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

Xuwen Chen; Justin Holmer

Displaying similar documents to “On the Klainerman–Machedon conjecture for the quantum BBGKY hierarchy with self-interaction”

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 $β \int B^{2}$ 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...

$C^{*}$ -basic construction between non-balanced quantum doubles

Qiaoling Xin, Tianqing Cao (2024)

Czechoslovak Mathematical Journal

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For finite groups $X$ , $G$ and the right $G$ -action on $X$ by group automorphisms, the non-balanced quantum double $D (X; G)$ is defined as the crossed product ${(ℂ X^{op})}^{*} ⋊ ℂ G$ . We firstly prove that $D (X; G)$ is a finite-dimensional Hopf $C^{*}$ -algebra. For any subgroup $H$ of $G$ , $D (X; H)$ can be defined as a Hopf $C^{*}$ -subalgebra of $D (X; G)$ in the natural way. Then there is a conditonal expectation from $D (X; G)$ onto $D (X; H)$ and the index is $[G; H]$ . Moreover, we prove that an associated natural inclusion of non-balanced quantum doubles is the crossed product by the...

The basic construction from the conditional expectation on the quantum double of a finite group

Qiaoling Xin, Lining Jiang, Zhenhua Ma (2015)

Czechoslovak Mathematical Journal

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Let $G$ be a finite group and $H$ a subgroup. Denote by $D (G; H)$ (or $D (G)$ ) the crossed product of $C (G)$ and $ℂ H$ (or $ℂ G$ ) with respect to the adjoint action of the latter on the former. Consider the algebra $〈 D (G), e 〉$ generated by $D (G)$ and $e$ , where we regard $E$ as an idempotent operator $e$ on $D (G)$ for a certain conditional expectation $E$ of $D (G)$ onto $D (G; H)$ . Let us call $〈 D (G), e 〉$ the basic construction from the conditional expectation $E : D (G) \to D (G; H)$ . The paper constructs a crossed product algebra $C (G / H \times G) ⋊ ℂ G$ , and proves that there is an algebra isomorphism between...

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...

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 $ℝ^{N}$ , $u \in H^{1} (ℝ^{N})$ , where $N \geq 2$ . Under natural conditions on the nonlinearity $f$ , we prove the existence of $𝑖𝑛𝑓𝑖𝑛𝑖𝑡𝑒𝑙𝑦𝑚𝑎𝑛𝑦𝑛𝑜𝑛𝑟𝑎𝑑𝑖𝑎𝑙𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛𝑠$ in any dimension $N \geq 2$ . Our result complements earlier works of Bartsch and Willem $(N = 4 𝚘𝚛 N \geq 6)$ and Lorca-Ubilla $(N = 5)$ where solutions invariant under the action of $O (2) \times O (N - 2)$ are constructed. In contrast, the solutions we construct are invariant under the action of $D_{k} \times O (N - 2)$ where $D_{k} \subset O (2)$ denotes the dihedral...

$H^{p}$ 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.

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 $u_{0} \in X$ , where $X \in {M_{2, q}^{s} (ℝ), H^{σ} (𝕋), H^{s_{1}} (ℝ) + H^{s_{2}} (𝕋)}$ and $q \in [1, 2]$ , $s \geq 0$ , or $σ \geq 0$ , or $s_{2} \geq s_{1} \geq 0$ . Moreover, if $M_{2, q}^{s} (ℝ) ↪ L^{3} (ℝ)$ , or if $σ \geq \frac{1}{6}$ , or if $s_{1} \geq \frac{1}{6}$ 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...

Quantum expanders and geometry of operator spaces

Gilles Pisier (2014)

Journal of the European Mathematical Society

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We show that there are well separated families of quantum expanders with asymptotically the maximal cardinality allowed by a known upper bound. This has applications to the “growth" of certain operator spaces: It implies asymptotically sharp estimates for the growth of the multiplicity of $M_{N}$ -spaces needed to represent (up to a constant $C > 1$ ) the $M_{N}$ -version of the $n$ -dimensional operator Hilbert space $O H_{n}$ as a direct sum of copies of $M_{N}$ . We show that, when $C$ is close to 1, this multiplicity grows...

Effective Hamiltonians and Quantum States

Lawrence C. Evans (2000-2001)

Séminaire Équations aux dérivées partielles

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We recount here some preliminary attempts to devise quantum analogues of certain aspects of Mather’s theory of minimizing measures [M1-2, M-F], augmented by the PDE theory from Fathi [F1,2] and from [E-G1]. This earlier work provides us with a Lipschitz continuous function $u$ solving the eikonal equation aėȧnd a probability measure $σ$ solving a related transport equation. We present some elementary formal identities relating certain quantum states $ψ$ and $u, σ$ . We show also how...

Capacitary estimates of positive solutions of semilinear elliptic equations with absorbtion

Moshe Marcus, Laurent Véron (2004)

Journal of the European Mathematical Society

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Let $Ω$ be a bounded domain of class $C^{2}$ in $ℝ$ N and let $K$ be a compact subset of $\partial Ω$ . Assume that $q \geq (N + 1) / (N - 1)$ and denote by $U_{K}$ the maximal solution of $- Δ u + u^{q} = 0$ in $Ω$ which vanishes on $\partial Ω ∖ K$ . We obtain sharp upper and lower estimates for $U_{K}$ in terms of the Bessel capacity $C_{2 / q, q^{'}}$ and prove that $U_{K}$ is $σ$ -moderate. In addition we describe the precise asymptotic behavior of $U_{K}$ at points $σ \in K$ , which depends on the “density” of $K$ at $σ$ , measured in terms of the capacity $C_{2 / q, q^{'}}$ .

Cluster ensembles, quantization and the dilogarithm

Vladimir V. Fock, Alexander B. Goncharov (2009)

Annales scientifiques de l'École Normale Supérieure

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A cluster ensemble is a pair $(𝒳, 𝒜)$ of positive spaces (i.e. varieties equipped with positive atlases), coming with an action of a symmetry group $Γ$ . The space $𝒜$ is closely related to the spectrum of a cluster algebra [12]. The two spaces are related by a morphism $p : 𝒜 \to 𝒳$ . The space $𝒜$ is equipped with a closed $2$ -form, possibly degenerate, and the space $𝒳$ has a Poisson structure. The map $p$ is compatible with these structures. The dilogarithm together with its motivic and quantum avatars plays a central...

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 $G_{1} \subseteq G$ or with $G_{2} \subseteq \bar{G}$ , where $\bar{G}$ is the complement of $G$ . For $t \geq 2$ and $0 \leq k \leq ⌊ \frac{t}{2} ⌋$ , let $K_{t}^{- k}$ be the graph...

On upper bounds for total $k$ -domination number via the probabilistic method

Saylí Sigarreta, Saylé Sigarreta, Hugo Cruz-Suárez (2023)

Kybernetika

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For a fixed positive integer $k$ and $G = (V, E)$ a connected graph of order $n$ , whose minimum vertex degree is at least $k$ , a set $S \subseteq V$ is a total $k$ -dominating set, also known as a $k$ -tuple total dominating set, if every vertex $v \in V$ has at least $k$ neighbors in $S$ . The minimum size of a total $k$ -dominating set for $G$ is called the total $k$ -domination number of $G$ , denoted by $γ_{k t} (G)$ . The total $k$ -domination problem is to determine a minimum total $k$ -dominating set of $G$ . Since the exact problem is in general quite difficult...