Semiclassical measures for the Schrödinger equation on the torus

Nalini Anantharaman; Fabricio Macià

Displaying similar documents to “Semiclassical measures for the Schrödinger equation on the torus”

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 $L = d i v (A (x) \nabla_{x}) - μ$ 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 $K ₙ^{\infty}$ considered by Zhao and Pinchover. As an application we study the cone $_{L} (ℝ ⁿ ₊)$ of all positive L-solutions continuously...

Estimates of capacity of self-similar measures

Jozef Myjak, Tomasz Szarek (2002)

Annales Polonici Mathematici

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We give lower and upper estimates of the capacity of self-similar measures generated by iterated function systems $(S_{i}, p_{i}) : i = 1, . . ., N$ where $S_{i}$ are bi-lipschitzean transformations.

Invariant weighted Wiener measures and almost sure global well-posedness for the periodic derivative NLS

Andrea R. Nahmod, Tadahiro Oh, Luc Rey-Bellet, Gigliola Staffilani (2012)

Journal of the European Mathematical Society

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We construct an invariant weighted Wiener measure associated to the periodic derivative nonlinear Schrödinger equation in one dimension and establish global well-posedness for data living in its support. In particular almost surely for data in a Fourier–Lebesgue space $ℱ L^{s, r} (T)$ with $s \geq \frac{1}{2}, 2 < r < 4, (s - 1) r < - 1$ and scaling like $H^{\frac{1}{2} - ϵ} (𝕋)$ , for small $ϵ > 0$ . We also show the invariance of this measure.

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), $x \in ℝ^{N}$ , $u \in H ¹ (ℝ^{N})$ , where λ is a positive parameter, a and b are positive functions, while $f : ℝ \to ℝ$ 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.

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 $ℝ^{d} \times ℝ ₊$ , consider the Hermite-Schrödinger equation i∂u/∂t = -Δu + |x|²u, with given boundary values on $ℝ^{d}$ . 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 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 $\infty$ as $ε \to 0$ . 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.

On Ordinary and Standard Lebesgue Measures on $ℝ^{\infty}$

Gogi Pantsulaia (2009)

Bulletin of the Polish Academy of Sciences. Mathematics

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New concepts of Lebesgue measure on $ℝ^{\infty}$ are proposed and some of their realizations in the ZFC theory are given. Also, it is shown that Baker’s both measures [1], [2], Mankiewicz and Preiss-Tišer generators [6] and the measure of [4] are not α-standard Lebesgue measures on $ℝ^{\infty}$ for α = (1,1,...).

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...

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: $- Δ u (x) + (ω + \frac{h^{2} (| x |)}{{| x |}^{2}} + \int_{| x |}^{+ \infty} \frac{h (s)}{s} u^{2} (s) d s) u (x) = {| u (x) |}^{p - 1} u (x)$ , where $h (r) = \frac{1}{2} \int_{0}^{r} s u^{2} (s) d s$ . 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 $p \in (1, 3)$ , the functional may be bounded from below or not, depending on $ω$ . Quite surprisingly, the threshold value for $ω$ is explicit....

Self-affine measures that are $L^{p}$ -improving

Kathryn E. Hare (2015)

Colloquium Mathematicae

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A measure is called $L^{p}$ -improving if it acts by convolution as a bounded operator from $L^{q}$ to L² for some q < 2. Interesting examples include Riesz product measures, Cantor measures and certain measures on curves. We show that equicontractive, self-similar measures are $L^{p}$ -improving if and only if they satisfy a suitable linear independence property. Certain self-affine measures are also seen to be $L^{p}$ -improving.

Convex Corson compacta and Radon measures

Grzegorz Plebanek (2002)

Fundamenta Mathematicae

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Assuming the continuum hypothesis, we show that (i) there is a compact convex subset L of $Σ (ℝ^{ω ₁})$ , and a probability Radon measure on L which has no separable support; (ii) there is a Corson compact space K, and a convex weak*-compact set M of Radon probability measures on K which has no $G_{δ}$ -points.

