اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدA unified approach to Schrodinger evolution of superoscillations and supershifts
100
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
Superoscillating functions and supershifts appear naturally in weak measurements in physics. Their evolution as initial conditions in the time dependent Schrodinger equation is an important and challenging problem in quantum mechanics and mathematical analysis. The concept that encodes the persistence of superoscillations during the evolution is the (more general) supershift property of the solution. In this paper we give a unified approach to determine the supershift property for the solution of the time dependent Schrodinger equation. The main advantage and novelty of our results is that they only require suitable estimates and regularity assumptions on the Greens function, but not its explicit form. With this efficient general technique we are able to treat various potentials.
قيم البحث
اقرأ أيضاً
In this paper we study the time dependent Schrodinger equation with all possible self-adjoint singular interactions located at the origin, which include the $delta$ and $delta$-potentials as well as boundary conditions of Dirichlet, Neumann, and Robi
n type as particular cases. We derive an explicit representation of the time dependent Greens function and give a mathematical rigorous meaning to the corresponding integral for holomorphic initial conditions, using Fresnel integrals. Superoscillatory functions appear in the context of weak measurements in quantum mechanics and are naturally treated as holomorphic entire functions. As an application of the Greens function we study the stability and oscillatory properties of the solution of the Schrodinger equation subject to a generalized point interaction when the initial datum is a superoscillatory function.
We propose a unified method for the large space-time scaling limit of emph{linear} collisional kinetic equations in the whole space. The limit is of emph{fractional} diffusion type for heavy tail equilibria with slow enough decay, and of diffusive ty
pe otherwise. The proof is constructive and the fractional/standard diffusion matrix is obtained. The equilibria satisfy a {em generalised} weighted mass condition and can have infinite mass. The method combines energy estimates and quantitative spectral methods to construct a `fluid mode. The method is applied to scattering models (without assuming detailed balance conditions), Fokker-Planck operators and L{e}vy-Fokker-Planck operators. It proves a series of new results, including the fractional diffusive limit for Fokker-Planck operators in any dimension, for which the characterization of the diffusion coefficient was not known, for L{e}vy-Fokker-Planck operators with general equilibria, and in cases where the equilibrium has infinite mass. It also unifies and generalises the results of ten previous papers with a quantitative method, and our estimates on the fluid approximation error seem novel in these cases.
In this survey, our aim is to emphasize the main known limitations to the use of Wigner measures for Schrodinger equations. After a short review of successful applications of Wigner measures to study the semi-classical limit of solutions to Schroding
er equations, we list some examples where Wigner measures cannot be a good tool to describe high frequency limits. Typically, the Wigner measures may not capture effects which are not negligible at the pointwise level, or the propagation of Wigner measures may be an ill-posed problem. In the latter situation, two families of functions may have the same Wigner measures at some initial time, but different Wigner measures for a larger time. In the case of systems, this difficulty can partially be avoided by considering more refined Wigner measures such as two-scale Wigner measures; however, we give examples of situations where this quadratic approach fails.
In this paper, the different operator forms of classical Yang-Baxter equation are given in the tensor expression through a unified algebraic method. It is closely related to left-symmetric algebras which play an important role in many fields in mathe
matics and mathematical physics. By studying the relations between left-symmetric algebras and classical Yang-Baxter equation, we can construct left-symmetric algebras from certain classical r-matrices and conversely, there is a natural classical r-matrix constructed from a left-symmetric algebra which corresponds to a parakahler structure in geometry. Moreover, the former in a special case gives an algebraic interpretation of the ``left-symmetry as a Lie bracket ``left-twisted by a classical r-matrix.
The aim of this paper is to study, in dimensions 2 and 3, the pure-power non-linear Schrodinger equation with an external uniform magnetic field included. In particular, we derive a general criteria on the initial data and the power of the non-linear
ity so that the corresponding solution blows up in finite time, and we show that the time for blow up to occur decreases as the strength of the magnetic field increases. In addition, we also discuss some observations about Strichartz estimates in 2 dimensions for the Mehler kernel, as well as similar blow-up results for the non-linear Pauli equation.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
