Vladimir Georgiev

Higher Order Fractional Leibniz Rule

The fractional Leibniz rule is generalized by the Coifman–Meyer estimate. It is shown that the arbitrary redistribution of fractional derivatives for higher order with the corresponding correction terms.

On the classification of the spectrally stable standing waves of the Hartree problem

Abstract We consider the fractional Hartree model, with general power non-linearity and arbitrary spatial dimension. We construct variationally the “normalized” solutions for the corresponding Choquard–Pekar model—in particular a number of key properties, like smoothness and bell-shapedness are established. As a consequence of the construction, we show that these solitons are spectrally stable as solutions to the time-dependent Hartree model. In addition, we analyze the spectral stability of the Moroz–Van Schaftingen solitons of the classical Hartree problem, in any dimensions and power non-linearity. A full classification is obtained, the main conclusion of which is that only and exactly the “normalized” solutions (which exist only in a portion of the range) are spectrally stable.

Zero resonances for localised potentials

This paper considers Hamiltonians with localised potentials and gives a variational characterisation of resonant coupling parameters, which allow to provide estimates for the first resonant parameter and in turn also to provide bounds for resonant free regions. As application we provide a constructive approach to calculate the first resonant parameter for Yukawa type potentials in

Existence and uniqueness of ground states for p-Choquard model

mathbb R^3

Dispersive Properties of Schrödinger Operators in the Absence of a Resonance at Zero Energy in 3D

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Long time dynamics for semi-relativistic NLS and half wave in arbitrary dimension

Abstract We study the p -Choquard equation in R n , n ≥ 3 and establish existence and uniqueness of ground states for the corresponding Weinstein functional. For proving the uniqueness of ground states, we use the radial symmetry to transform the equation into an ordinary differential system, and applying the Pohozaev identities and Gronwall lemma we show that any two Weinstein minimizers satisfying the p -Choquard equation coincide.

On fractional powers of singular perturbations of the Laplacian

In this paper we study spectral properties associated to the Schrodinger operator − Δ −Wwith potential W that is an exponentially decaying C 1 function. As applications we prove local energy decay for solutions to the perturbed wave equation and lack of resonances for the NLS.