相关论文: Functional derivatives, Schr\"{o}dinger equations,…
Formulas for Galerkin-Koornwinder (GK) approximations of delay differential equations are summarized. The functional analysis ingredients (semigroups, operators, etc.) are intentionally omitted to focus instead on the formulas required to…
A method for obtaining discretization formulas for the derivatives of a function is presented, which relies on a generalization of divided differences. These modified divided differences essentially correspond to a change of the dependent…
Considering fractional fast diffusion equations on bounded open polyhedral domains in $\mathbb{R}^N$, we give a fully Galerkin approximation of the solutions by $C^0$-piecewise linear finite elements in space and backward Euler…
We introduce the class of semiweak Cullen-regular quaternionic functions by interpreting Cullen-regular functions as solutions to an inhomogeneous PDE in terms of the Fueter operator.
In this paper, we introduce a new method for calculating fractional integrals and differentials. The method involves an equation that we have obtained from infinite applied integration by parts. The equation works for special class of…
Path integral representations for generalized Schr\"odinger operators obtained under a class of Bernstein functions of the Laplacian are established. The one-to-one correspondence of Bernstein functions with L\'evy subordinators is used,…
The Cauchy problem for fractional derivatives linear systems of ordinary differential equations with constant coefficients is considered, where at first the analytic expressions are given through the matrix exponent of its corresponding…
A set of functions is introduced which generalizes the famous Schur polynomials and their connection to Grasmannian manifolds. These functions are shown to provide a new method of constructing solutions to the KP hierarchy of nonlinear…
We review recent progress in the fractional Calder\'on problem, where one tries to determine an unknown coefficient in a fractional Schr\"odinger equation from exterior measurements of solutions. This equation enjoys remarkable uniqueness…
It is well known that certain fractional diffusion equations can be solved by the densities of stable L\'evy motions. In this paper we use the classical semigroup approach for L\'evy processes to define semi-fractional derivatives, which…
We investigate the existence of solutions to the fractional nonlinear Schr\"{o}dinger equation $(-\Delta)^s u = f(u)$ with prescribed $L^2$-norm $\int_{\mathbb{R}^N} |u|^2 \, dx =m$ in the Sobolev space $H^s(\mathbb{R}^N)$. Under fairly…
We explore the applicability of splitting methods involving complex coefficients to solve numerically the time-dependent Schr\"odinger equation. We prove that a particular class of integrators are conjugate to unitary methods for…
We discuss unique existence and microlocal regularity properties of Sobolev space solutions to the fractional Eringen wave equation, initially given in the form of a system of equations in which the classical non-local Eringen constitutive…
A relationship between the functional Schr\"odinger representation and the precanonical quantization of a scalar field theory is extended to an arbitrary curved space-time. The canonical functional derivative Schr\"odinger equation is…
In this paper we provide a different approach for existence of the variational solutions of the gradient flows associated to functionals on Sobolev spaces studied in \cite{BDDMS20}. The crucial condition is the convexity of the functional…
In the analysis of stochastic dynamical systems described by stochastic differential equations (SDEs), it is often of interest to analyse the sensitivity of the expected value of a functional of the solution of the SDE with respect to…
This article deals with the weak and strong unique continuation principle for fractional Schr\"odinger equations with scaling-critical and rough potentials via Carleman estimates. Our methods allow to apply the results to variable…
Transverse magnetic (TM) scattering of an electromagnetic wave from a periodic dielectric diffraction grating can mathematically be described by a volume integral equation. This volume integral equation, however, in general fails to feature…
We consider a partial data inverse problem with unbounded potentials. Rather than rely on functional analytic arguments or Carleman estimates, we construct an explicit Green's function with which we construct complex geometric optics (CGO)…
The Schr\"{o}dinger equation, in hyperspherical coordinates, is solved in closed form for a system of three particles on a line, interacting via pair delta functions. This is for the case of equal masses and potential strengths. The…