Displaying similar documents to “Optimal potentials for Schrödinger operators”

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.

Strichartz and smoothing estimates for Schrödinger operators with large magnetic potentials in 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 · + · 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 < 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...

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 .

Unconditional uniqueness of higher order nonlinear Schrödinger equations

Friedrich Klaus, Peer Kunstmann, Nikolaos Pattakos (2021)

Czechoslovak Mathematical Journal

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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 X , where X { M 2 , q s ( ) , H σ ( 𝕋 ) , H s 1 ( ) + H s 2 ( 𝕋 ) } and q [ 1 , 2 ] , s 0 , or σ 0 , or s 2 s 1 0 . Moreover, if M 2 , q s ( ) L 3 ( ) , or if σ 1 6 , or if s 1 1 6 and s 2 > 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...

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 t + ) 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 ) K T u L 2 ( [ 0 , T ] × Ω ) . We show that the same result remains true for ( i t + - V ) u = 0 where V 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 C ( 𝕋 2 ) and conjectured for V L ( 𝕋 2 ) . The higher dimensional generalization remains open for V L ( 𝕋 n ) .

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 .

Finite-energy sign-changing solutions with dihedral symmetry for the stationary nonlinear Schrödinger equation

Monica Musso, Frank Pacard, Juncheng Wei (2012)

Journal of the European Mathematical Society

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We address the problem of the existence of finite energy solitary waves for nonlinear Klein-Gordon or Schrödinger type equations Δ u - u + f ( u ) = 0 in N , u H 1 ( N ) , where N 2 . Under natural conditions on the nonlinearity f , we prove the existence of 𝑖𝑛𝑓𝑖𝑛𝑖𝑡𝑒𝑙𝑦𝑚𝑎𝑛𝑦𝑛𝑜𝑛𝑟𝑎𝑑𝑖𝑎𝑙𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛𝑠 in any dimension N 2 . Our result complements earlier works of Bartsch and Willem ( N = 4 𝚘𝚛 N 6 ) and Lorca-Ubilla ( N = 5 ) where solutions invariant under the action of O ( 2 ) × O ( N - 2 ) are constructed. In contrast, the solutions we construct are invariant under the action of D k × O ( N - 2 ) where D k O ( 2 ) denotes the dihedral...

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...

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

Yu Liu, Jing Zhang, Jie-Lai Sheng, Li-Juan Wang (2016)

Czechoslovak Mathematical Journal

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Let 1 = - Δ + V be a Schrödinger operator and let 2 = ( - Δ ) 2 + V 2 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 2 1 is defined in terms of the maximal function with respect to the semigroup { e - t 2 } and it is identical to the Hardy space H 1 1 established by Dziubański and Zienkiewicz. In this article, we prove the L p -boundedness of the commutator b = b f - ( b f ) generated by the Riesz transform = 2 2 - 1 / 2 , where b BMO θ ( ρ ) , which is larger...

On Schrödinger maps from T 1 to  S 2

Robert L. Jerrard, Didier Smets (2012)

Annales scientifiques de l'École Normale Supérieure

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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 ( T 1 , S 2 ) , equipped with the weak topology of  H 1 / 2 , to the space of distributions ( 𝒞 ( T 1 , 3 ) ) * . The argument relies...

Almost sure well-posedness for the periodic 3D quintic nonlinear Schrödinger equation below the energy space

Andrea R. Nahmod, Gigliola Staffilani (2015)

Journal of the European Mathematical Society

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We also prove a long time existence result; more precisely we prove that for fixed T > 0 there exists a set Σ T , ( Σ T ) > 0 such that any data φ ω ( x ) H γ ( 𝕋 3 ) , γ < 1 , ω Σ T , evolves up to time T into a solution u ( t ) with u ( t ) - e i t Δ φ ω C ( [ 0 , T ] ; H s ( 𝕋 3 ) ) , s = s ( γ ) > 1 . In particular we find a nontrivial set of data which gives rise to long time solutions below the critical space H 1 ( 𝕋 3 ) , that is in the supercritical scaling regime.

