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 $su{p}_{t>0}\left|T_{t}f\left(x\right)\right|$ belongs to $L^p\left({ℝ}^d\right)$, 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.

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 $\left|\right|su{p}_{t>0}|K_t{f\left(x\right)\left|\right||}_{L¹\left(dx\right)}<\infty$. We state conditions on V and $K_t$ which allow us to give an atomic characterization of the space $H{¹}_L$.

We consider the Schrödinger operator $-\Delta +V\left(x\right)$ on $H_0^1\left(\Omega \right)$, where $\Omega$ is a given domain of ${ℝ}^d$. Our goal is to study some optimization problems where an optimal potential $V\ge 0$ has to be determined in some suitable admissible classes and for some suitable optimization criteria, like the energy or the Dirichlet eigenvalues.

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), $x\in {ℝ}^N$, $u\in H¹\left({ℝ}^N\right)$, 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.

We present a novel approach for bounding the resolvent of $$H=-\Delta +i(A·\nabla +\nabla ·A)+V=:-\Delta +L1$$ for large energies. It is shown here that there exist a large integer $m$ and a large number ${\lambda }_0$ so that relative to the usual weighted $L^2$-norm, $$\parallel {\left(L{(-\Delta +(\lambda +i0))}^{-1}\right)}^m\parallel <\frac{1}{2}2$$ for all $\lambda >{\lambda }_0$. 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...

We show the existence of weak solutions in the extended sense of the Cauchy problem for the cubic fourth order nonlinear Schrödinger equation with the initial data $u_0\in X$, where $X\in \{M_{2,q}^s\left(ℝ\right),H^{\sigma }\left(𝕋\right),H^{s_1}\left(ℝ\right)+H^{s_2}\left(𝕋\right)\}$ and $q\in [1,2]$, $s\ge 0$, or $\sigma \ge 0$, or $s_2\ge s_1\ge 0$. Moreover, if $M_{2,q}^s\left(ℝ\right)\hookrightarrow L^3\left(ℝ\right)$, or if $\sigma \ge \frac{1}{6}$, or if $s_1\ge \frac{1}{6}$ and $s_2>\frac{1}{2}$ we show that the Cauchy problem is unconditionally wellposed in $X$. Similar results hold true for all higher order nonlinear Schrödinger equations and mixed order NLS due to a factorization property of the corresponding phase factors. For the proof we employ...

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

We deal with a class on nonlinear Schrödinger equations (NLS) with potentials $V\left(x\right)\sim {\left|x\right|}^{-\alpha }$, $0<\alpha <2$, and $K\left(x\right)\sim {\left|x\right|}^{-\beta }$, $\beta >0$. Working in weighted Sobolev spaces, the existence of ground states $v_{\epsilon }$ belonging to $W^{1,2}\left({ℝ}^N\right)$ is proved under the assumption that $\sigma <p<(N+2)/\left(N-2\right)$ for some $\sigma ={\sigma }_{N,\alpha ,\beta }$. Furthermore, it is shown that $v_{\epsilon }$ are spikes concentrating at a minimum point of $𝒜=V^{\theta }{K}^{-2/\left(p-1\right)}$, where $\theta =(p+1)/\left(p-1\right)-1/2$.

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 $H_V=-\Delta +V$ and (ii) the two-dimensional Dirac equations, as the large (but finite) time effective system of equations governing the evolution $e^{-i{H}_{V}t}{\psi }_0$, for data ${\psi }_0$, which is spectrally localized near a Dirac point. We conclude with a formal derivation and discussion...

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

We prove an estimate for the difference of two solutions of the Schrödinger map equation for maps from $T^1$ to $S^2.$ This estimate yields some continuity properties of the flow map for the topology of $L^2(T^1,S^2)$, provided one takes its quotient by the continuous group action of $T^1$ given by translations. We also prove that without taking this quotient, for any $t\&gt;0$ the flow map at time $t$ is discontinuous as a map from ${𝒞}^{\infty }(T^1,S^2)$, equipped with the weak topology of $H^{1/2},$ to the space of distributions ${\left({𝒞}^{\infty }(T^1,{ℝ}^3)\right)}^{*}.$ The argument relies...

