We prove a local version of the index theorem for Lorentzian Dirac-type operators on globally hyperbolic manifolds with Cauchy boundary. In case hypersurface is compact we do not assume self-adjointness Dirac operator spacetime or associated elliptic this integration our results in generalization previously known theorems spacetimes that allows twisting bundles non-compact gauge groups.

We show that the Dirac operator on a compact globally hyperbolic Lorentzian spacetime with spacelike Cauchy boundary is Fredholm if appropriate conditions are imposed. prove index of this given by same expression as in formula Atiyah-Patodi-Singer for Riemannian manifolds boundary. The also shown to equal certain constructed from evolution and spectral projection In case metric product type near Feynman parametrix constructed.

Abstract We discuss high energy properties of states for (possibly interacting) quantum fields in curved spacetimes. In particular, if the spacetime is real analytic, we show that an analogue timelike tube theorem and Reeh–Schlieder property hold with respect to satisfying a weak form microlocal analyticity condition. The former means von Neumann algebra observables spacelike equals significantly bigger region obtained by deforming boundary manner. This generalizes theorems Araki (Helv Phys...

This paper establishes trace-formulae for a class of operators defined in terms the functional calculus Laplace operator on divergence-free vector fields with relative and absolute boundary conditions Lipschitz domains $\mathbb{R}^3$. Spectral scattering theory Laplacian is equivalent to spectral analysis Maxwell equations. The allow unbounded functions that are not admissible Birman-Krein formula. In special cases trace-formula reduces determinant formula Casimir energy being used physics...

We show in this article that the Reeh–Schlieder property holds for states of quantum fields on real analytic curved space–times if they satisfy an microlocal spectrum condition. This result setting general field theory, i.e., without assuming to obey a specific equation motion. Moreover, quasifree Klein–Gordon are further investigated present work and (analytic) condition is shown be equivalent simpler conditions. also prove any ground or KMS state stationary space–time fulfills

We introduce a class of functions near zero on the logarithmic cover complex plane that have convergent expansions into generalized power series.The construction covers cases where noninteger powers z and also terms containing log can appear.We show that, under natural assumptions, some important theorems from analysis carry over to this functions.In particular, it is possible define field generalize meromorphic functions, one formulate an analytic Fredholm theorem in class.We modified be...

We consider the case of scattering by several obstacles in Rd for d≥2. In this setting, absolutely continuous part Laplace operator Δ with Dirichlet boundary conditions and free Δ0 are unitarily equivalent. For suitable functions that decay sufficiently fast, we have difference g(Δ)−g(Δ0) is a trace-class its trace described Krein spectral shift function. article, study contribution to (and hence function) arises from assembling relative setting where completely separated. two obstacles,...

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

Abstract There recently has been some interest in the space of functions on an interval satisfying heat equation for positive time interior this interval. Such were characterised as being analytic a square with original its diagonal. In short note we provide direct argument that analogue result holds any dimension. For bounded Lipschitz domain $$(\Omega \subset \mathbb {R}^{d})$$ <mml:math...

We analyze the semiclassical limit of spectral theory on manifolds whose metrics have jump-like discontinuities. Such systems are quite different from with smooth Riemannian because does not relate to a classical flow but rather branching (ray-splitting) billiard dynamics. In order describe this system we introduce dynamical space functions phase space. To identify quantum dynamics in compute principal symbols Fourier integral operators associated reflected and refracted geodesic rays...

We consider scattering theory of the Laplace Beltrami operator on differential forms a Riemannian manifold that is Euclidean at infinity. The may have several boundary components caused by obstacles which relative conditions are imposed. Scattering takes place because presence these and possible non-trivial topology geometry. Unlike in case functions eigenvalues generally exist bottom continuous spectrum corresponding eigenforms represent cohomology classes. show appear expansion resolvent,...

Computing the Casimir force and energy between objects is a classical problem of quantum theory going back to 1940s. Several different approaches have been developed in literature often based on physical principles. Most notably representation terms determinants boundary layer operators makes it accessible numerical approach. In this paper, we first give an overview various methods discuss connection Krein-spectral shift function computational aspects. We propose variants Krylov subspace for...

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