Asymptotic behavior of solutions to certain nonlinear Schrödinger-Hartree equations

The asymptotic behavior of solutions to the Cauchy problem for the equation $$i\psi _\imath = \frac{1}{2}\Delta \psi - \upsilon (\psi )\psi , \upsilon = r^{ - 1} *|\psi |^2 ,$$ and for systems of similar form, is studied. It is shown that the norms $$\parallel \psi (t)\parallel _{L_2 (|x| \leqq R)}^2 + \parallel \nabla \psi (t)\parallel _{L_2 (|x| \leqq R)}^2 $$ are integrable in time for any fixedR>0, from which it follows that $$\mathop {\lim }\limits_{t \to \infty } \parallel \psi (t)\parallel _{L_2 (|x| \leqq R)} = 0.$$ \] Nevertheless, it is established that anL ₂-scattering theory is impossible.     

Non-existence of spontaneous magnetization in a one-dimensional Ising ferromagnet

It is proved that an infinite linear chain of spins μ _j =±1, with an interaction energy $$H = - \Sigma J(i - j)\mu _i \mu _j $$ has zero spontaneous magnetization at all finite temperatures, provided thatJ (n) is non-negative and that $$H = - \Sigma J(i - j)\mu _i \mu _j $$ . This shows that a theorem ofRuelle, establishing the absence of long-range order when the sum Σn J (n) converges, is not the best possible.     

Time decay of solutions to the cauchy problem for time-dependent Schrödinger-Hartree equations

We consider the time-dependent Schrödinger-Hartree equation (1) $$iu_t + \Delta u = \left( {\frac{1}{r}*|u|^2 } \right)u + \lambda \frac{u}{r},(t, x) \in \mathbb{R} \times \mathbb{R}^3 ,$$ (2) $$u(0,x) = \phi (x) \in \Sigma ^{2,2} ,x \in \mathbb{R}^3 ,$$ where λ≧0 and \(\Sigma ^{2,2} = \{ g \in L^2 ;\parallel g\parallel _{\Sigma ^{2,2} }^2 = \sum\limits_{|a| \leqq 2} {\parallel D^a g\parallel _2^2 + \sum\limits_{|\beta | \leqq 2} {\parallel x^\beta g\parallel _2^2< \infty } } \} \) . We show that there exists a unique global solutionu of (1) and (2) such that $$u \in C(\mathbb{R};H^{1,2} ) \cap L^\infty (\mathbb{R};H^{2,2} ) \cap L_{loc}^\infty (\mathbb{R};\Sigma ^{2,2} )$$ with $$u \in L^\infty (\mathbb{R};L^2 ).$$ Furthermore, we show thatu has the following estimates: $$\parallel u(t)\parallel _{2,2} \leqq C,a.c. t \in \mathbb{R},$$ and $$\parallel u(t)\parallel _\infty \leqq C(1 + |t|)^{ - 1/2} ,a.e. t \in \mathbb{R}.$$      

Localization in the ground state of the ising model with a random transverse field

We study the zero-temperature behavior of the Ising model in the presence of a random transverse field. The Hamiltonian is given by $$H = - J\sum\limits_{\left\langle {x,y} \right\rangle } {\sigma _3 (x)\sigma _3 (y) - \sum\limits_x {h(x)\sigma _1 (x)} } $$ whereJ>0,x,y∈Z ^d, σ₁, σ₃ are the usual Pauli spin 1/2 matrices, andh={h(x),x∈Z ^d} are independent identically distributed random variables. We consider the ground state correlation function 〈σ₃(x)σ₃(y)〉 and prove:

1. Letd be arbitrary. For anym>0 andJ sufficiently small we have, for almost every choice of the random transverse fieldh and everyx∈Z ^d, that $$\left\langle {\sigma _3 (x)\sigma _3 (y)} \right\rangle \leqq C_{x,h} e^{ - m\left| {x - y} \right|} $$ for ally∈Z ^d withC _{x h} <∞.
2. Letd≧2. IfJ is sufficiently large, then, for almost every choice of the random transverse fieldh, the model exhibits long range order, i.e., $$\mathop {\overline {\lim } }\limits_{\left| y \right| \to \infty } \left\langle {\sigma _3 (x)\sigma _3 (y)} \right\rangle > 0$$ for anyx∈Z ^d.

