Dynamical Localization for the Random Dimer Schrödinger Operator

We study the one-dimensional random dimer model, with Hamiltonian H _ω =Δ+V _ω , where for all x∈ $\mathbb{Z}$ , V _ω(2x)=V _ω(2x+1) and where the V _ω(2x) are i.i.d. Bernoulli random variables taking the values ±V, V>0. We show that, for all values of Vand with probability one in ω, the spectrum of His pure point. If V≤1 and V≠1/ $\sqrt 2$ , the Lyapunov exponent vanishes only at the two critical energies given by E=±V. For the particular value V=1/ $\sqrt 2$ , respectively, V= $\sqrt 2$ , we show the existence of new additional critical energies at E=±3/ $\sqrt 2$ , respectively, E=0. On any compact interval Inot containing the critical energies, the eigenfunctions are then shown to be semi-uniformly exponentially localized, and this implies dynamical localization: for all q>0 and for all ψ∈ $\ell$ ²( $\mathbb{Z}$ ) with sufficiently rapid decrease $${\mathop {\sup }\limits_t} r_{\psi ,I}^{\left( q \right)} {\kern 1pt} \left( t \right): = {\mathop {\sup }\limits_t} \left\langle {P_I \left( {H\omega } \right)\psi _t ,\left| X \right|^q P_I \left( {H\omega } \right)\psi _t } \right\rangle < \infty $$ Here $\psi _t = e^{- iH_{\omega ^t}} \psi$ , and P _I(H _ω) is the spectral projector of H _ωonto the interval I. In particular, if V>1 and V≠ $\sqrt 2$ , these results hold on the entire spectrum [so that one can take I=σ(H _ω)].     

Universal lower bounds on eigenvalue splittings for one dimensional Schrödinger operators

We provide lower bounds on the eigenvalue splitting for ?d ²/dx ²+V(x) depending only on qualitative properties ofV. For example, ifV is C^∝ on [a, b] andE _n ,E _n?1 are two successive eigenvalues of ?d ²/dx ²+V withu(a)=u(b)=0 boundary conditions, and if \(\lambda = \mathop {\max }\limits_{E \in (E_{n - 1} ,E_n );x \in (a,b)} |E - V(x)|^{1/2} \) , then $$E_n - E_{n - 1} \geqq \pi \lambda ^2 \exp \left[ { - \lambda (b - a)} \right]$$ . The exponential factor in such bounds are saturated precisely in tunneling examples. Our results arenot restricted toV's of compact support, but only require \(E_n< \mathop {\lim }\limits_{\overline {x \to \infty } } V(x)\) .     

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.

     

General lower bounds for resonances in one dimension

Lower bounds are derived for the magnitude of the imaginary parts of the resonance eigenvalues of a Schrödinger operator $$ - \frac{{d^2 }}{{dx^2 }} + V(x)$$ on the line, depending only on the support and bounds ofV and on the real part of the resonance eigenvalue. For example, if the resonance eigenvalue is denotedE +i?, then there existC and ?₀ depending only on ‖E‖_∞ andE such that if the support ofV is contained in an interval of length ? > ?₀, then $$\left| \varepsilon \right| > \frac{{m^3 \sqrt E }}{{(m + \sqrt E )^2 }}\exp ( - m\ell )(1 - C\ell ^{ - 1} ),$$ wherem‖V(x)?E? _∞ ^1/2 .     

The local structure of the spectrum of the one-dimensional Schrödinger operator

Let \(H_V = - \frac{{d^{\text{2}} }}{{dt^{\text{2}} }} + q(t,\omega )\) be an one-dimensional random Schrödinger operator in ?²(?V,V) with the classical boundary conditions. The random potentialq(t, ω) has a formq(t, ω)=F(x _t ), wherex _t is a Brownian motion on the compact Riemannian manifoldK andF:K→R ¹ is a smooth Morse function, \(\mathop {\min }\limits_K F = 0\) . Let \(N_V (\Delta ) = \sum\limits_{Ei(V) \in \Delta } 1 \) , where Δ∈(0, ∞),E _i (V) are the eigenvalues ofH _V . The main result (Theorem 1) of this paper is the following. IfV→∞,E ₀>0,k∈Z ₊ anda>0 (a is a fixed constant) then $$P\left\{ {N_V \left( {E_0 - \frac{a}{{2V}},E_0 + \frac{a}{{2V}}} \right) = k} \right\}\xrightarrow[{V \to \infty }]{}e^{ - an(E_0 )} (an(E_0 ))^k |k!,$$ wheren(E ₀) is a limit state density ofH _V ,V→∞. This theorem mean that there is no repulsion between energy levels of the operatorH _V ,V→∞. The second result (Theorem 2) describes the phenomen of the repulsion of the corresponding wave functions.     