Simple fractions and linear decomposition of some convolutions of measures

Jolanta K. Misiewicz, Roger Cooke (2001)

Discussiones Mathematicae Probability and Statistics

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Every characteristic function φ can be written in the following way: φ(ξ) = 1/(h(ξ) + 1), where h(ξ) = ⎧ 1/φ(ξ) - 1 if φ(ξ) ≠ 0 ⎨ ⎩ ∞ if φ(ξ) = 0 This simple remark implies that every characteristic function can be treated as a simple fraction of the function h(ξ). In the paper, we consider a class C(φ) of all characteristic functions of the form $φ_{a} (ξ) = [a / (h (ξ) + a)]$ , where φ(ξ) is a fixed characteristic function. Using the well known theorem on simple fraction decomposition of rational functions we obtain...

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 $ℝ^{d}$ . Our goal is to study some optimization problems where an optimal potential $V \geq 0$ has to be determined in some suitable admissible classes and for some suitable optimization criteria, like the energy or the Dirichlet eigenvalues.

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 $Ω = ℝ^{N}$ 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 $H^{1} (ℝ^{N})$ , where $P_{a (N)} : = - Δ - {(N - 2)}^{2} / {(4 | x |}^{2})$ is nonnegative and selfadjoint in $L^{2} (ℝ^{N})$ . On the other hand, if $Ω$ is a smooth and bounded domain with $0 \in Ω$ , the Hardy-Poincaré inequality is proved in J. L. Vazquez, E. Zuazua...

A convolution property of some measures with self-similar fractal support

Denise Szecsei (2007)

Colloquium Mathematicae

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We define a class of measures having the following properties: (1) the measures are supported on self-similar fractal subsets of the unit cube $I^{M} = {[0, 1)}^{M}$ , with 0 and 1 identified as necessary; (2) the measures are singular with respect to normalized Lebesgue measure m on $I^{M}$ ; (3) the measures have the convolution property that $μ * L^{p} \subseteq L^{p + ε}$ for some ε = ε(p) > 0 and all p ∈ (1,∞). We will show that if (1/p,1/q) lies in the triangle with vertices (0,0), (1,1) and (1/2,1/3), then $μ * L^{p} \subseteq L^{q}$ for any measure μ in our...

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 $\tilde{s} < 1$ , such that the solution is global for $s > \tilde{s}$ .

On the isotropic constant of marginals

Grigoris Paouris (2012)

Studia Mathematica

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We show that if μ₁, ..., μₘ are log-concave subgaussian or supergaussian probability measures in $ℝ^{n_{i}}$ , i ≤ m, then for every F in the Grassmannian $G_{N, n}$ , where N = n₁ + ⋯ + nₘ and n< N, the isotropic constant of the marginal of the product of these measures, $π_{F} (μ ₁ \otimes \dots \otimes μ ₘ)$ , is bounded. This extends known results on bounds of the isotropic constant to a larger class of measures.

On Dirichlet-Schrödinger operators with strong potentials

Gabriele Grillo (1995)

Studia Mathematica

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We consider Schrödinger operators H = -Δ/2 + V (V≥0 and locally bounded) with Dirichlet boundary conditions, on any open and connected subdomain $D \subset ℝ^{n}$ which either is bounded or satisfies the condition $d (x, D^{c}) \to 0$ as |x| → ∞. We prove exponential decay at the boundary of all the eigenfunctions of H whenever V diverges sufficiently fast at the boundary ∂D, in the sense that $d {(x, D^{C})}^{2} V (x) \to \infty$ as $d (x, D^{C}) \to 0$ . We also prove bounds from above and below for Tr(exp[-tH]), and in particular we give criterions for the finiteness of...

Strichartz and smoothing estimates for Schrödinger operators with large magnetic potentials in $ℝ^{3}$

M. Burak Erdoğan, Michael Goldberg, Wilhelm Schlag (2008)

Journal of the European Mathematical Society

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We present a novel approach for bounding the resolvent of $H = - Δ + i (A \cdot \nabla + \nabla \cdot A) + V = : - Δ + L 1$ for large energies. It is shown here that there exist a large integer $m$ and a large number $λ_{0}$ so that relative to the usual weighted $L^{2}$ -norm, $∥ {(L {(- Δ + (λ + i 0))}^{- 1})}^{m} ∥ < \frac{1}{2} 2$ for all $λ > λ_{0}$ . This requires suitable decay and smoothness conditions on $A, V$ . The estimate (2) is trivial when $A = 0$ , but difficult for large $A$ since the gradient term exactly cancels the natural decay of the free resolvent. To obtain (2), we introduce a conical decomposition of the resolvent and...

