Displaying similar documents to “Strichartz inequalities for the Schrödinger equation with the full Laplacian on the Heisenberg group”

The gaps in the spectrum of the Schrödinger operator

Haizhong Li, Linlin Su (2005)

Banach Center Publications

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We obtain inequalities between the eigenvalues of the Schrödinger operator on a compact domain Ω of a submanifold M in R N with boundary ∂Ω, which generalize many existing inequalities for the Laplacian on a bounded domain of a Euclidean space. We also establish similar inequalities for a closed minimal submanifold in the unit sphere, which generalize and improve Yang-Yau’s result.

On the boundary convergence of solutions to the Hermite-Schrödinger equation

Peter Sjögren, J. L. Torrea (2010)

Colloquium Mathematicae

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In the half-space d × , consider the Hermite-Schrödinger equation i∂u/∂t = -Δu + |x|²u, with given boundary values on d . We prove a formula that links the solution of this problem to that of the classical Schrödinger equation. It shows that mixed norm estimates for the Hermite-Schrödinger equation can be obtained immediately from those known in the classical case. In one space dimension, we deduce sharp pointwise convergence results at the boundary by means of this link.

On the equivalence of Green functions for general Schrödinger operators on a half-space

Abdoul Ifra, Lotfi Riahi (2004)

Annales Polonici Mathematici

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We consider the general Schrödinger operator L = d i v ( A ( x ) x ) - μ on a half-space in ℝⁿ, n ≥ 3. We prove that the L-Green function G exists and is comparable to the Laplace-Green function G Δ provided that μ is in some class of signed Radon measures. The result extends the one proved on the half-plane in [9] and covers the case of Schrödinger operators with potentials in the Kato class at infinity K considered by Zhao and Pinchover. As an application we study the cone L ( ) of all positive L-solutions continuously...

Existence and multiplicity results for a nonlinear stationary Schrödinger equation

Danila Sandra Moschetto (2010)

Annales Polonici Mathematici

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We revisit Kristály’s result on the existence of weak solutions of the Schrödinger equation of the form -Δu + a(x)u = λb(x)f(u), x N , u H ¹ ( N ) , where λ is a positive parameter, a and b are positive functions, while f : is sublinear at infinity and superlinear at the origin. In particular, by using Ricceri’s recent three critical points theorem, we show that, under the same hypotheses, a much more precise conclusion can be obtained.

Square functions associated to Schrödinger operators

I. Abu-Falahah, P. R. Stinga, J. L. Torrea (2011)

Studia Mathematica

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We characterize geometric properties of Banach spaces in terms of boundedness of square functions associated to general Schrödinger operators of the form ℒ = -Δ + V, where the nonnegative potential V satisfies a reverse Hölder inequality. The main idea is to sharpen the well known localization method introduced by Z. Shen. Our results can be regarded as alternative proofs of the boundedness in H¹, L p and BMO of classical ℒ-square functions.

Dispersive and Strichartz estimates on H-type groups

Martin Del Hierro (2005)

Studia Mathematica

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Our purpose is to generalize the dispersive inequalities for the wave equation on the Heisenberg group, obtained in [1], to H-type groups. On those groups we get optimal time decay for solutions to the wave equation (decay as t - p / 2 ) and the Schrödinger equation (decay as t ( 1 - p ) / 2 ), p being the dimension of the center of the group. As a corollary, we obtain the corresponding Strichartz inequalities for the wave equation, and, assuming that p > 1, for the Schrödinger equation.

Critical case of nonlinear Schrödinger equations with inverse-square potentials on bounded domains

Toshiyuki Suzuki (2014)

Mathematica Bohemica

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Nonlinear Schrödinger equations (NLS) a with strongly singular potential a | x | - 2 on a bounded domain Ω are considered. If Ω = N and a > - ( N - 2 ) 2 / 4 , then the global existence of weak solutions is confirmed by applying the energy methods established by N. Okazawa, T. Suzuki, T. Yokota (2012). Here a = - ( N - 2 ) 2 / 4 is excluded because D ( P a ( N ) 1 / 2 ) is not equal to H 1 ( N ) , where P a ( N ) : = - Δ - ( N - 2 ) 2 / ( 4 | x | 2 ) is nonnegative and selfadjoint in L 2 ( N ) . On the other hand, if Ω is a smooth and bounded domain with 0 Ω , the Hardy-Poincaré inequality is proved in J. L. Vazquez, E. Zuazua...