On the Klainerman–Machedon conjecture for the quantum BBGKY hierarchy with self-interaction

Xuwen Chen, Justin Holmer (2016)

Journal of the European Mathematical Society

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We consider the 3D quantum BBGKY hierarchy which corresponds to the N -particle Schrödinger equation. We assume the pair interaction is N 3 β 1 V ( B β ) . For the interaction parameter β ( 0 , 2 / 3 ) , we prove that, provided an energy bound holds for solutions to the BBKGY hierarchy, the N 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 β ( 0 , 3 / 5 ) in [28]. This allows, in the case β ( 0 , 3 / 5 ) , for the application of the Klainerman–Machedon...

Stability and semiclassics in self-generated fields

László Erdős, Soren Fournais, Jan Philip Solovej (2013)

Journal of the European Mathematical Society

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We consider non-interacting particles subject to a fixed external potential V and a self-generated magnetic field B . The total energy includes the field energy β B 2 and we minimize over all particle states and magnetic fields. In the case of spin-1/2 particles this minimization leads to the coupled Maxwell-Pauli system. The parameter β tunes the coupling strength between the field and the particles and it effectively determines the strength of the field. We investigate the stability and...

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.

A radial estimate for the maximal operator associated with the free Schrödinger equation

Sichun Wang (2006)

Studia Mathematica

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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 ( S d f ) ( x , t ) = 1 / ( 2 π ) e i x · ξ e i t | ξ | d f ̂ ( ξ ) d ξ , f ∈ (ℝⁿ), x ∈ ℝⁿ, t ∈ ℝ, ( S * * d f ) ( x ) = s u p t | 1 / ( 2 π ) e i x · ξ e i t | ξ | d f ̂ ( ξ ) d ξ | , 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 ≥...

Optimization problem under two-sided (max, +)/(min, +) inequality constraints

Karel Zimmermann (2020)

Applications of Mathematics

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( max , + ) -linear functions are functions which can be expressed as the maximum of a finite number of linear functions of one variable having the form f ( x 1 , , x h ) = max j ( a j + x j ) , where a j , j = 1 , , h , are real numbers. Similarly ( min , + ) -linear functions are defined. We will consider optimization problems in which the set of feasible solutions is the solution set of a finite inequality system, where the inequalities have ( max , + ) -linear functions of variables x on one side and ( min , + ) -linear functions of variables y on the other side....

Semi-classical standing waves for nonlinear Schrödinger equations at structurally stable critical points of the potential

Jaeyoung Byeon, Kazunaga Tanaka (2013)

Journal of the European Mathematical Society

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We consider a singularly perturbed elliptic equation ϵ 2 Δ u - V ( x ) u + f ( u ) = 0 , u ( x ) > 0 on N , 𝚕𝚒𝚖 x u ( x ) = 0 , where V ( x ) > 0 for any x N . The singularly perturbed problem has corresponding limiting problems Δ U - c U + f ( U ) = 0 , U ( x ) > 0 on N , 𝚕𝚒𝚖 x U ( x ) = 0 , c > 0 . Berestycki-Lions found almost necessary and sufficient conditions on nonlinearity f for existence of a solution of the limiting problem. There have been endeavors to construct solutions of the singularly perturbed problem concentrating around structurally stable critical points of potential V under possibly general conditions...

On the potential theory of some systems of coupled PDEs

Abderrahim Aslimani, Imad El Ghazi, Mohamed El Kadiri, Sabah Haddad (2016)

Commentationes Mathematicae Universitatis Carolinae

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In this paper we study some potential theoretical properties of solutions and super-solutions of some PDE systems (S) of type L 1 u = - μ 1 v , L 2 v = - μ 2 u , on a domain D of d , where μ 1 and μ 2 are suitable measures on D , and L 1 , L 2 are two second order linear differential elliptic operators on D with coefficients of class 𝒞 . We also obtain the integral representation of the nonnegative solutions and supersolutions of the system (S) by means of the Green kernels and Martin boundaries associated with L 1 and L 2 , and...