arXiv subjects
Publications and source records attributed to Vitali Vougalter.
In the article we establish the global well-posedness in W^{1, 2, 2}(R\times R^{+}) of the integro-differential equation in the case of the anomalous diffusion when the one dimensional negative Laplace operator is raised to a fractional power in the presence of the transport term. The model is relevant to the cell population dynamics in the Mathematical Biology. Our proof relies on a fixed point technique.
The article is devoted to the existence of solutions of a certain system of quadratic integral equations in H^1(R, R^N). We show the existence of a perturbed solution by using a fixed point technique in the Sobolev space on the real line.
We prove the existence of solutions for some integro-differential systems containing equations with and without the drift terms in the H^2 spaces by virtue of the fixed point technique when the elliptic equations contain second order differential operators with and without the Fredholm property, on the whole real line or on a finite interval with periodic boundary conditions. Let us emphasize that the study of the systems case is more complicated than of the scalar situation and requires to overcome more cumbersome technicalities.
We show global well-posedness in energy norm of the semi-relativistic Schrödinger-Poisson system of equations with attractive Coulomb interaction in ${\mathbb R}^3$ in the presence of pseudo-relativistic diffusion. We also discuss sufficient conditions to have well-posedness in ${\mathcal L}^2$. In the absence of dissipation, we show that the solution corresponding to an initial condition with negative energy blows up in finite time, which is as expected, since the nonlinearity is critical.
The Cauchy problem for the semi-relativistic Schrödinger-Poisson system of equations is studied in ${\mathbb R}^n$, $n\geq 1$, for a wide class of nonlocal interactions. Furthermore, the asymptotic behavior of the solution as the mass tends to infinity is rigorously discussed, which corresponds to a non-relativistic limit.
We study the stationary states of the semi-relativistic Schrödinger-Poisson system in the repulsive (plasma physics) Coulomb case. In particular, we establish the existence and the nonlinear stability of a wide class of stationary states by means of the energy-Casimir method. We generalize the global well-posedness result of our previous work for the semi-relativistic Schrödinger-Poisson system to spaces with higher regularity.
We show global existence and uniqueness of strong solutions for the Schrodinger-Poisson system in the repulsive Coulomb case with relativistic kinetic energy.
In this paper, we determine the exact expression for the hydrogen binding energy in the Pauli-Fierz model up to the order $O(α^5\logα^{-1})$, where $α$ denotes the finestructure constant, and prove rigorous bounds on the remainder term of the order $o(α^5\logα^{-1})$. As a consequence, we prove that the binding energy is not a real analytic function of $α$, and verify the existence of logarithmic corrections to the expansion of the ground state energy in powers of $α$, as conjectured in the recent literature.
We obtain solvability conditions for some elliptic equations involving non Fredholm operators with the methods of spectral theory and scattering theory for Schrodinger type operators. Though the Fredholm property is not satisfied, the solvability conditions are formulated in terms of orthogonality of the right-hand side to solutions of the homogeneous adjoint equation.
In this note, we determine the ground state energy of the translation invariant Pauli-Fierz model to subleading order $O(α^3)$ with respect to powers of the finestructure constant $α$, and prove rigorous error bounds of order $O(α^{4})$. A main objective of our argument is its brevity.
We consider a spinless particle coupled to a photon field and prove that even if the Schrödinger operator $p^2 + V$ does not have eigenvalues the system can have a ground state. We describe the coupling by means of the Pauli-Fierz Hamiltonian and our result holds in the case where the coupling constant $α$ is small.
We consider a Pauli-Fierz Hamiltonian for a particle coupled to a photon field. We discuss the effects of the increase of the binding energy and enhanced binding through coupling to a photon field, and prove that both effects are the results of the existence of the ground state of the self-energy operator with total momentum $P = 0$.
We define the two dimensional Pauli operator and identify its core for magnetic fields that are regular Borel measures. The magnetic field is generated by a scalar potential hence we bypass the usual $\bA\in L^2_{loc}$ condition on the vector potential which does not allow to consider such singular fields. We extend the Aharonov-Casher theorem for magnetic fields that are measures with finite total variation and we present a counterexample in case of infinite total variation. One of the key technical tools is a weighted $L^2$ estimate on a singular integral operator.