Some estimates of Schrödinger type operators on the Heisenberg group

In this paper, we consider the Schrödinger type operator ${H = (-\Delta _{\mathbb {H}}^n)^2 +V ^{2}}In this paper, we consider the Schr?dinger type operator H = (-D_{\mathbb H}ⁿ)² +V ²{H = (-\Delta _{\mathbb {H}}^n)^2 +V ^{2}}, where the nonnegative potential V belongs to the reverse H?lder class B_q1 for q₁ 3 \frac Q 2,Q 3 6{B_{{q}_{1}}\, {\rm for}\, q_{1}\geq {\frac {Q}{ 2}},Q \geq 6}, and D_{\mathbb Hⁿ}{\Delta_{\mathbb {H}^n}} is the sublaplacian on the Heisenberg group \mathbb Hⁿ{\mathbb {H}^n}. An L ^p estimate and a weak type L ¹ estimate for the operator ?⁴_{\mathbb Hⁿ} H^-1{\nabla^4_{\mathbb {H}^n} H^{-1}} when V ? B_q1{V \in B_{{q}_{1}}} for 1 < p ￡ \fracq₁2{1 < p \leq \frac{q_{1}}{2}} are obtained.     

Positivity of eigenvalues of the two-particle Schrödinger operator on a lattice

We consider the family H(k) of two-particle discrete Schrödinger operators depending on the quasimomentum of a two-particle system k ∈ $\mathbb{T}^d $ , where $\mathbb{T}^d $ is a d-dimensional torus. This family of operators is associated with the Hamiltonian of a system of two arbitrary particles on the d-dimensional lattice ?^d, d ≥ 3, interacting via a short-range attractive pair potential. We prove that the eigenvalues of the Schrödinger operator H(k) below the essential spectrum are positive for all nonzero values of the quasimomentum k ∈ $\mathbb{T}^d $ if the operator H(0) is nonnegative. We establish a similar result for the eigenvalues of the Schrödinger operator H₊(k), k ∈ $\mathbb{T}^d $ , corresponding to a two-particle system with repulsive interaction.     

Hardy spaces associated to the Schrödinger operator on strongly Lipschitz domains of {\mathbb{R}^d}

Let L = ?Δ + V be a Schrödinger operator and Ω be a strongly Lipschitz domain of ${\mathbb R^{d}}Let L = −Δ + V be a Schr?dinger operator and Ω be a strongly Lipschitz domain of \mathbb R^d{\mathbb R^{d}} , where Δ is the Laplacian on \mathbb R^d{\mathbb R^{d}} and the potential V is a nonnegative polynomial on \mathbb R^d{\mathbb R^{d}} . In this paper, we investigate the Hardy spaces on Ω associated to the Schr?dinger operator L.     

Absolutely continuous spectrum for random Schrödinger operators on the Bethe strip

The Bethe strip of width m is the cartesian product $\mathbb {B}\times \lbrace 1,\ldots ,m\rbrace$, where $\mathbb {B}$ is the Bethe lattice (Cayley tree). We prove that Anderson models on the Bethe strip have “extended states” for small disorder. More precisely, we consider Anderson‐like Hamiltonians $H_\lambda =\frac{1}{2} \Delta \otimes 1 + 1 \otimes A\,+\,\lambda \mathcal {V}$ on a Bethe strip with connectivity K ≥ 2, where A is an m × m symmetric matrix, $\mathcal {V}$ is a random matrix potential, and λ is the disorder parameter. Given any closed interval $I\subset \big (\!-\!\sqrt{K}+a_{{\rm max}},\sqrt{K}+a_{\rm {min}}\big )$, where a_min and a_max are the smallest and largest eigenvalues of the matrix A, we prove that for λ small the random Schrödinger operator H_λ has purely absolutely continuous spectrum in I with probability one and its integrated density of states is continuously differentiable on the interval I.     

Blow-up Theory for the Coupled L 2-Critical Nonlinear Schrödinger System in the Plane

In this paper, we study the blow-up theory for the L ²-critical coupled nonlinear Schrödinger system in the Euclidean plane ${\mathbb{R}^{2}}In this paper, we study the blow-up theory for the L ²-critical coupled nonlinear Schr?dinger system in the Euclidean plane \mathbbR²{\mathbb{R}^{2}} .We show that at the finite time blow-up, similar results of Weinstein, Merle-Tsutsumi for scalar Schrodinger equations are true for the system.     

