Trace singularities in obstacle scattering and the Poisson relation for the relative trace

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 conditions only on one of the objects. The relative operator $$g(\Delta ) - g(\Delta _1) - g(\Delta _2) + g(\Delta _0)$$ g ( Δ ) - g ( Δ 1 ) - g ( Δ 2 ) + g ( Δ 0 ) was introduced in Hanisch, Waters and one of the authors in (A relative trace formula for obstacle scattering. arXiv:2002.07291, 2020) and shown to be trace-class for a large class of functions g, including certain functions of polynomial growth. When g is sufficiently regular at zero and fast decaying at infinity then, by the Birman–Krein formula, this trace can be computed from the relative spectral shift function $$\xi _\mathrm {rel}(\lambda ) = -\frac{1}{\pi } {\text {Im}}(\Xi (\lambda ))$$ ξ rel ( λ ) = - 1 π Im ( Ξ ( λ ) ) , where $$\Xi (\lambda )$$ Ξ ( λ ) is holomorphic in the upper half-plane and fast decaying. In this paper we study the wave-trace contributions to the singularities of the Fourier transform of $$\xi _\mathrm {rel}$$ ξ rel . In particular we prove that $${\hat{\xi }}_\mathrm {rel}$$ ξ ^ rel is real-analytic near zero and we relate the decay of $$\Xi (\lambda )$$ Ξ ( λ ) along the imaginary axis to the first wave-trace invariant of the shortest bouncing ball orbit between the obstacles. The function $$\Xi (\lambda )$$ Ξ ( λ ) is important in the physics of quantum fields as it determines the Casimir interactions between the objects.

Evaluating Possibility of Registering Scattered Gravitational Radiation on Wormholes

Possibility of experimental registration of gravitational radiation scattered on wormholes was evaluated. Scattered radiation registration could become the experimental evidence of the wormhole gas theory explaining the dark matter nature. The simplest model of the traversable static spherically symmetric wormhole was used, which is the limiting case for the Bronnikov --- Ellis wormhole. Equations for gravitational wave against the background of non-empty curved space--time were obtained in the gauge, where the trace of a gravitational wave is not equal to zero. It is shown that equation on the trace is reduced to the Klein --- Gordon --- Fock equation. Explicit expressions were obtained for the gravitational wave trace scattering cross section on a wormhole. It was assumed that the gravitational wave amplitude order was equal to its trace order, numerical simulation was carried out, and scattered gravitational radiation intensity and amplitude from wormholes on Earth were estimated. In the multiverse case, when the wormhole throat was leading to another universe, conclusion was made that it was currently impossible to register radiation scattered by wormholes taking into account the LIGO/VIRGO detector sensitivity

Plane-wave orthogonal polynomial transform for amplitude-preserving noise attenuation

SUMMARY Amplitude-preserving data processing is an important and challenging topic in many scientific fields. The amplitude-variation details in seismic data are especially important because the amplitude variation is directly related with the subsurface wave impedance and fluid characteristics. We propose a novel seismic noise attenuation approach that is based on local plane-wave assumption of seismic events and the amplitude preserving capability of the orthogonal polynomial transform (OPT). The OPT is a way for representing spatially correlative seismic data as a superposition of polynomial basis functions, by which the random noise is distinguished from the useful energy by the high orthogonal polynomial coefficients. The seismic energy is the most correlative along the structural direction and thus the OPT is optimally performed in a flattened gather. We introduce in detail the flattening operator for creating the flattened dimension, where the OPT can be applied subsequently. The flattening operator is created by deriving a plane-wave trace continuation relation following the plane-wave equation. We demonstrate that both plane-wave trace continuation and OPT can well preserve the strong amplitude variation existing in seismic data. In order to obtain a robust slope estimation performance in the presence of noise, a robust slope estimation approach is introduced to substitute the traditional method. A group of synthetic, pre-stack and post-stack field seismic data are used to demonstrate the potential of the proposed framework in realistic applications.