattering lengths in terms of geometric data like the volumes of certain homological systoles.

math.AP↗
arXiv2016

Heat Kernel estimates for general boundary problems

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 of the self-adjoint extension. The method of proof is very general and is based on finite propagation speed estimates and explicit Fourier Tauberian theorems obtained by Y. Safarov.

math.AP↗
arXiv2016

A rigorous geometric derivation of the chiral anomaly in curved backgrounds

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 in a mathematically rigorous manner. It contains a term identical to the integrand in the Atiyah-Singer index theorem and another term involving the $η$-invariant of the Cauchy hypersurfaces.

math-ph↗
arXiv2015

The local counting function of operators of Dirac and Laplace type

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 residues of the spectral zeta function as well as certain Wodzicki residues. We then use the Wodzicki residue to obtain results about the local counting function of operators of Dirac and Laplace type. In particular, we express the second term of the mollified spectral counting function of Dirac type operators in terms of geometric quantities and characterize those Dirac type operators for which this coefficient vanishes.

math.SP↗
arXiv2015

Riesz means of the counting function of the Laplace operator on compact manifolds of non-positive curvature

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 Bérard can be used to prove a logarithmic improvement for the remainder term of the Riesz means of the counting function of the Laplace operator.

math.SP↗
arXiv2014

On the Noncommutative Residue for Projective Pseudodifferential Operators

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 depends only on the principal part of the projection. This general, purely algebraic statement applied to the algebra of projective pseudodifferential operators implies that the noncommutative residue factors to a map from the twisted K-theory of the co-sphere bundle. We use arguments from twisted K-theory to show that this map vanishes, thus showing that the noncommutative residue of a projective pseudodifferential projection vanishes. This also gives a very short proof in the classical setting.

math.DG↗
arXiv2014

The semiclassical theory of discontinuous systems and ray-splitting billiards

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 system we introduce a dynamical system on the space of functions on phase space. To identify the quantum dynamics in the semiclassical limit we compute the principal symbols of the Fourier integral operators associated to reflected and refracted geodesic rays and identify the relation between classical and quantum dynamics. In particular we prove a quantum ergodicity theorem for discontinuous systems. In order to do this we introduce a new notion of ergodicity for the ray-splitting dynamics. The paper contains an Appendix written by Yves Colin de Verdiere in which a non-trivial class of examples is constructed.

math.AP↗
arXiv2013

The theory of Hahn meromorphic functions, a holomorphic Fredholm theorem and its applications

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 carry over to the class of these functions. In particular it is possible to define a field of functions that generalize meromorphic functions and one can formulate an analytic Fredholm theorem in this class. We show that this modified analytic Fredholm theorem can be applied in spectral theory to prove convergent expansions of the resolvent for Bessel type operators and Laplace-Beltrami operators for manifolds that are Euclidean at infinity. These results are important in scattering theory as they are the key step to establish analyticity of the scattering matrix and the existence of generalized eigenfunctions at points in the spectrum.

math.CA↗
arXiv2013

Explicit bounds on eigenfunctions and spectral functions on manifolds hyperbolic near a point

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 also derive bounds for the local heat trace. Our estimates are purely local and therefore also hold for any manifold at points near which the metric is locally hyperbolic.

math.SP↗
arXiv2011

Geometry of the high energy limit of differential operators on vector bundles

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 describing particles with spin such as the Dirac equation do not behave classically at high energies. Nonetheless, the dynamical properties of the classical frame flow determine the behavior of eigensections of the corresponding operator for large energies. We review what a high energy limit is and how it can be described for geometric operators.

math.SP↗