On the global wellposedness for the nonlinear Schrödinger equations with L p -large initial data

We consider the Cauchy problem for the nonlinear Schrödinger equations $ \begin{array}{l} iu_t + \triangle u \pm |u|^{p-1}u =0, \qquad x \in \mathbb{R}^d, \quad t \in \mathbb{R} \\ u(x,0)= u_0(x), \qquad x \in \mathbb{R}^d \end{array} $ for 1 < p < 1 + 4/d and prove that there is a ${\rho (p ,d) \in (1,2)}We consider the Cauchy problem for the nonlinear Schr?dinger equations

l iut + \triangle u ±|u|p-1u = 0,        x ? \mathbbRd,     t ? \mathbbR u(x,0) = u0(x),        x ? \mathbbRd \begin{array}{l} iu_t + \triangle u \pm |u|^{p-1}u =0, \qquad x \in \mathbb{R}^d, \quad t \in \mathbb{R} \\ u(x,0)= u_0(x), \qquad x \in \mathbb{R}^d \end{array}      7. Some estimates for the Schrödinger type operators with nonnegative potentials      Yu?LiuEmail author  Jianfeng?Dong 《Archiv der Mathematik》2012,98(5):467-475 In this paper we consider the Schrödinger operator ?Δ + V on \({\mathbb R^d}\), where the nonnegative potential V belongs to the reverse Hölder class \({B_{q_{_1}}}\) for some \({q_{_1}\geq \frac{d}{2}}\) with d ≥ 3. Let \({H^1_L(\mathbb R^d)}\) denote the Hardy space related to the Schrödinger operator L = ?Δ + V and \({BMO_L(\mathbb R^d)}\) be the dual space of \({H^1_L(\mathbb R^d)}\). We show that the Schrödinger type operator \({\nabla(-\Delta +V)^{-\beta}}\) is bounded from \({H^1_L(\mathbb R^d)}\) into \({L^p(\mathbb R^d)}\) for \({p=\frac{d}{d-(2\beta-1)}}\) with \({ \frac{1}{2}<\beta<\frac{3}{2} }\) and that it is also bounded from \({L^p(\mathbb R^d)}\) into \({BMO_L(\mathbb R^d)}\) for \({p=\frac{d}{2\beta-1}}\) with \({ \frac{1}{2}<\beta< 2}\).      8. Geometric relative Hardy inequalities and the discrete spectrum of Schrödinger operators on manifolds      Kazuo Akutagawa  Hironori Kumura 《Calculus of Variations and Partial Differential Equations》2013,48(1-2):67-88 The classical Hardy inequality for the Laplacian Δ on ${\mathbb{R}^n}$ shows the borderline-behavior of a potential V for the following question: whether the Schrödinger operator ?Δ + V has a finite or infinite number of the discrete spectrum. In this paper, we will give a sharp generalization of this inequality on ${\mathbb{R}^n}$ to a relative version of that on large classes of complete noncompact manifolds. Replacing ${\mathbb{R}^n}$ by some specific classes of complete noncompact manifolds, including hyperbolic spaces, we also establish some sharp criteria for the above-type question.      9. Existence of semiclassical states for a coupled Schrödinger system with potentials and nonlocal nonlinearities      Minbo Yang  Yuanhong Wei  Yanheng Ding 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2014,65(1):41-68 In this paper, we first study a Schrödinger system with nonlocal coupling nonlinearities of Hartree type $$\left\{\begin{array}{ll} -\varepsilon^{2}\Delta u +V_1(x)u = \left ( \int \limits_{\mathbb{R}^{3}} \frac{u^{2}}{|x-y|}{\rm d}y \right)u\,+\, {\beta} \left ( \int \limits_{\mathbb{R}^{3}} \frac{v^{2}}{|x-y|}{\rm d} y \right)u,\\ -\varepsilon^{2} \Delta v +V_2(x)v = \left(\int \limits_{\mathbb{R}^{3}} \frac{v^{2}}{|x-y|}{\rm d}y \right)v \,+ \, {\beta} \left ( \int \limits_{\mathbb{R}^{3}} \frac{u^{2}}{|x-y|}{\rm d}y \right)v. \end{array}\right.$$ Using variational methods, we prove the existence of purely vector ground state solutions for the Schrödinger system if the parameter ${\varepsilon}$ is small enough. Secondly, we also establish some existence results for the coupled Schrödinger system with critical exponents.      10. Carleson measures,BMO spaces and balayages associated to Schrdinger operators      Peng Chen  Xuan Thinh Duong  Ji Li  Liang Song  LiXin Yan 《中国科学 数学(英文版)》2017,60(11):2077-2092 Let L be a Schrdinger operator of the form L =-? + V acting on L~2(R~n), n≥3, where the nonnegative potential V belongs to the reverse Hlder class B_q for some q≥n. Let BMO_L(R~n) denote the BMO space associated to the Schrdinger operator L on R~n. In this article, we show that for every f ∈ BMO_L(R~n) with compact support, then there exist g ∈ L~∞(R~n) and a finite Carleson measure μ such that f(x) = g(x) + S_(μ,P)(x) with ∥g∥∞ + |||μ|||c≤ C∥f∥BMO_L(R~n), where S_(μ,P)=∫(R_+~(n+1))Pt(x,y)dμ(y, t),and Pt(x, y) is the kernel of the Poisson semigroup {e-~(t(L)~(1/2))}t0 on L~2(R~n). Conversely, if μ is a Carleson measure, then S_(μ,P) belongs to the space BMO_L(R~n). This extends the result for the classical John-Nirenberg BMO space by Carleson(1976)(see also Garnett and Jones(1982), Uchiyama(1980) and Wilson(1988)) to the BMO setting associated to Schrdinger operators.      