Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userDirect methods for pseudo-relativistic Schr{o}dinger operators
140
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
In this paper, we establish various maximal principles and develop the direct moving planes and sliding methods for equations involving the physically interesting (nonlocal) pseudo-relativistic Schr{o}dinger operators $(-Delta+m^{2})^{s}$ with $sin(0,1)$ and mass $m>0$. As a consequence, we also derive multiple applications of these direct methods. For instance, we prove monotonicity, symmetry and uniqueness results for solutions to various equations involving the operators $(-Delta+m^{2})^{s}$ in bounded domains, epigraph or $mathbb{R}^{N}$, including pseudo-relativistic Schrodinger equations, 3D boson star equations and the equations with De Giorgi type nonlinearities.
rate research
Read More
This article is devoted to the construction of numerical methods which remain insensitive to the smallness of the semiclassical parameter for the linear Schr{o}dinger equation in the semiclassical limit. We specifically analyse the convergence behavior of the first-order splitting. Our main result is a proof of uniform accuracy. We illustrate the properties of our methods with simulations.
We study the inverse scattering for Schr{o}dinger operators on locally perturbed periodic lattices. We show that the associated scattering matrix is equivalent to the Dirichlet-to-Neumann map for a boundary value problem on a finite part of the graph, and reconstruct scalar potentials as well as the graph structure from the knowledge of the S-matrix. In particular, we give a procedure for probing defects in hexagonal lattices (graphene).
We consider the large time behavior in two types of equations, posed on the whole space R^d: the Schr{o}dinger equation with a logarithmic nonlinearity on the one hand; compressible, isothermal, Euler, Korteweg and quantum Navier-Stokes equations on the other hand. We explain some connections between the two families of equations, and show how these connections may help having an insight in all cases. We insist on some specific aspects only, and refer to the cited articles for more details, and more complete statements. We try to give a general picture of the results, and present some heuristical arguments that can help the intuition, which are not necessarily found in the mentioned articles.
We analyze dynamical properties of the logarithmic Schr{o}dinger equation under a quadratic potential. The sign of the nonlinearity is such that it is known that in the absence of external potential, every solution is dispersive, with a universal asymptotic profile. The introduction of a harmonic potential generates solitary waves, corresponding to generalized Gaussons. We prove that they are orbitally stable, using an inequality related to relative entropy, which may be thought of as dual to the classical logarithmic Sobolev inequality. In the case of a partial confinement, we show a universal dispersive behavior for suitable marginals. For repulsive harmonic potentials, the dispersive rate is dictated by the potential, and no universal behavior must be expected.
In this paper, we introduce two new families of generalised Hermite polynomials/functions (GHPs/GHFs) in arbitrary dimensions, and develop efficient and accurate generalised Hermite spectral algorithms for PDEs with integral fractional Laplacian (IFL) and/or Schr{o}dinger operators in $mathbb R^d.$ As a generalisation of the G. Szeg{o}s family in 1D (1939), the first family of GHPs (resp. GHFs) are orthogonal with respect to $|bx|^{2mu} e^{-|bx|^2}$ (resp. $|bx |^{2mu}$) in $mathbb R^d$. We further define adjoint generalised Hermite functions (A-GHFs) which have an interwoven connection with the corresponding GHFs through the Fourier transform, and which are orthogonal with respect to the inner product $[u,v]_{H^s(mathbb R^d)}=((-Delta)^{s/ 2}u, (-Delta)^{s/2} v )_{mathbb R^d}$ associated with the IFL of order $s>0$. Thus, the spectral-Galerkin method using A-GHFs as basis functions leads to a diagonal stiffness matrix for the IFL (which is known to be notoriously difficult and expensive to discretise). The new basis also finds efficient and accurate in solving PDEs with the fractional Schr{o}dinger operator: $(-Delta)^s +|bs x|^{2mu}$ with $sin (0,1]$ and $mu>-1/2.$ Following the same spirit, we construct the second family of GHFs, dubbed as Muntz-type generalised Hermite functions (M-GHFs), which are orthogonal with respect to an inner product associated with the underlying Schr{o}dinger operator, and are tailored to the singularity of the solution at the origin. We demonstrate that the Muntz-type GHF spectral method leads to sparse matrices and spectrally accurate to some Schr{o}dinger eigenvalue problems.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
