Related papers: Translation Representations and Scattering By Two …
We prove that the scattering operators and wave operators are well-defined in the energy space for the system of defocusing Schr\"odinger equations $$ \begin{cases} i\partial_t u_\mu + \Delta u_\mu - \sum_{\mu,\nu=1 }^N…
It often goes unnoticed that, even for a finite number of degrees of freedom, the canonical commutation relations have many inequivalent irreducible unitary representations; the free particle and a particle in a box provide examples that…
We develop a new method of representation of quantum states in terms of the displaced number states. We call it representation, where is an amplitude of the base displaced states. In particular, representation was obtained for set of the…
For the scattering system given by the Laplacian in a half-space with a periodic boundary condition, we derive resolvent expansions at embedded thresholds, we prove the continuity of the scattering matrix, and we establish new formulas for…
In this paper we consider the Higher Spin Six Vertex Model on the lattice $\mathbb{Z}_{\geq 2} \times \mathbb{Z}_{\geq 1}$. We first identify a family of translation invariant measures and subsequently we study the one point distribution of…
It is shown that the potential perturbation that shifts a chosen standing wave in space is a block of potential barrier and well for every wave bump between neighbouring knots. The algorithms shifting the range of the primary localization…
Using a scheme involving a lifting of a row contraction we introduce a toy model of repeated interactions between quantum systems. In this model there is an outgoing Cuntz scattering system involving two wandering subspaces. We associate to…
We develop a new mathematical approach to diffeomorphism invariant quantum states for the quantisation of general field theories such as general relativity and modified gravity. Treating quantum fields as fibre bundles, we discuss operators…
We introduce the transition-density formalism, an efficient and general method for calculating the interaction of external probes with light nuclei. One- and two-body transition densities that encode the nuclear structure of the target are…
The multichannel scattering problem for the stationary Schr\"{o}dinger equation on a line with different thresholds at both infinities is investigated. The analytical structure of the Jost solutions and of the transition matrix relating the…
We introduce a new exactly solvable model in quantum mechanics that describes the propagation of particles through a potential field created by regularly spaced $\delta'$-type point interactions, which model the localized dipoles often…
It has been shown that quantum paradoxes have followed from one special assumption, i.e., from attributing basic physical meaning to Hamiltonian eigenfunctions and representing all physical states by vectors of the Hilbert space spanned on…
We discuss some basic properties of Lie group representations in rigged Hilbert spaces. In particular, we show that a differentiable representation in a rigged Hilbert space may be obtained as the projective limit of a family of continuous…
We prove that the polynomial form of the scattering equations is a Macaulay H-basis. We demonstrate that this H-basis facilitates integrand reduction and global residue computations in a way very similar to using a Gr\"obner basis, but…
Collective coherent scattering of laser light by an ensemble of polarizable point particles creates long range interactions, whose properties can be tailored by choice of injected laser powers, frequencies and polarizations. We use a…
We show that under natural and quite general assumptions, a large part of a matrix for a bounded linear operator on a Hilbert space can be preassigned. The result is obtained in a more general setting of operator tuples leading to…
We give an operator theoretic approach to the constructions of multiresolutions as they are used in a number of basis constructions with wavelets, and in Hilbert spaces on fractals. Our approach starts with the following version of the…
We investigate scattering phenomena in a one-dimensional attractive Bose-Hubbard model with a time-periodically modulated impurity. We analyze both single-particle and pair (doublon) transmission, exploring a range of interaction strengths…
Due to spectral obstructions, a scattering theory in the Lax-Phillips sense for the wave equation for differential p-forms on H^{n+1} cannot be developed. As a consequence, Huygens' principle for the wave equation in this context does not…
The scattering transform is a wavelet-based model of Convolutional Neural Networks originally introduced by S. Mallat. Mallat's analysis shows that this network has desirable stability and invariance guarantees and therefore helps explain…