11. An asymptotic of the negative discrete spectrum of the Schrödinger operator      G. V. Rozenblyum 《Mathematical Notes》1977,21(3):222-227 The Schrödinger operator Hu = -Δu + V(x)u, where V(x) → 0 as ¦x¦ → ∞, is considered in L2(Rm) for m?3. The asymptotic formula $$N(\lambda ,V) \sim \Upsilon _m \int {(\lambda - V(x))_ + ^{{m \mathord{\left/ {\vphantom {m {2_{dx} }}} \right. \kern-\nulldelimiterspace} {2_{dx} }}} ,} \lambda \to ---0,$$ is established for the number N(λ, V) of the characteristic values of the operator H which are less than λ. It is assumed about the potential V that V = Vo + V1; Vo < 0, ¦Vo =o (¦Vo¦3/2) as ¦x¦ → ∞; σ (t/2, Vo) ?cσ (t. Vo) and V1∈Lm/2,loc, σ(t, V1) =o (σ (t, Vo)), where σ (t,f)= mes {x:¦f (x) ¦ > t).      12. On Isomorphisms of Hardy Spaces Associated with Schrödinger Operators      Jacek Dziubański  Jacek Zienkiewicz 《Journal of Fourier Analysis and Applications》2013,19(3):447-456 Let L=?Δ+V is a Schrödinger operator on $\mathbb{R}^{d}$ , d≥3, V≥0. Let $H^{1}_{L}$ denote the Hardy space associated with L. We shall prove that there is an L-harmonic function w, 0<δ≤w(x)≤C, such that the mapping $$H_L^1 \ni f\mapsto wf\in H^1\bigl(\mathbb{R}^d\bigr) $$ is an isomorphism from the Hardy space $H_{L}^{1}$ onto the classical Hardy space $H^{1}(\mathbb{R}^{d})$ if and only if $\Delta^{-1}V(x)=-c_{d}\int_{\mathbb{R}^{d}} |x-y|^{2-d} V(y) dy$ belongs to $L^{\infty}(\mathbb{R}^{d})$ .      13. A Spectral Multiplier Theorem Associated with a Schrödinger Operator      Younghun Hong 《Journal of Fourier Analysis and Applications》2016,22(3):591-622 We establish a Hörmander type spectral multiplier theorem for a Schrödinger operator \(H=-\Delta +V(x)\) in \(\mathbb {R}^3\), provided V is contained in a large class of short range potentials. This result does not require the Gaussian heat kernel estimate for the semigroup \(e^{-tH}\), and indeed the operator H may have negative eigenvalues. As an application, we show local well-posedness of a 3d quintic nonlinear Schrödinger equation with a potential.      14. Positive solutions of Schrödinger equations and fine regularity of boundary points      Alano Ancona 《Mathematische Zeitschrift》2012,272(1-2):405-427 Given a Lipschitz domain Ω in ${{\mathbb R}^N}$ and a nonnegative potential V in Ω such that V(x) d(x, ?Ω)2 is bounded we study the fine regularity of boundary points with respect to the Schrödinger operator L V := Δ ? V in Ω. Using potential theoretic methods, several conditions are shown to be equivalent to the fine regularity of ${z \in \partial \Omega}$ . The main result is a simple (explicit if Ω is smooth) necessary and sufficient condition involving the size of V for ${z \in \partial \Omega}$ to be finely regular. An intermediate result consists in a majorization of ${\int_A \vert{\frac{ u} {d(.,\partial \Omega)}}\vert^2\, dx}$ for u positive harmonic in Ω and ${A \subset \Omega}$ . Conditions for almost everywhere regularity in a subset A of ?Ω are also given as well as an extension of the main results to a notion of fine ${\mathcal{ L}_1 \vert \mathcal{L}_0}$ -regularity, if ${\mathcal{L}_j = \mathcal{L} - V_j, V_0,\, V_1}$ being two potentials, with V 0 ≤ V 1 and ${\mathcal{L}}$ a second order elliptic operator.      15. Some estimates for commutators of Riesz transform associated with Schrödinger type operators      Yu Liu  Jing Zhang  Jie-Lai Sheng  Li-Juan Wang 《Czechoslovak Mathematical Journal》2016,66(1):169-191 Let L1 = ?Δ + V be a Schr:dinger operator and let L2 = (?Δ)2 + V2 be a Schrödinger type operator on ?n (n ? 5), where V≠ 0 is a nonnegative potential belonging to certain reverse Hölder class Bs for s ? n/2. The Hardy type space \(H_{{L_2}}^1\) is defined in terms of the maximal function with respect to the semigroup \(\left\{ {{e^{ - t{L_2}}}} \right\}\) and it is identical to the Hardy space \(H_{{L_1}}^1\) established by Dziubański and Zienkiewicz. In this article, we prove the Lp-boundedness of the commutator Rb = bRf - R(bf) generated by the Riesz transform \(R = {\nabla ^2}L_2^{ - 1/2}\), where \(b \in BM{O_\theta }(\varrho )\), which is larger than the space BMO(?n). Moreover, we prove that Rb is bounded from the Hardy space \(H_{\mathcal{L}_1 }^1 \) into weak \(L_{weak}^1 (\mathbb{R}^n )\).      