On the number of positive solutions of singularly perturbed 1D nonlinear Schrödinger equations

Patricio Felmer, Salomé Martínez, Kazunaga Tanaka (2006)

Journal of the European Mathematical Society

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We study singularly perturbed 1D nonlinear Schrödinger equations (1.1). When V ( x ) has multiple critical points, (1.1) has a wide variety of positive solutions for small ε and the number of positive solutions increases to as ε 0 . We give an estimate of the number of positive solutions whose growth order depends on the number of local maxima of V ( x ) . Envelope functions or equivalently adiabatic profiles of high frequency solutions play an important role in the proof.

Semiclassical measures for the Schrödinger equation on the torus

Nalini Anantharaman, Fabricio Macià (2014)

Journal of the European Mathematical Society

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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,...

What is a Sobolev space for the Laguerre function systems?

B. Bongioanni, J. L. Torrea (2009)

Studia Mathematica

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We discuss the concept of Sobolev space associated to the Laguerre operator L α = - y d ² / d y ² - d / d y + y / 4 + α ² / 4 y , y ∈ (0,∞). We show that the natural definition does not agree with the concept of potential space defined via the potentials ( L α ) - s . An appropriate Laguerre-Sobolev space is defined in order to achieve that coincidence. An application is given to the almost everywhere convergence of solutions of the Schrödinger equation. Other Laguerre operators are also considered.

A variational analysis of a gauged nonlinear Schrödinger equation

Alessio Pomponio, David Ruiz (2015)

Journal of the European Mathematical Society

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This paper is motivated by a gauged Schrödinger equation in dimension 2 including the so-called Chern-Simons term. The study of radial stationary states leads to the nonlocal problem: - Δ u ( x ) + ω + h 2 ( | x | ) | x | 2 + | x | + h ( s ) s u 2 ( s ) d s u ( x ) = | u ( x ) | p - 1 u ( x ) , where h ( r ) = 1 2 0 r s u 2 ( s ) d s . This problem is the Euler-Lagrange equation of a certain energy functional. In this paper the study of the global behavior of such functional is completed. We show that for p ( 1 , 3 ) , the functional may be bounded from below or not, depending on ω . Quite surprisingly, the threshold value for ω is explicit....

Time-dependent Schrödinger perturbations of transition densities

Krzysztof Bogdan, Wolfhard Hansen, Tomasz Jakubowski (2008)

Studia Mathematica

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We construct the fundamental solution of t - Δ y - q ( t , y ) for functions q with a certain integral space-time relative smallness, in particular for those satisfying a relative Kato condition. The resulting transition density is comparable to the Gaussian kernel in finite time, and it is even asymptotically equal to the Gaussian kernel (in small time) under the relative Kato condition. The result is generalized to arbitrary strictly positive and finite time-nonhomogeneous transition densities on measure...

Global existence of solutions to Schrödinger equations on compact riemannian manifolds below H 1

Sijia Zhong (2010)

Bulletin de la Société Mathématique de France

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In this paper, we will study global well-posedness for the cubic defocusing nonlinear Schrödinger equations on the compact Riemannian manifold without boundary, below the energy space, i.e. s < 1 , under some bilinear Strichartz assumption. We will find some s ˜ < 1 , such that the solution is global for s > s ˜ .

Waves in Honeycomb Structures

Charles L. Fefferman, Michael I. Weinstein (2012)

Journées Équations aux dérivées partielles

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We review recent work of the authors on the non-relativistic Schrödinger equation with a honeycomb lattice potential, V . In particular, we summarize results on (i) the existence of Dirac points, conical singularities in dispersion surfaces of H V = - Δ + V and (ii) the two-dimensional Dirac equations, as the large (but finite) time effective system of equations governing the evolution e - i H V t ψ 0 , for data ψ 0 , which is spectrally localized near a Dirac point. We conclude with a formal derivation and discussion...

Sharp trace asymptotics for a class of 2 D -magnetic operators

Horia D. Cornean, Søren Fournais, Rupert L. Frank, Bernard Helffer (2013)

Annales de l’institut Fourier

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In this paper we prove a two-term asymptotic formula for the spectral counting function for a 2 D magnetic Schrödinger operator on a domain (with Dirichlet boundary conditions) in a semiclassical limit and with strong magnetic field. By scaling, this is equivalent to a thermodynamic limit of a 2 D Fermi gas submitted to a constant external magnetic field. The original motivation comes from a paper by H. Kunz in which he studied, among other things, the boundary correction for...

