Uniqueness properties for discrete equations and Carleman estimates

Using Carleman estimates, we give a lower bound for solutions to the discrete Schrödinger equation in both dynamic and stationary settings that allows us to prove uniqueness results, under some assumptions on the decay of the solutions

A dynamical uncertainty principle in von Neumann algebras by operator monotone functions

Suppose that A(1),..., A(N) are observables (selfadjoint matrices) and rho is a state (density matrix). In this case the standard uncertainty principle, proved by Robertson, gives a bound for the quantum generalized variance, namely for det{Cov(rho) (A(j), A(k) )}, using the commutators [A(j), A(k)]; this bound is trivial when N is odd. Recently a different inequality of Robertson-type has been proved by the authors with the help of the theory of operator monotone functions. In this case the bound makes use of the commutators [rho, A(j)] and is non-trivial for any N. In the present paper we generalize this new result to the von Neumann algebra case. Nevertheless the proof appears to simplify all the existing ones

Some lower bounds for solutions of Schrodinger evolutions

We present some lower bounds for regular solutions of Schr odinger equations with bounded and time dependent complex potentials. Assuming that the solution has some positive mass at time zero within a ball of certain radius, we prove that this mass can be observed if one looks at the solution and its gradient in space-time regions outside of that ball

The Schrödinger Equaton and Uncertainty Principles

101 p.The main task of this thesis is the analysis of the initial data u0 of Schrödinger’s initial value problem in order to determine certain properties of its dynamical evolution. First, we consider the elliptic Schrödinger problem in its perturbative form. The idea is to find lower bounds for the solution giving conditions at time t = 0 together with a size condition on the potential. After analyzing the elliptic case, we give a similar result for the hyperbolic Schrödinger operator. Next, we focus on the free particle case; this is the case where no potential is involved. The goal here is to quantify the L2 norm of the solution in a space-time cylinder. Following the same idea as before we want to find conditions at time t = 0 to ensure this. To carry out this task we define the Σδ space where δ is a parameter on the interval (0, 1]. We see that if belongs in this space then so does its evolution in time and use this fact to give lower bounds for the L2 norm of the solution. For δ = 1 we give a different approach and make use of the Virial Theorem. We will see that this case has particular properties. Finally, we study dynamical uncertainty principles derived from the previous study. The key point will be to write the solution as u = ρeiθ, where ρ and θ are real functions. Thus, we give uncertainty principles in terms of these functions and find explicit expressions for them so that u becomes a minimizer of the problem

Uncertainty principles for the Schrödinger equation on Riemannian symmetric spaces of the noncompact type

International audienceLet X be a Riemannian symmetric space of the noncompact type. We prove that the solution of the time-dependent Schrödinger equation on X with square integrable initial condition f is identically zero at all times t whenever f and the solution at a time t 0 >0 are simultaneously very rapidly decreasing. The stated condition of rapid decrease is of Beurling type. Conditions respectively of Gelfand-Shilov, Cowling-Price and Hardy type are deduced

The Schrödinger Equaton and Uncertainty Principles

101 p.The main task of this thesis is the analysis of the initial data u0 of Schrödinger’s initial value problem in order to determine certain properties of its dynamical evolution. First, we consider the elliptic Schrödinger problem in its perturbative form. The idea is to find lower bounds for the solution giving conditions at time t = 0 together with a size condition on the potential. After analyzing the elliptic case, we give a similar result for the hyperbolic Schrödinger operator. Next, we focus on the free particle case; this is the case where no potential is involved. The goal here is to quantify the L2 norm of the solution in a space-time cylinder. Following the same idea as before we want to find conditions at time t = 0 to ensure this. To carry out this task we define the Σδ space where δ is a parameter on the interval (0, 1]. We see that if belongs in this space then so does its evolution in time and use this fact to give lower bounds for the L2 norm of the solution. For δ = 1 we give a different approach and make use of the Virial Theorem. We will see that this case has particular properties. Finally, we study dynamical uncertainty principles derived from the previous study. The key point will be to write the solution as u = ρeiθ, where ρ and θ are real functions. Thus, we give uncertainty principles in terms of these functions and find explicit expressions for them so that u becomes a minimizer of the problem

The Schrödinger equation and Uncertainty Principles

The main task of this thesis is the analysis of the initial data u0 of Schrödinger’s initial value problem in order to determine certain properties of its dynamical evolution. First we consider the elliptic Schrödinger problem in its perturbative form. The idea is to find lower bounds for the solution giving conditions at time t = 0 together with a size condition on the potential. After analyzing the elliptic case we give a similar result for the hyperbolic Schrödinger operator. Next we focus on the free particle case, this is, the case where no potential is involved. The goal here is to quantify the L2 norm of the solution in a space-time cylinder. Following the same idea as before we want to find conditions at time t = 0 to ensure this. To carry out this task we define the Σδ space where δ is a parameter on the interval (0,1]. We see that if u0 belongs in this space then so does its evolution in time and use this fact to give lower bounds for the L2 norm of the solution. For δ = 1 we give a different approach and make use of the Virial Theorem. We will see that this case has particular properties. Finally we study dynamical uncertainty principles derived from the previous study. The key point will be to write the solution as u = ρeiθ, where ρ and θ are real functions. Thus we give uncertainty principles in terms of these functions and find explicit expressions for them so that u becomes a minimizer of the problem

Monod and the spirit of molecular biology

International audienceThe founders of molecular biology shared views on the place of biology within science, as well as on the relations of molecular biology to Darwinism. Jacques Monod was no exception, but the study of his writings is particularly interesting because he expressed his point of view very clearly and pushed the implications of some of his choices further than most of his contemporaries. The spirit of molecular biology is no longer the same as in the 1960s but, interestingly, Monod anticipated some recent evolutions of this discipline