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The set of Multi-level Amplitude Damping (MAD) quantum channels is introduced as a generalization of the standard qubit Amplitude Damping Channel to quantum systems of finite dimension $d$. In the special case of $d=3$, by exploiting…
Unique transformation properties under the hyperspherical inversion of a partial differential equation describing a stationary scalar wave in an $N$-dimensional ($N\geqslant2$) Maxwell fish-eye medium are exploited to construct a closed…
Quantum theory is awash in multidimensional integrals that contain exponentials in the integration variables, their inverses, and inverse polynomials of those variables. The present paper introduces a means to reduce pairs of such integrals…
Integral representations are derived for the parabolic cylinder functions $U(a,x)$, $V(a,x)$ and $W(a,x)$ and their derivatives. The new integrals will be used in numerical algorithms based on quadrature. They follow from contour integrals…
The primary goal of this paper is to introduce and investigate generalized incomplete exponential functions with matrix parameters. Integral representation, differential formula, addition formula, multiplication formula, and recurrence…
A recent development in the theory of fractional differential equations with variable coefficients has been a method for obtaining an exact solution in the form of an infinite series involving nested fractional integral operators. This…
A new method is presented for obtaining indefinite integrals of common special functions. The approach is based on a Lagrangian formulation of the general homogeneous linear ordinary differential equation of second order. A general integral…
We suggest a closed form expression for the path integral of quantum transition amplitudes. We introduce a quantum action with renormalized parameters. We present numerical results for the $V \sim x^{4}$ potential. The renormalized action…
A generalization of the Catalan numbers is considered. New results include binomial identities, recursive relations and a close formula for the multivariate generating function. A simple expression for the Catalan determinant is derived.
We obtain simple formulas for the matrix elements of the resolvent operator (the Green's function) in any finite set of square integrable basis. These formulas are suitable for numerical computations whether the basis elements are…
We derive inclusion regions for the eigenvalues of matrix polynomials expressed in a general polynomial basis, which can lead to significantly better results than traditional bounds. We present several applications to engineering problems.
A general algebraic method of quantum corrections evaluations is presented. Quantum corrections to a few classical solutions of Landau-Ginzburg model (phi-in-quadro) are calculated in arbitrary dimensions. The Green function for heat…
A careful study of the classical/quantum connection with the aid of coherent states offers new insights into various technical problems. This analysis includes both canonical as well as closely related affine quantization procedures. The…
The quantum integer [n]_q is the polynomial 1 + q + q^2 + ... + q^{n-1}, and the sequence of polynomials { [n]_q }_{n=1}^{\infty} is a solution of the functional equation f_{mn}(q) = f_m(q)f_n(q^m). In this paper, semidirect products of…
This work is devoted to infinite-energy solutions of semi-linear wave equations in unbounded smooth domains of $\mathbb{R}^3$ with fractional damping of the form $(-\Delta_x+1)^\frac{1}{2}\partial_t u$. The work extends previously known…
We present simple and stable formulas for computing power (including absorbed/radiated, scattered and extinction power) in current-based volume integral equation formulations. The proposed formulas are given in terms of vector-matrix-vector…
In this work we calculate the exact Green's function for arbitrary rectangular potentials. Specifically we focus on Green's function for rectangular quantum wells enlarging the knowledge of exact solutions for Green's functions and also…
We describe some "unrestricted" algorithms which are useful for the computation of elementary and special functions when the precision required is not known in advance. Several general classes of algorithms are identified and illustrated by…
The study gives a brief overview of existing modifications of the method of functional separation of variables for nonlinear PDEs. It proposes a more general approach to the construction of exact solutions to nonlinear equations of applied…
This paper deals with a general class of integrals, the particular cases of which are connected to outstanding problems in astronomy and physics. Reaction rate probability integrals in the theory of nuclear reaction rates, Kr\"atzel…