Displaying similar documents to “On Schrödinger maps from T 1 to  S 2

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.

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...

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...

Finite-dimensional maps and dendrites with dense sets of end points

Hisao Kato, Eiichi Matsuhashi (2006)

Colloquium Mathematicae

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The first author has recently proved that if f: X → Y is a k-dimensional map between compacta and Y is p-dimensional (0 ≤ k, p < ∞), then for each 0 ≤ i ≤ p + k, the set of maps g in the space C ( X , I p + 2 k + 1 - i ) such that the diagonal product f × g : X Y × I p + 2 k + 1 - i is an (i+1)-to-1 map is a dense G δ -subset of C ( X , I p + 2 k + 1 - i ) . In this paper, we prove that if f: X → Y is as above and D j (j = 1,..., k) are superdendrites, then the set of maps h in C ( X , j = 1 k D j × I p + 1 - i ) such that f × h : X Y × ( j = 1 k D j × I p + 1 - i ) is (i+1)-to-1 is a dense G δ -subset of C ( X , j = 1 k D j × I p + 1 - i ) for each 0 ≤ i ≤ p.

Σ s -products revisited

Reynaldo Rojas-Hernández (2015)

Commentationes Mathematicae Universitatis Carolinae

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We show that any Σ s -product of at most 𝔠 -many L Σ ( ω ) -spaces has the L Σ ( ω ) -property. This result generalizes some known results about L Σ ( ω ) -spaces. On the other hand, we prove that every Σ s -product of monotonically monolithic spaces is monotonically monolithic, and in a similar form, we show that every Σ s -product of Collins-Roscoe spaces has the Collins-Roscoe property. These results generalize some known results about the Collins-Roscoe spaces and answer some questions due to Tkachuk [Lifting the Collins-Roscoe...

𝒞 k -regularity for the ¯ -equation with a support condition

Shaban Khidr, Osama Abdelkader (2017)

Czechoslovak Mathematical Journal

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Let D be a 𝒞 d q -convex intersection, d 2 , 0 q n - 1 , in a complex manifold X of complex dimension n , n 2 , and let E be a holomorphic vector bundle of rank N over X . In this paper, 𝒞 k -estimates, k = 2 , 3 , , , for solutions to the ¯ -equation with small loss of smoothness are obtained for E -valued ( 0 , s ) -forms on D when n - q s n . In addition, we solve the ¯ -equation with a support condition in 𝒞 k -spaces. More precisely, we prove that for a ¯ -closed form f in 𝒞 0 , q k ( X D , E ) , 1 q n - 2 , n 3 , with compact support and for ε with 0 < ε < 1 there...

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...

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...

The potential-Ramsey number of K n and K t - k

Jin-Zhi Du, Jian Hua Yin (2022)

Czechoslovak Mathematical Journal

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A nonincreasing sequence π = ( d 1 , ... , d n ) of nonnegative integers is a graphic sequence if it is realizable by a simple graph G on n vertices. In this case, G is referred to as a realization of π . Given two graphs G 1 and G 2 , A. Busch et al. (2014) introduced the potential-Ramsey number of G 1 and G 2 , denoted by r pot ( G 1 , G 2 ) , as the smallest nonnegative integer m such that for every m -term graphic sequence π , there is a realization G of π with G 1 G or with G 2 G ¯ , where G ¯ is the complement of G . For t 2 and 0 k t 2 , let K t - k be the graph...

Hardness of embedding simplicial complexes in d

Jiří Matoušek, Martin Tancer, Uli Wagner (2011)

Journal of the European Mathematical Society

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Let 𝙴𝙼𝙱𝙴𝙳 k d be the following algorithmic problem: Given a finite simplicial complex K of dimension at most k , does there exist a (piecewise linear) embedding of K into d ? Known results easily imply polynomiality of 𝙴𝙼𝙱𝙴𝙳 k 2 ( k = 1 , 2 ; the case k = 1 , d = 2 is graph planarity) and of 𝙴𝙼𝙱𝙴𝙳 k 2 k for all k 3 . We show that the celebrated result of Novikov on the algorithmic unsolvability of recognizing the 5-sphere implies that 𝙴𝙼𝙱𝙴𝙳 d d and 𝙴𝙼𝙱𝙴𝙳 ( d - 1 ) d are undecidable for each d 5 . Our main result is NP-hardness of 𝙴𝙼𝙱𝙴𝙳 2 4 and, more generally, of 𝙴𝙼𝙱𝙴𝙳 k d for all...

Sum-product theorems and incidence geometry

Mei-Chu Chang, Jozsef Solymosi (2007)

Journal of the European Mathematical Society

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In this paper we prove the following theorems in incidence geometry. 1. There is δ > 0 such that for any P 1 , , P 4 , and Q 1 , , Q n 2 , if there are n ( 1 + δ ) / 2 many distinct lines between P i and Q j for all i , j , then P 1 , , P 4 are collinear. If the number of the distinct lines is < c n 1 / 2 then the cross ratio of the four points is algebraic. 2. Given c > 0 , there is δ > 0 such that for any P 1 , P 2 , P 3 2 noncollinear, and Q 1 , , Q n 2 , if there are c n 1 / 2 many distinct lines between P i and Q j for all i , j , then for any P 2 { P 1 , P 2 , P 3 } , we have δ n distinct lines between P and Q j . 3. Given...