SUMMARYWe consider parabolic Dirac operators which do not involve fractional derivatives and use them to show the solvability of the in-stationary Navier-Stokes equations over time-varying domains.
SUMMARYIn this paper we prove a L p -decomposition where one of the components is the kernel of a first-order differential operator that factorizes the non-stationary Schrödinger operator − −i* t .
We study the boundary behavior of discrete monogenic functions, i.e. null-solutions of a discrete Dirac operator, in the upper and lower half space. Calculating the Fourier symbol of the boundary operator we construct the corresponding discrete Hilbert transforms, the projection operators arising from them, and discuss the notion of discrete Hardy spaces. Hereby, we focus on the 3D-case with the generalization to the n-dimensional case being straightforward.
The aim of this work is to study the numerical solution of the nonlinear Schrödinger problem using a combination between Witt basis and finite difference approximations. We construct a discrete fundamental solution for the nonstationary Schrödinger operator and we show the convergence of the numerical scheme. Numerical examples are given at the end of the article.
The theory of quaternionic operators has applications in several different fields such as quantum mechanics, fractional evolution problems, and quaternionic Schur analysis, just to name a few. The main difference between complex and quaternionic operator theory is based on the definition of spectrum. In fact, in quaternionic operator theory the classical notion of resolvent operator and the one of spectrum need to be replaced by the two Sresolvent operators and the S-spectrum. This is a consequence of the non-commutativity of the quaternionic setting. Indeed, the S-spectrum of a quaternionic linear operator T is given by the non invertibility of a second order operator. This presents new challenges which makes our approach to perturbation theory of quaternionic operators different from the classical case. In this paper we study the problem of perturbation of a quaternionic normal operator in a Hilbert space by making use of the concepts of S-spectrum and of slice hyperholomorphicity of the S-resolvent operators. For this new setting we prove results on the perturbation of quaternionic normal operators by operators belonging to a Schatten class and give conditions which guarantee the existence of a nontrivial hyperinvariant subspace of a quaternionic linear operator.
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