Global well-posedness for the Klein-Gordon-Schrödinger system with generalized higher order coupling, which is a system of PDEs in two variables arising from quantum physics, is proven. It is shown that the system is globally well-posed in $(u,n)\in L^2\times L^2$ under some conditions on the nonlinearity (the coupling term), by using the $L^2$ conservation law for $u$ and controlling the growth of $n$ via the estimates in the local theory. In particular, this extends the well-posedness results for such a system in...

We also prove a long time existence result; more precisely we prove that for fixed $T>0$ there exists a set ${\Sigma }_T$, $\mathbb{P}\left({\Sigma }_T\right)>0$ such that any data ${\phi }^{\omega }\left(x\right)\in H^{\gamma }\left({𝕋}^3\right),\gamma <1,\omega \in {\Sigma }_T$, evolves up to time $T$ into a solution $u\left(t\right)$ with $u\left(t\right)-e^{it\Delta }{\phi }^{\omega }\in C([0,T];H^s\left({𝕋}^3\right))$, $s=s\left(\gamma \right)>1$. In particular we find a nontrivial set of data which gives rise to long time solutions below the critical space $H^1\left({𝕋}^3\right)$, that is in the supercritical scaling regime.

We address the problem of the existence of finite energy solitary waves for nonlinear Klein-Gordon or Schrödinger type equations $\Delta u-u+f\left(u\right)=0$ in ${ℝ}^N$, $u\in H^1\left({ℝ}^N\right)$, where $N\ge 2$. Under natural conditions on the nonlinearity $f$, we prove the existence of $\mathrm{𝑖𝑛𝑓𝑖𝑛𝑖𝑡𝑒𝑙𝑦𝑚𝑎𝑛𝑦𝑛𝑜𝑛𝑟𝑎𝑑𝑖𝑎𝑙𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛𝑠}$ in any dimension $N\ge 2$. Our result complements earlier works of Bartsch and Willem $(N=4\mathrm{𝚘𝚛}N\ge 6)$ and Lorca-Ubilla $(N=5)$ where solutions invariant under the action of $O\left(2\right)\times O(N-2)$ are constructed. In contrast, the solutions we construct are invariant under the action of $D_k\times O(N-2)$ where $D_k\subset O\left(2\right)$ denotes the dihedral...

Let ${ℒ}_1=-\Delta +V$ be a Schrödinger operator and let ${ℒ}_2={(-\Delta )}^2+V^2$ be a Schrödinger type operator on ${ℝ}^n$ $(n\ge 5)$, where $V\ne 0$ is a nonnegative potential belonging to certain reverse Hölder class $B_s$ for $s\ge n/2$. The Hardy type space $H_{{ℒ}_2}^1$ is defined in terms of the maximal function with respect to the semigroup $\left\{{\mathrm{e}}^{-t{ℒ}_2}\right\}$ and it is identical to the Hardy space $H_{{ℒ}_1}^1$ established by Dziubański and Zienkiewicz. In this article, we prove the $L^p$-boundedness of the commutator ${ℛ}_b=bℛf-ℛ\left(bf\right)$ generated by the Riesz transform $ℛ={\nabla }^2{ℒ}_2^{-1/2}$, where $b\in {\mathrm{BMO}}_{\theta }\left(\rho \right)$, which is larger...

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...

Let L = -Δ + V be a Schrödinger operator in ${ℝ}^d$ and $H{¹}_L\left({ℝ}^d\right)$ be the Hardy type space associated to L. We investigate the bilinear operators T⁺ and T¯ defined by $T^{\pm }(f,g)\left(x\right)=\left(T₁f\right)\left(x\right)\left(T₂g\right)\left(x\right)\pm \left(T₂f\right)\left(x\right)\left(T₁g\right)\left(x\right)$, where T₁ and T₂ are Calderón-Zygmund operators related to L. Under some general conditions, we prove that either T⁺ or T¯ is bounded from $L^p\left({ℝ}^d\right)\times L^q\left({ℝ}^d\right)$ to $H{¹}_L\left({ℝ}^d\right)$ for 1 < p,q < ∞ with 1/p + 1/q = 1. Several examples satisfying these conditions are given. We also give a counterexample for which the classical Hardy space estimate fails. ...