Optimal potentials for Schrödinger operators

Giuseppe Buttazzo, Augusto Gerolin, Berardo Ruffini, Bozhidar Velichkov (2014)

Journal de l’École polytechnique — Mathématiques

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We consider the Schrödinger operator - Δ + V ( x ) on H 0 1 ( Ω ) , where Ω is a given domain of d . Our goal is to study some optimization problems where an optimal potential V 0 has to be determined in some suitable admissible classes and for some suitable optimization criteria, like the energy or the Dirichlet eigenvalues.

Hardy spaces H¹ for Schrödinger operators with certain potentials

Jacek Dziubański, Jacek Zienkiewicz (2004)

Studia Mathematica

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Let K t t > 0 be the semigroup of linear operators generated by a Schrödinger operator -L = Δ - V with V ≥ 0. We say that f belongs to H ¹ L if | | s u p t > 0 | K t f ( x ) | | | L ¹ ( d x ) < . We state conditions on V and K t which allow us to give an atomic characterization of the space H ¹ L .

Lieb–Thirring inequalities on the half-line with critical exponent

Tomas Ekholm, Rupert Frank (2008)

Journal of the European Mathematical Society

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We consider the operator - d 2 / d r 2 - V in L 2 ( + ) with Dirichlet boundary condition at the origin. For the moments of its negative eigenvalues we prove the bound tr ( - d 2 / d r 2 - V ) - γ C γ , α + ( V ( r ) - 1 / ( 4 r 2 ) ) + γ + ( 1 + α ) / 2 r α d r for any α [ 0 , 1 ) and γ ( 1 - α ) / 2 . This includes a Lieb-Thirring inequality in the critical endpoint case.

Unique continuation for Schrödinger operators in dimension three or less

Eric T. Sawyer (1984)

Annales de l'institut Fourier

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We show that the differential inequality | Δ u | v | u | has the unique continuation property relative to the Sobolev space H l o c 2 , 1 ( Ω ) , Ω R n , n 3 , if v satisfies the condition ( K n loc ) lim r 0 sup x K | x - y | &lt; r | x - y | 2 - n v ( y ) d y = 0 for all compact K Ω , where if n = 2 , we replace | x - y | 2 - n by - log | x - y | . This resolves a conjecture of B. Simon on unique continuation for Schrödinger operators, H = - Δ + v , in the case n 3 . The proof uses Carleman’s approach together with the following pointwise inequality valid for all N = 0 , 1 , 2 , ... and any u H c 2 , 1 ( R 3 - { 0 } ) , | u ( x ) | | x | N C R 3 | x - y | - 1 | Δ u ( y ) | | y | N d y for a.e. x in R 3 .

Lower bounds for Schrödinger operators in H¹(ℝ)

Ronan Pouliquen (1999)

Studia Mathematica

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We prove trace inequalities of type | | u ' | | L 2 2 + j k j | u ( a j ) | 2 λ | | u | | L 2 2 where u H 1 ( ) , under suitable hypotheses on the sequences a j j and k j j , with the first sequence increasing and the second bounded.

H p spaces associated with Schrödinger operators with potentials from reverse Hölder classes

Jacek Dziubański, Jacek Zienkiewicz (2003)

Colloquium Mathematicae

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Let A = -Δ + V be a Schrödinger operator on d , d ≥ 3, where V is a nonnegative potential satisfying the reverse Hölder inequality with an exponent q > d/2. We say that f is an element of H A p if the maximal function s u p t > 0 | T t f ( x ) | belongs to L p ( d ) , where T t t > 0 is the semigroup generated by -A. It is proved that for d/(d+1) < p ≤ 1 the space H A p admits a special atomic decomposition.

Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity

Antonio Ambrosetti, Veronica Felli, Andrea Malchiodi (2005)

Journal of the European Mathematical Society

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We deal with a class on nonlinear Schrödinger equations (NLS) with potentials V ( x ) | x | α , 0 < α < 2 , and K ( x ) | x | β , β > 0 . Working in weighted Sobolev spaces, the existence of ground states v ε belonging to W 1 , 2 ( N ) is proved under the assumption that σ < p < ( N + 2 ) / ( N 2 ) for some σ = σ N , α , β . Furthermore, it is shown that v ε are spikes concentrating at a minimum point of 𝒜 = V θ K 2 / ( p 1 ) , where θ = ( p + 1 ) / ( p 1 ) 1 / 2 .