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We will present versions of the Rellich-Kondrachov theorem for pseudo-differential operators acting on localizable Hardy spaces. One of the techniques includes boundedness properties for pseudodifferential operators with symbols in the…
We are looking at a Dirac electron in the electromagnetic field of a plane monochrome polarized X-ray. It will be attempted to link the terms of a certain (joint) asymptotic expansion of the Heisenberg propagations of momentum- and…
We introduce a new type of convergence in probability theory, which we call ``mod-Gaussian convergence''. It is directly inspired by theorems and conjectures, in random matrix theory and number theory, concerning moments of values of…
Fractional calculus provides a rigorous mathematical framework to describe anomalous stochastic processes by generalizing the notion of classical differential equations to their fractional-order counterparts. By introducing the fractional…
We consider non-self-adjoint electromagnetic Schr\"odinger operators on arbitrary open sets with complex scalar potentials whose real part is not necessarily bounded from below. Under a suitable sufficient condition on the electromagnetic…
We consider Schr\"odinger operators on a class of periodic quantum graphs with randomly distributed Kirchhoff coupling constants at all vertices. Using the technique of self-adjoint extensions we obtain conditions for localization on…
We introduce the quadratic Fermi algebra, which is a Lie algebra, and show that the vacuum distributions of the associated Hamiltonians define the fermionic Meixner probability distributions. In order to emphasize the difference with the…
In this paper, we present an approach for explicitly constructing quasi-periodic Schr\"odinger operators with Cantor spectrum with $C^k$ potential. Additionally, we provide polynomial asymptotics on the size of spectral gaps.
A family of probability distributions attached to a class of generalized weighted Bergman spaces on the Poincar\'e disk are introduced by constructing a kind of generalized coherent states. Their main statistical parameters are obtained…
The new numerical version of the Wigner approach to quantum mechanics for treatment thermodynamic properties of strongly coupled systems of particles has been developed for extreme conditions, when analytical approximations obtained in…
In this paper, we give a sharp sparse domination of pseudodifferential operators associated with symbols belonging to the H\"{o}rmander class, and fundamental solutions of dispersive equations. Furthermore, we give boundedness results of…
The real theory of the Dunkl operators has been developed very extensively, while there still lacks the corresponding complex theory. In this paper we introduce the complex Dunkl operators for certain Coxeter groups. These complex Dunkl…
In this paper we consider the space-fractional Schr\"odinger equation with a singular potential for a wide class of fractional hypoelliptic operators. Such analysis can be conveniently realised in the setting of graded Lie groups. The paper…
We consider physical interpretations of non-trivial boundary conditions of self-adjoint extensions for one-dimensional Schr\"odinger operator of free spinless particle. Despite its model and rather abstract character this question is worth…
Functionals of Brownian motion have diverse applications in physics, mathematics, and other fields. The probability density function (PDF) of Brownian functionals satisfies the Feynman-Kac formula, which is a Schrodinger equation in…
Using a Fourier spectral method, we provide a detailed numerically investigation of dispersive Schr\"odinger type equations involving a fractional Laplacian. By an appropriate choice of the dispersive exponent, both mass and energy sub- and…
We introduce a stochastic process and functional that should describe the semigroup generated by the stochastic Bessel operator. Recently Gorin and Shkolnikov showed that the largest eigenvalues for certain random matrix ensembles with soft…
We analyze quasi probability distributions in discrete phase space related to the discrete Heisenberg-Weyl group. In particular, we discuss the relation between the Discrete Wigner and Q- functions.
We present a new approach to absolute continuity of laws of Poisson functionals. The theoretical framework is that of local Dirichlet forms as a tool to study probability spaces. The method gives rise to a new explicit calculus that we show…
Weak measurements of photon position can be used to obtain direct experimental evidence of the wavefunction of a photon between generation and ultimate detection. Significantly, these measurement results can also be understood as complex…