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Register a new userObservability and Controllability for the Schr{o}dinger Equation on Quotients of Groups of Heisenberg Type
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We give necessary and sufficient conditions for the controllability of a Schrodinger equation involving the sub-Laplacian of a nilmanifold obtained by taking the quotient of a group of Heisenberg type by one of its discrete sub-groups.This class of nilpotent Lie groups is a major example of stratified Lie groups of step 2. The sub-Laplacian involved in these Schrodinger equations is subelliptic, and, contrarily to what happens for the usual elliptic Schrodinger equation for example on flat tori or on negatively curved manifolds, there exists a minimal time of controllability. The main tools used in the proofs are (operator-valued) semi-classical measures constructed by use of representation theory and a notion of semi-classical wave packets that we introduce here in the context of groups of Heisenberg type.
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In this survey paper, we report on recent works concerning exact observability (and, by duality, exact controllability) properties of subelliptic wave and Schr{o}dinger-type equations. These results illustrate the slowdown of propagation in directions transverse to the horizontal distribution. The proofs combine sub-Riemannian geometry, semi-classical analysis, spectral theory and non-commutative harmonic analysis.
This analysis is concerned with the controllability of quantum systems in the case where the standard dipolar approximation, involving the permanent dipole moment of the system, is corrected with a polarizability term, involving the field induced dipole moment. Sufficient conditions for approximate controllability are given. For transfers between eigenstates of the free Hamiltonian, the control laws are explicitly given. The results apply also for unbounded or non-regular potentials.
The blowup is studied for the nonlinear Schr{o}dinger equation $iu_{t}+Delta u+ |u|^{p-1}u=0$ with $p$ is odd and $pge 1+frac 4{N-2}$ (the energy-critical or energy-supercritical case). It is shown that the solution with negative energy $E(u_0)<0$ blows up in finite or infinite time. A new proof is also presented for the previous result in cite{HoRo2}, in which a similar result but more general in a case of energy-subcritical was shown.
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 investigate normalized solutions for the Schr{o}dinger equation with combined Hartree type and power nonlinearities, namely begin{equation*} left{ begin{array}{ll} -Delta u+lambda u=gamma (I_{alpha }ast leftvert urightvert ^{p})|u|^{p-2}u+mu |u|^{q-2}u & quad text{in}quad mathbb{R}^{N}, int_{mathbb{R}^{N}}|u|^{2}dx=c, & end{array}% right. end{equation*} where $Ngeq 2$ and $c>0$ is a given real number. Under different assumptions on $gamma ,mu ,p$ and $q$, we prove several nonexistence, existence and multiplicity results. In particular, we are more interested in the cases when the competing effect of Hartree type and power nonlinearities happens, i.e. $gamma mu <0,$ including the cases $gamma <0,mu >0$ and $% gamma >0,mu <0.$ Due to the different strength of two types of nonlinearities, we find some differences in results and in the geometry of the corresponding functionals between these two cases.
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