Related papers: Strong and weak semiclassical limits for some roug…
In this paper we obtain a stabilization result for the Schr\"odinger equation under generic assumptions on the potential. Then we consider the Schr\"odinger equation with a potential which has a random time-dependent amplitude. We show that…
In this note, we prove weighted resolvent estimates for the semiclassical Schr\"odinger operator $-h^2 \Delta + V(x) : L^2(\mathbb{R}^n) \to L^2(\mathbb{R}^n)$, $n \neq 2$. The potential $V$ is real-valued, and assumed to either decay at…
In this paper we investigate the well-posedness of the Cauchy problem for a Schr\"odinger operator with singular lower order terms. We allow distributional coefficients and we approach this problem via the regularising methods at the core…
Semiclassical transition probabilities characterize transfer of energy between "hard" and "soft" modes in various physical systems. We establish the boundary problem for singular euclidean solutions used to calculate such probabilities.…
We study a class of delta-like perturbations of the Laplacian on the half-line, characterized by Robin boundary conditions at the origin. Using the formalism of nonstandard analysis, we derive a simple connection with a suitable family of…
In this paper, we study the semiclassical limit for the stationary magnetic nonlinear Schr\"odinger equation \begin{align}\label{eq:initialabstract}\left( i \hbar \nabla + A(x) \right)^2 u + V(x) u = |u|^{p-2} u, \quad x\in…
We study the stability of the Schr\"odinger equation generated by time-dependent Hamiltonians with constant form domain. That is, we bound the difference between solutions of the Schr\"odinger equation by the difference of their…
In this paper, we study an inverse scattering problem associated with the time-harmonic Schr\"odinger equation where both the potential and the source terms are unknown. The source term is assumed to be a generalised Gaussian random…
An alternative method is proposed for deriving the time dependent Schroedinger equation from the pictures of wave and matrix mechanics. The derivation is of a mixed classical quantum character, since time is treated as a classical variable,…
We establish uniform bounds for the solutions $e^{it\Delta}u$ of the Schr\"{o}dinger equation on arithmetic flat tori, generalising earlier results by J. Bourgain. We also study the regularity properties of weak-* limits of sequences of…
We discuss some subtleties which arise in the semiclassical approximation to quantum gravity. We show that integrability conditions prevent the existence of Tomonaga-Schwinger time functions on the space of three-metrics but admit them on…
We revisit a recent proposal for a definition of time in quantum cosmology, to investigate the effects of having more than one possible type of clock "at the same time". We use as test tube an extension of Einstein gravity with a massless…
We attack the specific time-dependent Hamiltonian problem H=-{1/2} (t_o/t)^a \partial_{xx} + (1/2) \omega^2 (t/t_o)^b x^2. This corresponds to a time-dependent mass (TM) Schr\"odinger equation. We give the specific transformations to a…
Problems posed by semirelativistic Hamiltonians of the form H = sqrt{m^2+p^2} + V(r) are studied. It is shown that energy upper bounds can be constructed in terms of certain related Schroedinger operators; these bounds include free…
Aharonov-Albert-Vaidman's weak values are investigated by a semiclassical method. Examples of the semiclassical calculation that reproduces "anomalous" weak values are shown. Furthermore, a complex extension of Ehrenfest's quantum-classical…
We study how the singular behaviour of classical systems at bifurcations is reflected by their quantum counterpart. The semiclassical contributions of individual periodic orbits to trace formulae of Gutzwiller type are known to diverge when…
It is shown how the essentials of quantum theory, i.e., the Schroedinger equation and the Heisenberg uncertainty relations, can be derived from classical physics. Next to the empirically grounded quantisation of energy and momentum, the…
Constrained Hamiltonian description of the classical limit is utilized in order to derive consistent dynamical equations for hybrid quantum-classical systems. Starting with a compound quantum system in the Hamiltonian formulation conditions…
In this article we prove a reducibility result for the linear Schr\"odinger equation on a Zoll manifold with quasi-periodic in time pseudo-differential perturbation of order less or equal than $1/2$. As far as we know, this is the first…
We consider the spatially inhomogeneous non-cutoff Boltzmann equation with hard potentials in the non-perturbative setting. For initial data with polynomial decay in the velocity variable, we establish the local-in-time existence and…