Growth of Sobolev norms in linear Schrödinger equations as a dispersive phenomenon

Growth of Sobolev norms of solutions of linear Schrödinger equations on some compact manifolds

We give a new proof of a theorem of Bourgain [4], asserting that solutions of linear Schrödinger equations on the torus, with smooth time dependent potential, have Sobolev norms growing at most like t when t→ +∞, for any > 0. Our proof extends to Schrödinger equations on other examples of compact riemannian manifolds.

Logarithmic Bounds on Sobolev Norms for Time Dependent Linear Schrödinger Equations

Partial Differential Equations Γ -convergence and Sobolev norms

We study a Γ -convergence problem related to a new characterization of Sobolev spaces W1,p(RN) (p > 1) established in H.-M. Nguyen [H.-M. Nguyen, Some new characterizations of Sobolev spaces, J. Funct. Anal. 237 (2006) 689–720] and J. Bourgain and H.-M. Nguyen [J. Bourgain, H.-M. Nguyen, A new characterization of Sobolev spaces, C. R. Acad. Sci. Paris, Ser. I 343 (2006) 75–80]. We can also hand...

Supersymmetry of Generalized Linear Schrödinger Equations in (1+1) Dimensions

We review recent results on how to extend the supersymmetry SUSY formalism in Quantum Mechanics to linear generalizations of the time-dependent Schrödinger equation in (1+1) dimensions. The class of equations we consider contains many known cases, such as the Schrödinger equation for position-dependent mass. By evaluating intertwining relations, we obtain explicit formulas for the interrelation...

Sobolev Norms of Automorphic Functionals

It is well known that Frobenius reciprocity is one of the central tools in the representation theory. In this paper, we discuss Frobenius reciprocity in the theory of automorphic functions. This Frobenius reciprocity was discovered by Gel’fand, Fomin, and PiatetskiShapiro in the 1960s as the basis of their interpretation of the classical theory of automorphic functions in terms of the represent...