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...

$H^{p}$ 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 $ℝ^{d}$ , 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 $L^{p} (ℝ^{d})$ , 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.

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 $| | s u p_{t > 0} | K_{t} {f (x) | | |}_{L ¹ (d x)} < \infty$ . We state conditions on V and $K_{t}$ which allow us to give an atomic characterization of the space $H ¹_{L}$ .

Continuous linear functionals on the space of Borel vector measures

Pola Siwek (2008)

Annales Polonici Mathematici

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We study properties of the space ℳ of Borel vector measures on a compact metric space X, taking values in a Banach space E. The space ℳ is equipped with the Fortet-Mourier norm ${| | \cdot | |}_{ℱ}$ and the semivariation norm ||·||(X). The integral introduced by K. Baron and A. Lasota plays the most important role in the paper. Investigating its properties one can prove that in most cases the space $(ℳ, | | \cdot | |_{ℱ}) *$ is contained in but not equal to the space (ℳ,||·||(X))*. We obtain a representation of the continuous functionals...

Quasi-periodic solutions with Sobolev regularity of NLS on $𝕋^{d}$ with a multiplicative potential

Massimiliano Berti, Philippe Bolle (2013)

Journal of the European Mathematical Society

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We prove the existence of quasi-periodic solutions for Schrödinger equations with a multiplicative potential on $𝕋^{d}, d \geq 1$ , finitely differentiable nonlinearities, and tangential frequencies constrained along a pre-assigned direction. The solutions have only Sobolev regularity both in time and space. If the nonlinearity and the potential are $C^{\infty}$ then the solutions are $C^{\infty}$ . The proofs are based on an improved Nash-Moser iterative scheme, which assumes the weakest tame estimates for the inverse linearized...

Erratum to “Geometric rigidity of $\times m$ invariant measures” (J. Eur. Math. Soc. 14, 1539–1563 (2012))

Michael Hochman (2013)

Journal of the European Mathematical Society

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Control for Schrödinger operators on 2-tori: rough potentials

Jean Bourgain, Nicolas Burq, Maciej Zworski (2013)

Journal of the European Mathematical Society

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For the Schrödinger equation, $(i \partial_{t} + \nabla) u = 0$ on a torus, an arbitrary non-empty open set $Ω$ provides control and observability of the solution: $∥u {|_{t = 0}∥}_{L^{2} (𝕋^{2})} \leq K_{T} {∥u∥}_{L^{2} ([0, T] \times Ω)}$ . We show that the same result remains true for $(i \partial_{t} + \nabla - V) u = 0$ where $V \in L^{2} (𝕋^{2})$ , and $𝕋^{2}$ is a (rational or irrational) torus. That extends the results of [1], and [8] where the observability was proved for $V \in C (𝕋^{2})$ and conjectured for $V \in L^{\infty} (𝕋^{2})$ . The higher dimensional generalization remains open for $V \in L^{\infty} (𝕋^{n})$ .

On inhomogeneous self-similar measures and their $L^{q}$ spectra

Przemysław Liszka (2013)

Annales Polonici Mathematici

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Let $S_{i} : ℝ^{d} \to ℝ^{d}$ for i = 1,..., N be contracting similarities, let $(p ₁, . . ., p_{N}, p)$ be a probability vector and let ν be a probability measure on $ℝ^{d}$ with compact support. It is well known that there exists a unique inhomogeneous self-similar probability measure μ on $ℝ^{d}$ such that $μ = \sum_{i = 1}^{N} p_{i} μ \circ S_{i}^{- 1} + p ν$ . We give satisfactory estimates for the lower and upper bounds of the $L^{q}$ spectra of inhomogeneous self-similar measures. The case in which there are a countable number of contracting similarities and probabilities is considered. In particular,...