16. Infinitesimal form boundedness and Trudinger’s subordination for the Schrödinger operator      V.G.?Maz’ya  I.E.?VerbitskyEmail author 《Inventiones Mathematicae》2005,162(1):81-136 We give explicit analytic criteria for two problems associated with the Schrödinger operator H=-Δ+Q on L2(? n ) where Q∈D’(? n ) is an arbitrary real- or complex-valued potential. First, we obtain necessary and sufficient conditions on Q so that the quadratic form \(\langle{Q}\cdot,\ \cdot\rangle\) has zero relative bound with respect to the Laplacian. For Q∈L1loc(? n ), this property can be expressed in the form of the integral inequality: $\left\vert\int_{\mathbb{R}^n} |u(x)|^2 Q(x) dx \right\vert\leq\epsilon\| \nabla u \|^2_{L^2(\mathbb{R}^n)} + C(\epsilon) \|u \|^2_{L^2(\mathbb{R}^n)}, \quad\forall u \in C^{\infty}_0(\mathbb{R}^n),$ for an arbitrarily small ε>0 and some C(ε)>0. One of the major steps here is the reduction to a similar inequality with nonnegative function \(|\nabla(1-\Delta)^{-1} Q|^2 + |(1-\Delta)^{-1} Q|\) in place of Q. This provides a complete solution to the infinitesimal form boundedness problem for the Schrödinger operator, and leads to new broad classes of admissible distributional potentials Q, which extend the usual L p and Kato classes, as well as those based on the well-known conditions of Fefferman–Phong and Chang–Wilson–Wolff. Secondly, we characterize Trudinger’s subordination property where C(ε) in the above inequality is subject to the condition C(ε)≤cε-β(β>0) as ε→+0. Such quadratic form inequalities can be understood entirely in the framework of Morrey–Campanato spaces, using mean oscillations of \(\nabla(1-\Delta)^{-1}Q\) and \((1-\Delta)^{-1}Q\) on balls or cubes. A version of this condition where ε∈(0,+∞) is equivalent to the multiplicative inequality: $\left\vert\int_{\mathbb{R}^n} |u(x)|^2Q(x)dx\right\vert\leq{C}\|\nabla{u}\|^{2p}_{L^2(\mathbb{R}^n)}\|u\|^{2(1-p)}_{L^2(\mathbb{R}^n)},\quad\forall{u}\in{C}^\infty_0(\mathbb{R}^n),$ with \(p=\frac\beta{1 + \beta}\in(0,1)\). We show that this inequality holds if and only if \(\nabla\Delta^{-1} Q \in{BMO}(\mathbb{R}^n)\) if \(p=\frac{1}{2}\). For \(0 < p < \frac{1}{2}\), it is valid whenever \(\nabla\Delta^{-1}Q\) is Hölder-continuous of order 1-2p, or respectively lies in the Morrey space \(\mathcal{L}^{2,\lambda}\) with λ=n+2-4p if \(\frac{1}{2} < p < 1\). As a consequence, we characterize completely the class of those Q which satisfy an analogous multiplicative inequality of Nash’s type, with \(\|u\|_{L^1(\mathbb{R}^n)}\) in placeof \(\|u\|_{L^2(\mathbb{R}^n)}\). These results are intimately connected with spectral theory and dynamics of the Schrödinger operator, and elliptic PDE theory.      17. Groundstates for the nonlinear Schrödinger equation with potential vanishing at infinity      Denis Bonheure  Jean Van Schaftingen 《Annali di Matematica Pura ed Applicata》2010,189(2):273-301 Groundstates of the stationary nonlinear Schrödinger equation $-\Delta u +V u =K u^{p-1}$ , are studied when the nonnegative function V and K are neither bounded away from zero, nor bounded from above. A special attention is paid in the case of a potential V that goes to 0 at infinity. Conditions on compact embeddings that allow to prove in particular the existence of groundstates are established. The fact that the solution is in ${L^2(\mathbb R^N)}Groundstates of the stationary nonlinear Schr?dinger equation -Du +V u = K up-1-\Delta u +V u =K u^{p-1}      18. On the well-posedness for stochastic fourth-order Schrödinger equations      Dao-yuan Fang  Lin-zi Zhang  Ting Zhang 《高校应用数学学报(英文版)》2011,26(3):307-318 The influence of the random perturbations on the fourth-order nonlinear Schrödinger equations, $iu_t + \Delta ^2 u + \varepsilon \Delta u + \lambda |u|^{p - 1} u = \dot \xi ,(t,x) \in \mathbb{R}^ + \times \mathbb{R}^n ,n \geqslant 1,\varepsilon \in \{ - 1,0, + 1\} ,$ , is investigated in this paper. The local well-posedness in the energy space H 2(? n ) are proved for \(p > \tfrac{{n + 4}}{{n + 2}}\), and p ≤ 2# ? 1 if n ≥ 5. Global existence is also derived for either defocusing or focusing L 2-subcritical nonlinearities.       19. Global attractor for the weakly damped driven Schrödinger equation in $ H^2 (\mathbb{R}) $      Nikos I. Karachalios  Nikos M. Stavrakakis 《NoDEA : Nonlinear Differential Equations and Applications》2002,9(3):347-360 We discuss the asymptotic behaviour of the Schrödinger equation ¶¶$ iu_{t} + u_{xx} +i\alpha u -k\sigma(|u|^{2})u\, = f, \;\; x \in \mathbb{R}, \;\; t \geq 0,\;\;\;\alpha,\;k>0 $ iu_{t} + u_{xx} +i\alpha u -k\sigma(|u|^{2})u\, = f, \;\; x \in \mathbb{R}, \;\; t \geq 0,\;\;\;\alpha,\;k>0 ¶¶ with the initial condition u(x,0) = u0 (x) u(x,0) = u_0 (x) . We prove existence of a global attractor in\ H2 (\mathbbR) H^2 (\mathbb{R}) , by using a decomposition of the semigroup in weighted Sobolev spaces to overcome the noncompactness of the classical Sobolev embeddings.      20. Essential spectrum of Schrödinger operators with δ-interactions on unbounded hypersurfaces      V. S. Rabinovich 《Mathematical Notes》2017,102(5-6):698-709

