Narrow Results

Two non-commutative dynamical entropies are studied in connection with the classical limit. For systems with a strongly chaotic classical limit, the Kolmogorov–Sinai invariant is recovered on time scales that are logarithmic in the quantization parameter. The model of the quantized hyperbolic automorphisms of the 2-torus is examined in detail.

In the limit $\hslash \to 0$, we analyze a class of Schrödinger operators $H_{\hslash }={\hslash }^{2}L+\hslash W+V\cdot {\mathrm{\text{id}}}_{{ℰ}}$ acting on sections of a vector bundle ${ℰ}$ over a Riemannian manifold $M$ where $L$ is a Laplace type operator, $W$ is an endomorphism field and the potential energy $V$ has a non-degenerate minimum at some point $p\in M$. We construct quasimodes of WKB-type near $p$ for eigenfunctions associated with the low-lying eigenvalues of $H_{\hslash }$. These are obtained from eigenfunctions of the associated harmonic oscillator $H_{p,\hslash }$ at $p$, acting on smooth functions on the tangent space.

We give a method to obtain estimates on distorted resolvents which we apply to the Stark effect with general potentials. This paper also contains a geometric perturbation theory which together with the estimates mentioned above, shows the existence of resonances and gives an upper bound on the width of resonances in various situations.

We consider the semi-classical limit of nonlinear Schrödinger equations in the presence of both a polynomial nonlinearity and the derivative in space of a polynomial nonlinearity. By working in a class of analytic initial data, we do not have to assume any hyperbolic structure on the (limiting) phase/amplitude system. The solution, its approximation, and the error estimates are considered in time-dependent analytic regularity.

In this article we study standing wave solutions for the nonlinear Schrödinger equation, which correspond to solutions of the equation

We consider the standing wave solutions of the three dimensional semilinear Schrödinger equation with competing potential functions V and K and under the action of an external electromagnetic field B. We establish some necessary conditions for a sequence of such solutions to concentrate, in two different senses, around a given point. In the particular but important case of nonlinearities of power type, the spikes locate at the critical points of a smooth ground energy map independent of B.

We justify the WKB analysis for generalized nonlinear Schrödinger equations (NLS), including the hyperbolic NLS and the Davey–Stewartson II system. Since the leading order system in this analysis is not hyperbolic, we work with analytic regularity, with a radius of analyticity decaying with time, in order to obtain better energy estimates. This provides qualitative information regarding equations for which global well-posedness in Sobolev spaces is widely open.

We consider a semi-classical nonlinear Schrödinger equation. For initial data causing focusing at one point in the linear case, we study a nonlinearity which is super-critical in terms of asymptotic effects near the caustic. We prove the existence of infinitely many phase shifts appearing at the approach of the critical time. This phenomenon is suggested by a formal computation. The rigorous proof shows a quantitatively different asymptotic behavior. We explain these aspects, and discuss some problems left open.

This paper deals with an example of a semi-classical consideration for heat equations. It shows that some geometric theorems are corollaries of such a semi-classical consideration.