     

Energy weighting for the upgrade of the CMS HCAL

In these simulation studies an energy weighting method is applied to the signals of the CMS hadronic calorimeter readout with a longitudinal segmentation for a possible future upgrade. Tabulated weighting factors are used to compensate for the different response of hadronic and electromagnetic energy depositions of simulated pion showers in the hadronic calorimeter. The weighting improves the relative energy resolution: $$ (\sigma _E /E)^2 = \left[ {((92.2 \pm 0.6)\% /\sqrt E )^2 + ((6.5 \pm 0.1)\% )^2 } \right] $$ (before weighting), $$ (\sigma _{E,weight} /E)^2 = \left[ {((85.4 \pm 0.5)\% /\sqrt E )^2 + ((4.4 \pm 0.1)\% )^2 } \right] $$ (after weighting), where E in the square root has units of GeV.     

Pesin’s Entropy Formula for Systems Between C^1 and C^{1+\alpha }

In this article we give a new observation of Pesin’s entropy formula, motivated from Mañé’s proof of (Ergod Theory Dyn Sys 1:95–102, 1981). Let \(M\) be a compact Riemann manifold and \(f:\,M\rightarrow M\) be a \(C^1\) diffeomorphism on \(M\) . If \(\mu \) is an \(f\) -invariant probability measure which is absolutely continuous relative to Lebesgue measure and nonuniformly-H \(\ddot{\text {o}}\) lder-continuous(see Definition 1.1), then we have Pesin’s entropy formula, i.e., the metric entropy \(h_\mu (f)\) satisfies $$\begin{aligned} h_{\mu }(f)=\int \sum _{\lambda _i(x)> 0}\lambda _i(x)d\mu , \end{aligned}$$ where \(\lambda _1(x)\ge \lambda _2(x)\ge \cdots \ge \lambda _{dim\,M}(x)\) are the Lyapunov exponents at \(x\) with respect to \(\mu .\) Nonuniformly-H \(\ddot{\text {o}}\) lder-continuous is a new notion from probabilistic perspective weaker than \(C^{1+\alpha }.\)      

BV estimates fail for most quasilinear hyperbolic systems in dimensions greater than one

We show that for most non-scalar systems of conservation laws in dimension greater than one, one does not have BV estimates of the form $$\begin{gathered} \parallel \overline V u(\overline t )\parallel _{T.V.} \leqq F(\parallel \overline V u(0)\parallel _{T.V.} ), \hfill \\ F \in C(\mathbb{R}),F(0) = 0,F Lipshitzean at 0, \hfill \\ \end{gathered} $$ even for smooth solutions close to constants. Analogous estimates forL ^p norms $$\parallel u(\overline t ) - \overline u \parallel _{L^p } \leqq F(\parallel u(0) - \overline u \parallel _{L^p } ),p \ne 2$$ withF as above are also false. In one dimension such estimates are the backbone of the existing theory.     

A global attracting set for the Kuramoto-Sivashinsky equation

New bounds are given for the L²-norm of the solution of the Kuramoto-Sivashinsky equation $$\partial _t U(x,t) = - (\partial _x^2 + \partial _x^4 )U(x,t) - U(x,t)\partial _x U(x,t)$$ , for initial data which are periodic with periodL. There is no requirement on the antisymmetry of the initial data. The result is $$\mathop {\lim \sup }\limits_{t \to \infty } \left\| {U( \cdot ,t)} \right\|_2 \leqslant const. L^{8/5} $$ .     

A limit law for the ground state of Hill's equation

It is proved that the ground state Λ(L) of (?1)x the Schrödinger operator with white noise potential, on an interval of lengthL, subject to Neumann, periodic, or Dirichlet conditions, satisfies the law $$\mathop {\lim }\limits_{L \uparrow \infty } P[(L/\pi )\Lambda ^{1/2} \exp ( - \tfrac{8}{3}\Lambda ^{3/2} ) > x] = \left\{ {\begin{array}{*{20}c} {1forx< 0} \\ {e^{ - x} forx \geqslant 0} \\ \end{array} } \right.$$      

Analyticity of the scattering operator for the nonlinear Klein-Gordon equation with cubic nonlinearity

The wave and scattering operators for the equation $$\left( {\square + m^2 } \right)\varphi + \lambda \varphi ^2 = 0$$ withm>0 and λ>0 on four-dimensional Minkowski space are analytic on the space of finite-energy Cauchy data, i.e.L ₂ ¹ (R ³)⊕L ₂(R ³).     

