Related papers: The trace formula for a point scatterer on a compa…
In this article we extend previous semiclassical studies by including more general perturbative potentials of the harmonic oscillator in arbitrary spatial dimensions. Our starting point is a radial harmonic potential with an arbitrary even…
Let $\Sigma\subset\mathbb{R}^d$ be a $C^\infty$-smooth closed compact hypersurface, which splits the Euclidean space $\mathbb{R}^d$ into two domains $\Omega_\pm$. In this note self-adjoint Schr\"odinger operators with $\delta$ and…
We give an explicit geometric formula for the twisted orbital integrals using the method of the hypoelliptic Laplacian developed by Bismut. Combining with the twisted trace formula, we can evaluate the equivariant trace of the heat…
Two-dimensional dielectric microcavities are of widespread use in microoptics applications. Recently, a trace formula has been established for dielectric cavities which relates their resonance spectrum to the periodic rays inside the…
We consider self-adjoint Dirac operators $\ham{D}=\ham{D}_0 + V(x)$, where $\ham{D}_0$ is the free three-dimensional Dirac operator and $V(x)$ is a smooth compactly supported Hermitian matrix. We define resonances of $\ham{D}$ as poles of…
We consider the one dimensional Schr\"odinger operator with properly connecting generalized point interaction at the origin. We derive a trace formula for trace of difference of resolvents of perturbed and unperturbed Schr\"odinger…
The paper is devoted to investigation of the spectrum of a perturbed Laplace-Beltrami operator on manifolds with closed geodesics of the same length and where metric is a perturbation of the metric of the unit sphere. As the result, we give…
The Gutzwiller trace formula is extended to include diffraction effects. The new trace formula involves periodic rays which have non-geometrical segments as a result of diffraction on the surfaces and edges of the scatter.
We present several formulas for the traces of elements in complex hyperbolic triangle groups generated by complex reflections. The space of such groups of fixed signature is of real dimension one. We parameterise this space by a real…
We consider the case of scattering by several obstacles in $\mathbb{R}^d$ for $d \geq 2$. We establish a relative trace formula for Neumann and transmission boundary conditions analogous to the one obtained in arXiv:2002.07291 for Dirichlet…
The functional determinant of Laplace-type operators on the 3-dimensional non-compact hyperbolic manifold with invariant fundamental domain of finite volume is computed by quadratures and making use of the related terms of the Selberg trace…
We consider the case of scattering of several obstacles in $\mathbb{R}^d$ for $d \geq 2$. In this setting the absolutely continuous part of the Laplace operator $\Delta$ with Dirichlet boundary conditions and the free Laplace operator…
Two trace formulas for the spectra of arbitrary Hermitian matrices are derived by transforming the given Hermitian matrix $H$ to a unitary analogue. In the first type the unitary matrix is $e^{i(\lambda\II - H)}$ where $\lambda$ is the…
We use invariance theory to compute the divergence term $a_{m+2,m}^{d+\delta}$ in the super trace for the twisted de Rham complex for a closed Riemannian manifold.
We calculate the one-loop effective potential of a self-interacting scalar field on the spacetime of the form $\reals^2\times H^2/\Gamma$. The Selberg trace formula associated with a co-compact discrete group $\Gamma$ in $PSL(2,\reals )$…
Inspired by a recent paper of G. Liu and X. Tan (2023), we would like to measure how the magnetic effect appears in the heat trace formula associated with the magnetic Laplacian and the magnetic Dirichlet-to-Neumann operator. We propose to…
Our goal is to extend the theory of the spectral shift function to the case where only the difference of some powers of the resolvents of self-adjoint operators belongs to the trace class. As an example, we consider a couple of Dirac…
We stabilize the full Arthur-Selberg trace formula for the metaplectic covering of symplectic groups over a number field. This provides a decomposition of the invariant trace formula for metaplectic groups, which encodes information about…
Trace formulae for d-regular graphs are derived and used to express the spectral density in terms of the periodic walks on the graphs under consideration. The trace formulae depend on a parameter w which can be tuned continuously to assign…
We derive dispersion estimates for solutions of a one-dimensional discrete Dirac equations with a potential. In particular, we improve our previous result, weakening the conditions on the potential. To this end we also provide new results…