arXiv subjects

L. Sharma

Publications and source records attributed to L. Sharma.

5 recordsLinked to original sources

arXiv2026

Normalized solutions to an exponential growth Choquard equation driven by mixed local-nonlocal operator in $\mathbb{R}^2$

In this paper, we study the existence of normalized solutions to the mixed nonlinear Choquard equation with exponential growth \begin{equation} \left\{ \mathcal{L}u+λu = Λ(I_α*F(u))f(u) \text{ in }\mathbb{R}^{2}; \;\; \int_{\mathbb{R}^{2}}|u|^{2}\,dx = a^{2}, \right. \end{equation} where $\mathcal{L}=-Δ+(-Δ)^s$, $0 0$, $I_α$ is the Riesz potential of order $α\in(0,2)$, $Λ>0$, and $λ\in\mathbb{R}$ arises as a Lagrange multiplier. The nonlinearity has critical exponential growth in the sense of the Trudinger-Moser inequality. Using variational methods, the associated Pohožaev manifold, and suitable Moser-type estimates, we establish the existence of a normalized solution of mountain-pass type with associated non zero Lagrange multiplier. Under an additional monotonicity assumption on the nonlinearity, we further show that the obtained solution is a normalized ground state. We also establish the regularity properties required for the proof of the corresponding Pohožaev identity.

math.AP↗

arXiv2026

Critical Sobolev Inequalities and Extremals for a Partially Fractional Grushin Operator

We study the partially fractional Grushin-type operator \begin{equation} \mathcal L_{α,β}u=(-Δ_x)^αu-(1+β)^2|x|^{2β}Δ_y u, \qquad (x,y)\in\mathbb{R}^n\times\mathbb{R}^m,\quad 0<α<1,\ β\ge 0, \end{equation} which couples fractional diffusion in $x$ with degenerate local diffusion in $y$. Working with the anisotropic dilations $(x,y)\mapsto(ρx,ρ^{α+β}y)$ and an adapted quasi-distance, we prove a uniform fractional Grushin-Poincaré inequality on anisotropic cylinders and, via a Riesz-type potential built from a telescoping sum over dyadic cylinders, a global Sobolev inequality at the sharp critical exponent $2^*_{α,β}=2Q_{α,β}/(Q_{α,β}-2α)$, where $Q_{α,β}=n+(α+β)m$. We then establish a concentration-compactness principle for the associated energy space. For $β>0$ the weight $|x|^{2β}$ destroys translation invariance in $x$, and we show that concentration is confined to the degenerate hyperplane $ Σ= \{(0,y):y\in\mathbb{R}^m\}$; this localization yields the existence of extremals for the sharp constant $S_{α,β}(n,m)$.

math.AP↗

arXiv2024

Inelastic scattering of vortex electrons beyond the Born approximation

We present a theoretical study of the inelastic scattering of vortex electrons by a hydrogen atom. In our study, special emphasis is placed on the effects of the Coulomb interaction between a projectile electron and a target atom. To understand these effects, we construct vortex electron wave functions both from free space and distorted solutions of the Schrödinger equation. These wave functions give rise to the first Born and distorted wave scattering amplitudes, respectively. The derived theory has been employed to investigate the $1s \rightarrow 2p$ transition of a hydrogen atom induced by electrons with the kinetic energies in the range from 20 to 100 eV. The results of the calculations have clearly indicated that the Coulomb interaction can significantly affect the phase pattern and probability density of a vortex electron beam as well as the squared transition amplitudes. For the latter, the most pronounced effect was found for the excitation to the $\ket{2p\, m_f=0}$ sublevel and large scattering angles.

physics.atom-ph↗

arXiv2018

Comprehensive Study of Properties of a Endohedrally Confined Ca Atom using Relativistic Many-body Methods

We have carried out theoretical investigations of electron correlation effects on the atomic properties of the Ca atom trapped inside an attractive spherically symmetric potential well of an endohedral fullerene C$_{60}$ cluster. Relativistic coupled-cluster (RCC) theory has been employed to obtain electron correlation energy, ionization potential and dipole polarizability of this atom. We have also performed calculations using the Dirac-Hartree-Fock (DF), relativistic second-order many-body perturbation theory (RMBPT(2) method) and relativistic random phase approximation (RRPA) to demonstrate propagation of the correlation effects in these properties. Our results are compared with the reported calculations employing multi-configuration Hartree-Fock (MCHF) method in Phys. Rev. A {\bf 87}, 013409 (2016). We found trends in correlation energy with respect to the potential depth are same, but magnitudes are very large in the relativistic calculations. We have also determined the differential and total cross-sections for elastic scattering of electrons from the free and confined Ca atoms using the electronic charge densities from the Dirac-Hartree core-potential (DFCP) and RCC methods to demonstrate role of potential depth in these properties.

physics.atom-ph↗

arXiv2017

Application of Relativistic Coupled-cluster Theory to Electron Impact Excitations of Mg$^+$ in the Plasma Environment

A relativistic coupled-cluster (RCC) theory is implemented to study electron impact excitations of atomic species. As a test case, the electron impact excitations of the $3s ~ ^2S_{1/2} - 3p ~ ^2P_{1/2;3/2}$ resonance transitions are investigated in the singly charged magnesium (Mg$^+$) ion using this theory. Accuracies of wave functions of Mg$^+$ are justified by evaluating its attachment energies of the relevant states and compared with the experimental values. The continuum wave function of the projectile electron are obtained by solving Dirac equations assuming distortion potential as static potential of the ground state of Mg$^+$. Comparison of the calculated electron impact excitation differential and total cross-sections with the available measurements are found to be in very good agreements at various incident electron energies. Further, calculations are carried out in the plasma environment in the Debye Hückel model framework, which could be useful in the astrophysics. Influence of plasma strength on the cross-sections as well as linear polarization of the photon emission in the $3p ~ ^2P_{3/2} - 3s ~ ^2S_{1/2}$ transition is investigated for different incident electron energies.

physics.atom-ph↗