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Semiclassical measures for the Schrödinger equation on the torus

Nalini Anantharaman, Fabricio Macià (2014)

Journal of the European Mathematical Society

In this article, the structure of semiclassical measures for solutions to the linear Schrödinger equation on the torus is analysed. We show that the disintegration of such a measure on every invariant lagrangian torus is absolutely continuous with respect to the Lebesgue measure. We obtain an expression of the Radon-Nikodym derivative in terms of the sequence of initial data and show that it satisfies an explicit propagation law. As a consequence, we also prove an observability inequality, saying...

Semiclassical spectral estimates for Toeplitz operators

David Borthwick, Thierry Paul, Alejandro Uribe (1998)

Annales de l'institut Fourier

Let X be a compact Kähler manifold with integral Kähler class and L → X a holomorphic Hermitian line bundle whose curvature is the symplectic form of X . Let H ∈ C ∞ ( X , ℝ ) be a Hamiltonian, and let T k be the Toeplitz operator with multiplier H acting on the space ℋ k = H 0 ( X , L ⊗ k ) . We obtain estimates on the eigenvalues and eigensections of T k as k → ∞ , in terms of the classical Hamilton flow of H . We study in some detail the case when X is an integral coadjoint orbit of a Lie group.

Semiclassical states for weakly coupled nonlinear Schrödinger systems

Eugenio Montefusco, Benedetta Pellacci, Marco Squassina (2008)

Journal of the European Mathematical Society

We consider systems of weakly coupled Schrödinger equations with nonconstant potentials and investigate the existence of nontrivial nonnegative solutions which concentrate around local minima of the potentials. We obtain sufficient and necessary conditions for a sequence of least energy solutions to concentrate.

Semiclassical states of nonlinear Schrödinger equations with bounded potentials

Antonio Ambrosetti, Marino Badiale, Silvia Cingolani (1996)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Using some perturbation results in critical point theory, we prove that a class of nonlinear Schrödinger equations possesses semiclassical states that concentrate near the critical points of the potential V .

Semiclassics of the quantum current in very strong magnetic fields

Soren Fournais (2002)

Annales de l’institut Fourier

We prove a formula for the current in an electron gas in a semiclassical limit corresponding to strong, constant, magnetic fields. Little regularity is assumed for the scalar potential V . In particular, the result can be applied to the mean field from magnetic Thomas-Fermi theory V MTF . The proof is based on an estimate on the density of states in the second Landau band.

Semi-infinite cohomology and superconformal algebras

Elena Poletaeva (2001)

Annales de l’institut Fourier

We describe representations of certain superconformal algebras in the semi-infinite Weil complex related to the loop algebra of a complex finite-dimensional Lie algebra and in the semi-infinite cohomology. We show that in the case where the Lie algebra is endowed with a non-degenerate invariant symmetric bilinear form, the relative semi-infinite cohomology of the loop algebra has a structure, which is analogous to the classical structure of the de Rham cohomology in Kähler...

Semilinear relations and *-representations of deformations of so(3)

Yuriĭ Samoĭlenko, Lyudmila Turowska (1997)

Banach Center Publications

We study a family of commuting selfadjoint operators = ( A k ) k = 1 n , which satisfy, together with the operators of the family = ( B j ) j = 1 n , semilinear relations ⅀ i f i j ( ) B j g i j ( ) = h ( ) , ( f i j , g i j , h j : ℝ n → ℂ are fixed Borel functions). The developed technique is used to investigate representations of deformations of the universal enveloping algebra U(so(3)), in particular, of some real forms of the Fairlie algebra U q ' ( s o ( 3 ) ) .

Shape optimization problems for metric graphs

Giuseppe Buttazzo, Berardo Ruffini, Bozhidar Velichkov (2014)

ESAIM: Control, Optimisation and Calculus of Variations

Γ):Γ ∈ 𝒜, ℋ1(Γ) = l}, where ℋ1D1,...,Dk }  ⊂ Rd . The cost functional ℰ(Γ) is the Dirichlet energy of Γ defined through the Sobolev functions on Γ vanishing on the points Di. We analyze the existence of a solution in both the families of connected sets and of metric graphs. At the end, several explicit examples are discussed.

Currently displaying 41 – 60 of 241