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Some estimates for the Schrödinger type operators with nonnegative potentials

In this paper we consider the Schrödinger operator ?Δ + V on \({\mathbb R^d}\), where the nonnegative potential V belongs to the reverse Hölder class \({B_{q_{_1}}}\) for some \({q_{_1}\geq \frac{d}{2}}\) with d ≥ 3. Let \({H^1_L(\mathbb R^d)}\) denote the Hardy space related to the Schrödinger operator L = ?Δ + V and \({BMO_L(\mathbb R^d)}\) be the dual space of \({H^1_L(\mathbb R^d)}\). We show that the Schrödinger type operator \({\nabla(-\Delta +V)^{-\beta}}\) is bounded from \({H^1_L(\mathbb R^d)}\) into \({L^p(\mathbb R^d)}\) for \({p=\frac{d}{d-(2\beta-1)}}\) with \({ \frac{1}{2}<\beta<\frac{3}{2} }\) and that it is also bounded from \({L^p(\mathbb R^d)}\) into \({BMO_L(\mathbb R^d)}\) for \({p=\frac{d}{2\beta-1}}\) with \({ \frac{1}{2}<\beta< 2}\).     

Geometric relative Hardy inequalities and the discrete spectrum of Schrödinger operators on manifolds

The classical Hardy inequality for the Laplacian Δ on ${\mathbb{R}^n}$ shows the borderline-behavior of a potential V for the following question: whether the Schrödinger operator ?Δ + V has a finite or infinite number of the discrete spectrum. In this paper, we will give a sharp generalization of this inequality on ${\mathbb{R}^n}$ to a relative version of that on large classes of complete noncompact manifolds. Replacing ${\mathbb{R}^n}$ by some specific classes of complete noncompact manifolds, including hyperbolic spaces, we also establish some sharp criteria for the above-type question.     