Let d > 0 be a positive real number and n ≥ 1 a positive integer and define the operator $S_d$ and its associated global maximal operator $S*{*}_d$ by $\left(S_{d}f\right)(x,t)=1/\left(2\pi \right)ⁿ{\int }_{ℝⁿ}{e}^{ix·\xi }{e}^{{it\left|\xi \right|}^d}f̂\left(\xi \right)d\xi$, f ∈ (ℝⁿ), x ∈ ℝⁿ, t ∈ ℝ, $(S*{*}_{d}f)\left(x\right)=su{p}_{t\in ℝ}|1/\left(2\pi \right)ⁿ{\int }_{ℝⁿ}{e}^{ix·\xi }{e}^{{it\left|\xi \right|}^d}f̂\left(\xi \right)d\xi |$, f ∈ (ℝⁿ), x ∈ ℝⁿ, where f̂ is the Fourier transform of f and (ℝⁿ) is the Schwartz class of rapidly decreasing functions. If d = 2, $S_{d}f$ is the solution to the initial value problem for the free Schrödinger equation (cf. (1.3) in this paper). We prove that for radial functions f ∈ (ℝⁿ), if n ≥ 3, 0 < d ≤ 2, and p ≥...

We prove trace inequalities of type $\left|\right|u^{\text{'}}{\left|\right|}_{L^2}^2+{\sum }_{j\in \mathbb{Z}}{k}_j|u\left(a_j\right){|}^2{\ge \lambda \left|\right|u\left|\right|}_{L^2}^2$ where $u\in H^1\left(ℝ\right)$, under suitable hypotheses on the sequences ${a_j}_{j\in \mathbb{Z}}$ and ${k_j}_{j\in \mathbb{Z}}$, with the first sequence increasing and the second bounded.

We consider the 3D quantum BBGKY hierarchy which corresponds to the $N$-particle Schrödinger equation. We assume the pair interaction is $N^{3\beta –1}V\left(B^{\beta }\right)$. For the interaction parameter $\beta \in (0,2/3)$, we prove that, provided an energy bound holds for solutions to the BBKGY hierarchy, the $N\to \infty$ limit points satisfy the space-time bound conjectured by S. Klainerman and M. Machedon [45] in 2008. The energy bound was proven to hold for $\beta \in (0,3/5)$ in [28]. This allows, in the case $\beta \in (0,3/5)$, for the application of the Klainerman–Machedon...

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: $-\Delta u\left(x\right)+\left(\omega +\frac{h^2\left(\right|x\left|\right)}{{\left|x\right|}^2}+{\int }_{\left|x\right|}^{+\infty }\frac{h\left(s\right)}{s}{u}^2\left(s\right)ds\right)u\left(x\right)={\left|u\left(x\right)\right|}^{p-1}u\left(x\right)$, where $h\left(r\right)=\frac{1}{2}{\int }_0^{r}s{u}^2\left(s\right)ds$. 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 $\omega$. Quite surprisingly, the threshold value for $\omega$ is explicit....

We consider the cubic defocusing nonlinear Schrödinger equation in the two dimensional torus. Fix $s>1$. Recently Colliander, Keel, Staffilani, Tao and Takaoka proved the existence of solutions with $s$-Sobolev norm growing in time. We establish the existence of solutions with polynomial time estimates. More exactly, there is $c>0$ such that for any $𝒦\gg 1$ we find a solution $u$ and a time $T$ such that ${\parallel u\left(T\right)\parallel }_{H^s}\ge 𝒦{\parallel u\left(0\right)\parallel }_{H^s}$. Moreover, the time $T$ satisfies the polynomial bound $0<T<{𝒦}^C$.