Positivity and monotonicity properties ofC 0-semigroups. I

If exp {?tH}, exp {?tK}, are self-adjoint, positivity preserving, contraction semigroups on a Hilbert space ?=L ²(X;dμ) we write (*) $$e^{ - tH} \succcurlyeq e^{ - tK} \succcurlyeq 0$$ whenever exp {?tH}-exp {?tK} is positivity preserving for allt≧0 and then we characterize the class of positive functions for which (*) always implies $$e^{ - tf(H)} \succcurlyeq e^{ - tf(K)} \succcurlyeq 0.$$ This class consists of thef∈C ^∞(0, ∞) with $$( - 1)^n f^{(n + 1)} (x) \geqq 0,x \in (0,\infty ),n = 0,1,2, \ldots .$$ In particular it contains the class of monotone operator functions. Furthermore if exp {?tH} isL ^p (X;dμ) contractive for allp∈[1, ∞] and allt>0 (or, equivalently, forp=∞ andt>0) then exp {?tf(H)} has the same property. Various applications to monotonicity properties of Green's functions are given.     

Spectral properties of one dimensional quasi-crystals

In this paper we prove that the one dimensional Schrödinger operator onl ²(?) with potential given by: $$\upsilon (n) = \lambda \chi _{[1 - \alpha , 1[} (x + n\alpha )\alpha \notin \mathbb{Q}$$ has a Cantor spectrum of zero Lebesgue measure for any irrationalα and any λ>0. We can thus extend the Kotani result on the absence of absolutely continuous spectrum for this model, to all .     

Strongly positive semigroups and faithful invariant states

Let (?, τ, ω) denote aW*-algebra ?, a semigroupt>0?τ _t of linear maps of ? into ?, and a faithful τ-invariant normal state ω over ?. We assume that τ is strongly positive in the sense that $$\tau _t (A^ * A) \geqq \tau _t (A)^ * \tau _t (A)$$ for allA∈? andt>0. Therefore one can define a contraction semigroupT on ?= \(\overline {\mathcal{M}\Omega } \) by $$T_t A\Omega = \tau _t (A)\Omega ,{\rm A} \in \mathcal{M},$$ where Ω is the cyclic and separating vector associated with ω. We prove 1. the fixed points ?(τ) of τ are given by ?(τ)=?∩T′=?∩E′, whereE is the orthogonal projection onto the subspace ofT-invariant vectors, 2. the state ω has a unique decomposition into τ-ergodic states if, and only if, ?(τ) or {?υE}′ is abelian or, equivalently, if (?, τ, ω) is ?-abelian, 3. the state ω is τ-ergodic if, and only if, ?υE is irreducible or if $$\mathop {\inf }\limits_{\omega '' \in Co\omega 'o\tau } \left\| {\omega '' - \omega '} \right\| = 0$$ for all normal states ω′ where Coω′°τ denotes the convex hull of {ω′°τ _t } _t>0. Subsequently we assume that τ is 2-positive,T is normal, andT* _t ?₊Ω \( \subseteqq \overline {\mathcal{M}_ + \Omega } \) , and then prove 4. there exists a strongly positive semigroup |τ| which commutes with τ and is determined by $$\left| \tau \right|_t \left( A \right)\Omega = \left| {T_t } \right|A\Omega ,$$ 5. results similar to 1 and 2 apply to |τ| but the τ-invariant state ω is |τ|-ergodic if, and only if, $$\mathop {\lim }\limits_{t \to \infty } \left\| {\omega 'o\tau _t - \omega } \right\| = 0$$ for all normal states ω′.     

Note on trace inequalities

Under conditions which are sufficiently general for physical applications the trace inequalities $$tr e^{ - (A + B)} \leqq tr e^{ - A} e^{ - B} $$ and $$|tr e^{ - (A + iB)} | \leqq tr e^{ - A} $$ withA andB self adjoint are shown in a rigorous manner.     

On the freund-witten adelic formula for Veneziano amplitudes

On the basis of the analysis of the adele group (Tate's formula), a regularization for the divergent infinite product ofp-adic Г-functions $$\Gamma _p (\alpha ) = \frac{{1 - p^{\alpha - 1} }}{{[ - p^{ - \alpha } }}$$ is proposed, and the adelic formula is proved $$reg\coprod\limits_{p = 2}^\infty {\Gamma _p (\alpha )} = \frac{{\zeta (\alpha )}}{{\zeta (1 - \alpha )}}$$ whereζ(α) is the Riemannζ-function.     