Existence of semiclassical states for a coupled Schrödinger system with potentials and nonlocal nonlinearities

In this paper, we first study a Schrödinger system with nonlocal coupling nonlinearities of Hartree type $$\left\{\begin{array}{ll} -\varepsilon^{2}\Delta u +V_1(x)u = \left ( \int \limits_{\mathbb{R}^{3}} \frac{u^{2}}{|x-y|}{\rm d}y \right)u\,+\, {\beta} \left ( \int \limits_{\mathbb{R}^{3}} \frac{v^{2}}{|x-y|}{\rm d} y \right)u,\\ -\varepsilon^{2} \Delta v +V_2(x)v = \left(\int \limits_{\mathbb{R}^{3}} \frac{v^{2}}{|x-y|}{\rm d}y \right)v \,+ \, {\beta} \left ( \int \limits_{\mathbb{R}^{3}} \frac{u^{2}}{|x-y|}{\rm d}y \right)v. \end{array}\right.$$ Using variational methods, we prove the existence of purely vector ground state solutions for the Schrödinger system if the parameter ${\varepsilon}$ is small enough. Secondly, we also establish some existence results for the coupled Schrödinger system with critical exponents.     

Carleson measures,BMO spaces and balayages associated to Schrdinger operators

Let L be a Schrdinger operator of the form L =-? + V acting on L~2(R~n), n≥3, where the nonnegative potential V belongs to the reverse Hlder class B_q for some q≥n. Let BMO_L(R~n) denote the BMO space associated to the Schrdinger operator L on R~n. In this article, we show that for every f ∈ BMO_L(R~n) with compact support, then there exist g ∈ L~∞(R~n) and a finite Carleson measure μ such that f(x) = g(x) + S_(μ,P)(x) with ∥g∥∞ + |||μ|||c≤ C∥f∥BMO_L(R~n), where S_(μ,P)=∫(R_+~(n+1))Pt(x,y)dμ(y, t),and Pt(x, y) is the kernel of the Poisson semigroup {e-~(t(L)~(1/2))}t0 on L~2(R~n). Conversely, if μ is a Carleson measure, then S_(μ,P) belongs to the space BMO_L(R~n). This extends the result for the classical John-Nirenberg BMO space by Carleson(1976)(see also Garnett and Jones(1982), Uchiyama(1980) and Wilson(1988)) to the BMO setting associated to Schrdinger operators.     

An asymptotic of the negative discrete spectrum of the Schrödinger operator

The Schrödinger operator Hu = -Δu + V(x)u, where V(x) → 0 as ¦x¦ → ∞, is considered in L₂(R^m) for m?3. The asymptotic formula $$N(\lambda ,V) \sim \Upsilon _m \int {(\lambda - V(x))_ + ^{{m \mathord{\left/ {\vphantom {m {2_{dx} }}} \right. \kern-\nulldelimiterspace} {2_{dx} }}} ,} \lambda \to ---0,$$ is established for the number N(λ, V) of the characteristic values of the operator H which are less than λ. It is assumed about the potential V that V = V_o + V₁; V_o < 0, ¦V_o =o (¦V_o¦^3/2) as ¦x¦ → ∞; σ (t/2, V_o) ?cσ (t. V_o) and V₁∈L_m/2,loc, σ(t, V₁) =o (σ (t, V_o)), where σ (t,f)= mes {x:¦f (x) ¦ > t).     

On Isomorphisms of Hardy Spaces Associated with Schrödinger Operators

Let L=?Δ+V is a Schrödinger operator on $\mathbb{R}^{d}$ , d≥3, V≥0. Let $H^{1}_{L}$ denote the Hardy space associated with L. We shall prove that there is an L-harmonic function w, 0<δ≤w(x)≤C, such that the mapping $$H_L^1 \ni f\mapsto wf\in H^1\bigl(\mathbb{R}^d\bigr) $$ is an isomorphism from the Hardy space $H_{L}^{1}$ onto the classical Hardy space $H^{1}(\mathbb{R}^{d})$ if and only if $\Delta^{-1}V(x)=-c_{d}\int_{\mathbb{R}^{d}} |x-y|^{2-d} V(y) dy$ belongs to $L^{\infty}(\mathbb{R}^{d})$ .     

A Spectral Multiplier Theorem Associated with a Schrödinger Operator

We establish a Hörmander type spectral multiplier theorem for a Schrödinger operator \(H=-\Delta +V(x)\) in \(\mathbb {R}^3\), provided V is contained in a large class of short range potentials. This result does not require the Gaussian heat kernel estimate for the semigroup \(e^{-tH}\), and indeed the operator H may have negative eigenvalues. As an application, we show local well-posedness of a 3d quintic nonlinear Schrödinger equation with a potential.     

Positive solutions of Schrödinger equations and fine regularity of boundary points

Given a Lipschitz domain Ω in ${{\mathbb R}^N}$ and a nonnegative potential V in Ω such that V(x) d(x, ?Ω)² is bounded we study the fine regularity of boundary points with respect to the Schrödinger operator L _V := Δ ? V in Ω. Using potential theoretic methods, several conditions are shown to be equivalent to the fine regularity of ${z \in \partial \Omega}$ . The main result is a simple (explicit if Ω is smooth) necessary and sufficient condition involving the size of V for ${z \in \partial \Omega}$ to be finely regular. An intermediate result consists in a majorization of ${\int_A \vert{\frac{ u} {d(.,\partial \Omega)}}\vert^2\, dx}$ for u positive harmonic in Ω and ${A \subset \Omega}$ . Conditions for almost everywhere regularity in a subset A of ?Ω are also given as well as an extension of the main results to a notion of fine ${\mathcal{ L}_1 \vert \mathcal{L}_0}$ -regularity, if ${\mathcal{L}_j = \mathcal{L} - V_j, V_0,\, V_1}$ being two potentials, with V ₀ ≤ V ₁ and ${\mathcal{L}}$ a second order elliptic operator.     

Some estimates for commutators of Riesz transform associated with Schrödinger type operators

Let L₁ = ?Δ + V be a Schr:dinger operator and let L² = (?Δ)² + V² be a Schrödinger type operator on ?ⁿ (n ? 5), where V≠ 0 is a nonnegative potential belonging to certain reverse Hölder class B_s for s ? n/2. The Hardy type space \(H_{{L_2}}^1\) is defined in terms of the maximal function with respect to the semigroup \(\left\{ {{e^{ - t{L_2}}}} \right\}\) 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(bf) generated by the Riesz transform \(R = {\nabla ^2}L_2^{ - 1/2}\), where \(b \in BM{O_\theta }(\varrho )\), which is larger than the space BMO(?ⁿ). Moreover, we prove that R_b is bounded from the Hardy space \(H_{\mathcal{L}_1 }^1 \) into weak \(L_{weak}^1 (\mathbb{R}^n )\).     

Infinitesimal form boundedness and Trudinger’s subordination for the Schrödinger operator

We give explicit analytic criteria for two problems associated with the Schrödinger operator H=-Δ+Q on L²(? ⁿ ) where Q∈D’(? ⁿ ) is an arbitrary real- or complex-valued potential.

First, we obtain necessary and sufficient conditions on Q so that the quadratic form \(\langle{Q}\cdot,\ \cdot\rangle\) has zero relative bound with respect to the Laplacian. For Q∈L¹_loc(? ⁿ ), this property can be expressed in the form of the integral inequality:

$\left\vert\int_{\mathbb{R}^n} |u(x)|^2 Q(x) dx \right\vert\leq\epsilon\| \nabla u \|^2_{L^2(\mathbb{R}^n)} + C(\epsilon) \|u \|^2_{L^2(\mathbb{R}^n)}, \quad\forall u \in C^{\infty}_0(\mathbb{R}^n),$

for an arbitrarily small ε>0 and some C(ε)>0. One of the major steps here is the reduction to a similar inequality with nonnegative function \(|\nabla(1-\Delta)^{-1} Q|^2 + |(1-\Delta)^{-1} Q|\) in place of Q. This provides a complete solution to the infinitesimal form boundedness problem for the Schrödinger operator, and leads to new broad classes of admissible distributional potentials Q, which extend the usual L ^p and Kato classes, as well as those based on the well-known conditions of Fefferman–Phong and Chang–Wilson–Wolff.

Secondly, we characterize Trudinger’s subordination property where C(ε) in the above inequality is subject to the condition C(ε)≤cε^-β(β>0) as ε→+0. Such quadratic form inequalities can be understood entirely in the framework of Morrey–Campanato spaces, using mean oscillations of \(\nabla(1-\Delta)^{-1}Q\) and \((1-\Delta)^{-1}Q\) on balls or cubes. A version of this condition where ε∈(0,+∞) is equivalent to the multiplicative inequality:

$\left\vert\int_{\mathbb{R}^n} |u(x)|^2Q(x)dx\right\vert\leq{C}\|\nabla{u}\|^{2p}_{L^2(\mathbb{R}^n)}\|u\|^{2(1-p)}_{L^2(\mathbb{R}^n)},\quad\forall{u}\in{C}^\infty_0(\mathbb{R}^n),$

with \(p=\frac\beta{1 + \beta}\in(0,1)\). We show that this inequality holds if and only if \(\nabla\Delta^{-1} Q \in{BMO}(\mathbb{R}^n)\) if \(p=\frac{1}{2}\). For \(0 < p < \frac{1}{2}\), it is valid whenever \(\nabla\Delta^{-1}Q\) is Hölder-continuous of order 1-2p, or respectively lies in the Morrey space \(\mathcal{L}^{2,\lambda}\) with λ=n+2-4p if \(\frac{1}{2} < p < 1\). As a consequence, we characterize completely the class of those Q which satisfy an analogous multiplicative inequality of Nash’s type, with \(\|u\|_{L^1(\mathbb{R}^n)}\) in placeof \(\|u\|_{L^2(\mathbb{R}^n)}\).