A remark on tensor representations

The identity $$\sum\limits_{v = 0} {\left( {\begin{array}{*{20}c} {n + 1} \\ v \\ \end{array} } \right)\left[ {\left( {\begin{array}{*{20}c} {n - v} \\ v \\ \end{array} } \right) - \left( {\begin{array}{*{20}c} {n - v} \\ {v - 1} \\ \end{array} } \right)} \right] = ( - 1)^n } $$ is proved and, by means of it, the coefficients of the decomposition ofD ₁ ⁿ into irreducible representations are found. It holds: ifD ₁ ⁿ \(\mathop {\sum ^n }\limits_{m = 0} A_{nm} D_m \) , then $$A_{nm} = \mathop \sum \limits_{\lambda = 0} \left( {\begin{array}{*{20}c} n \\ \lambda \\ \end{array} } \right)\left[ {\left( {\begin{array}{*{20}c} \lambda \\ {n - m - \lambda } \\ \end{array} } \right) - \left( {\begin{array}{*{20}c} \lambda \\ {n - m - \lambda - 1} \\ \end{array} } \right)} \right].$$      

The Lifshitz Tail and Relaxation to Equilibrium in the One-Dimensional Disordered Ising Model

We study spectral properties of the generator of the Glauber dynamics for a 1D disordered stochastic Ising model with random bounded couplings. By an explicit representation for the upper branch of the generator we get an asymptotic formula for the integrated density of states of the generator near the upper edge of the spectrum. This asymptotic behavior appears to have the form of the Lifshitz tail, which is typical for random operators near fluctuation boundaries. As a consequence we find the asymptotics for the average over the disorder of the time-autocorrelation function to be $$\langle \langle \sigma _{\text{0}}^\omega (t),\sigma _0 (0)\rangle _{P(\omega ) = } {\text{exp\{ }} - gt - kt^{1/3} {\text{(1 + }}o(1){\text{)\} as }}t \to \infty $$ with constants g, k depending on the distribution of the random couplings.     

Scattering of electrons due to screw dislocations

The phase dismatching effect on the scattering due to screw dislocations is reformulated to take the discreteness of lattice sites into account. Thet-matrix for an electron scattered from the statep top′ is $$\begin{gathered} t\left( {p,p'} \right) = ip_z T\exp \left\{ {i\left( {p - p'} \right) \cdot m_A } \right\}\exp \left\{ {i\left( {p - p'} \right) \cdot \left( {i + j} \right)/2} \right\} \hfill \\ \cdot \frac{{\left[ {\exp \left( { - ip_y } \right) - \exp \left( {ip'_y } \right)} \right] + \left( {\upsilon _y /\upsilon _x } \right)\left[ {\exp \left( {ip_x } \right) - \exp \left( { - ip'_x } \right)} \right]}}{{1 - \exp \left[ {i\left\{ {\left( {p_x - p'_x } \right) + \left( {\upsilon _y /\upsilon _x } \right)\left( {p_y - p'_y } \right)} \right\}} \right]}} \hfill \\ \end{gathered}$$ for 0≦v _y ≦v _x ≦1 and |p _y |, |p′ _y |?1. Here,v is the group velocity of the incident electron andm _A is the position of the dislocation axis. All vector notations represent vectors in two-dimensional space, the unit vectors of which are represented byi andj. Expressions for |p _y |, |p′ _y |?π and other values ofv are obtained through simple modifications. As an application, the resistivity due to screw dislocations is discussed qualitatively.     

Indecomposable representations with invariant inner product

Consequences of the existence of an invariant (necessarily indefinite) non-degenerate inner product for an indecomposable representation π of a groupG on a space \(\mathfrak{H}\) are studied. If π has an irreducible subrepresentation π₁ on a subspace \(\mathfrak{H}_1 \) , it is shown that there exists an invariant subspace \(\mathfrak{H}_2 \) of \(\mathfrak{H}\) containing \(\mathfrak{H}_1 \) and satisfying the following conditions: (1) the representation π ₁ ^# =π mod \(\mathfrak{H}_2 \) on \(\mathfrak{H}\) mod \(\mathfrak{H}_2 \) is conjugate to the representation (π₁, \(\mathfrak{H}_1 \) ), (2) \(\mathfrak{H}_1 \) is a null space for the inner product, and (3) the induced inner product on \(\mathfrak{H}_2 \) mod \(\mathfrak{H}_1 \) is non-degenerate and invariant for the representation $$\pi _2 = (\pi _2 |_{\mathfrak{H}_2 } )\bmod \mathfrak{H}_1 ,$$ a special example being the Gupta-Bleuler triplet for the one-particle space of the free classical electromagnetic field with \(\mathfrak{H}_1 \) =space of longitudinal photons and \(\mathfrak{H}_2 \) =the space defined by the subsidiary condition.     

Conformally covariant field equations I. first-order equations with non-vanishing mass

The connection of conformal fields with the Mackey theory is discussed. The necessary and sufficient conditions for the finite or infinite component field equation $$\left( {L_\mu \partial ^\mu + m} \right)\psi \left( x \right) = 0$$ to be conformally covariant, are derived. The conditions are then explicitly solved under very general assumptions and thus conformally covariant equations of the above type are explicitly found (Theorem 5). The circumstances under which the equation may be obtained from a Lagrangian are discussed.