These results are intimately connected with spectral theory and dynamics of the Schrödinger operator, and elliptic PDE theory.     

Groundstates for the nonlinear Schrödinger equation with potential vanishing at infinity

Groundstates of the stationary nonlinear Schrödinger equation $-\Delta u +V u =K u^{p-1}$ , are studied when the nonnegative function V and K are neither bounded away from zero, nor bounded from above. A special attention is paid in the case of a potential V that goes to 0 at infinity. Conditions on compact embeddings that allow to prove in particular the existence of groundstates are established. The fact that the solution is in ${L^2(\mathbb R^N)}Groundstates of the stationary nonlinear Schr?dinger equation

-Du +V u = K up-1-\Delta u +V u =K u^{p-1}      18. On the well-posedness for stochastic fourth-order Schrödinger equations      Dao-yuan Fang  Lin-zi Zhang  Ting Zhang 《高校应用数学学报(英文版)》2011,26(3):307-318 The influence of the random perturbations on the fourth-order nonlinear Schrödinger equations, $iu_t + \Delta ^2 u + \varepsilon \Delta u + \lambda |u|^{p - 1} u = \dot \xi ,(t,x) \in \mathbb{R}^ + \times \mathbb{R}^n ,n \geqslant 1,\varepsilon \in \{ - 1,0, + 1\} ,$ , is investigated in this paper. The local well-posedness in the energy space H 2(? n ) are proved for \(p > \tfrac{{n + 4}}{{n + 2}}\), and p ≤ 2# ? 1 if n ≥ 5. Global existence is also derived for either defocusing or focusing L 2-subcritical nonlinearities.       19. Global attractor for the weakly damped driven Schrödinger equation in $ H^2 (\mathbb{R}) $      Nikos I. Karachalios  Nikos M. Stavrakakis 《NoDEA : Nonlinear Differential Equations and Applications》2002,9(3):347-360 We discuss the asymptotic behaviour of the Schrödinger equation ¶¶$ iu_{t} + u_{xx} +i\alpha u -k\sigma(|u|^{2})u\, = f, \;\; x \in \mathbb{R}, \;\; t \geq 0,\;\;\;\alpha,\;k>0 $ iu_{t} + u_{xx} +i\alpha u -k\sigma(|u|^{2})u\, = f, \;\; x \in \mathbb{R}, \;\; t \geq 0,\;\;\;\alpha,\;k>0 ¶¶ with the initial condition u(x,0) = u0 (x) u(x,0) = u_0 (x) . We prove existence of a global attractor in\ H2 (\mathbbR) H^2 (\mathbb{R}) , by using a decomposition of the semigroup in weighted Sobolev spaces to overcome the noncompactness of the classical Sobolev embeddings.      20. Essential spectrum of Schrödinger operators with δ-interactions on unbounded hypersurfaces      V. S. Rabinovich 《Mathematical Notes》2017,102(5-6):698-709

On the well-posedness for stochastic fourth-order Schrödinger equations

The influence of the random perturbations on the fourth-order nonlinear Schrödinger equations,

$iu_t + \Delta ^2 u + \varepsilon \Delta u + \lambda |u|^{p - 1} u = \dot \xi ,(t,x) \in \mathbb{R}^ + \times \mathbb{R}^n ,n \geqslant 1,\varepsilon \in \{ - 1,0, + 1\} ,$

, is investigated in this paper. The local well-posedness in the energy space H ²(? ⁿ ) are proved for \(p > \tfrac{{n + 4}}{{n + 2}}\), and p ≤ 2^# ? 1 if n ≥ 5. Global existence is also derived for either defocusing or focusing L ²-subcritical nonlinearities.

     

Global attractor for the weakly damped driven Schrödinger equation in $ H^2 (\mathbb{R}) $

We discuss the asymptotic behaviour of the Schrödinger equation ¶¶$ iu_{t} + u_{xx} +i\alpha u -k\sigma(|u|^{2})u\, = f, \;\; x \in \mathbb{R}, \;\; t \geq 0,\;\;\;\alpha,\;k>0 $ iu_{t} + u_{xx} +i\alpha u -k\sigma(|u|^{2})u\, = f, \;\; x \in \mathbb{R}, \;\; t \geq 0,\;\;\;\alpha,\;k>0 ¶¶ with the initial condition u(x,0) = u₀ (x) u(x,0) = u_0 (x) . We prove existence of a global attractor in\ H² (\mathbbR) H^2 (\mathbb{R}) , by using a decomposition of the semigroup in weighted Sobolev spaces to overcome the noncompactness of the classical Sobolev embeddings.