Absence of Negative Energy Spectrum for N-Particle Hamiltonians

We consider the spectrum of the quantum Hamiltonian H for a system of N one-dimensional particles. H is given by $H = \sum\nolimits_{i = 1}^n { - \frac{1}{{2m_i }}\frac{{\partial ^2 }}{{\partial x_i^2 }}} + \sum {_{1 \leqslant i < j \leqslant N} } V_{ij} \left( {x_i - x_j } \right)$ acting in L ²(R ^N ). We assume that each pair potential is a sum of a hard core for |x|≤a, a>0, and a function V _ij (x), |x|>a, with $\smallint _a^\infty \left| {x - a} \right|\left| {V_{ij} \left( x \right)} \right|dx < \infty $ . We give conditions on V ^? _ij (x), the negative part of V _ij (x), which imply that H has no negative energy spectrum for all N. For example, this is the case if V ^? _ij (x) has finite range 2a and $$2m_i \smallint _a^{2a} \left| {x - a} \right|\left| {V_{ij}^ - \left( x \right)} \right|dx < 1.$$ If V ^? _ij is not necessarily small we also obtain a thermodynamic stability bound inf?σ(H)≥?cN, where 0<c<∞, is an N-independent constant.     

Borel summability in the disorder parameter of the averaged Green's function for Gaussian disorder

In this note we prove Borel summability in the disorder parameter of the averaged Green's function <G(E,x,y>) _y of tight binding models $$H_V = - \Delta + V$$ with Gaussian disorder $$d\lambda (V) = (2\pi \gamma )^{ - 1/2} \exp ( - V^2 /2\gamma )dV$$ forγ→0 and fixed large |E|. Using this, we can reconstruct the density of states ?(E)γ from the Borel sums of <G(E,x,x>) _y with ImE↗0 and ImE↘0.     

Perturbation theory for shape resonances and large barrier potentials

We develop a systematic perturbation and resonance theory for the one-dimensional Schrödinger equation of the form $$( - d^2 /dx^2 + U(x) + \lambda V(x) - E)\psi (x) = 0,0 \leqq x< \infty ,$$ where the barrier potentialV(x) is supported only wherex≧1 and is non-negative there, and λ is a real parameter tending to infinity. We prove that every λ=∞ eigenvalue turns into a resonance or an eigenvalue for finite λ.     

Equations of electrodynamics with noninvariant velocity of light

A five-dimensional spaceV ^{5(t, x, c)}, in which the velocity of light c is the fifth coordinate of the event point together with the radius vectorx and time t is is introduced. The equations rot $$rot E = - \frac{1}{c}\frac{{\partial H}}{{\partial t}}; div E = 4\pi \rho ;$$ $$rot H = - \frac{1}{c}\frac{{\partial E}}{{\partial t}} + 4\pi \rho \frac{{o - Nx\dot c/c}}{{c + (1 - N)\dot ct}}; div H = 0,$$ invariant relative to the symmetry group c ₁₆₀₀ ^tf of D'Alembert's equation are constructed in the space V⁵ (E andH are the electric and magnetic fields;ν= dx/dt, c=dc/dt,N and N is a numerical parameter). The transformation properties of the electromagnetic field and the electric charge density ρ are examined for the finite group of transformationsP ₁₀ ^N ×Δ ₁ ^N ·T ₁ ^N . A theorem for adding velocities is formulated. The limiting transition to a Minkowski space M⁴(t,x)?V⁵(t, x,c) is examined.     

A Weighted Dispersive Estimate for Schrödinger Operators in Dimension Two

Let H = ?Δ + V, where V is a real valued potential on ${\mathbb {R}^2}$ satisfying ${\|V(x)|\lesssim \langle x \rangle^{-3-}}$ . We prove that if zero is a regular point of the spectrum of H = ?Δ + V, then $${\| w^{-1} e^{itH}P_{ac}f\|_{L^\infty(\mathbb{R}^2)} \lesssim \frac{1}{|t|\log^2(|t|)} \| w f\|_{L^1(\mathbb{R}^2)},\,\,\,\,\,\,\,\, |t| \geq 2}$$ , with w(x) = (log(2 + |x|))². This decay rate was obtained by Murata in the setting of weighted L ² spaces with polynomially growing weights.     

Asymptotic inverse spectral problem for anharmonic oscillators

We study perturbationsL=A+B of the harmonic oscillatorA=1/2(??²+x ²?1) on ?, when potentialB(x) has a prescribed asymptotics at ∞,B(x)～|x|^?α V(x) with a trigonometric even functionV(x)=Σa _mcosω _m x. The eigenvalues ofL are shown to be λ _k =k+μ _k with small μ _k =O(k ^?γ), γ=1/2+1/4. The main result of the paper is an asymptotic formula for spectral fluctuations {μ _k }, $$\mu _k \sim k^{ - \gamma } \tilde V(\sqrt {2k} ) + c/\sqrt {2k} ask \to \infty ,$$ whose leading term \(\tilde V\) represents the so-called “Radon transform” ofV, $$\tilde V(x) = const\sum {\frac{{a_m }}{{\sqrt {\omega _m } }}\cos (\omega _m x - \pi /4)} .$$ as a consequence we are able to solve explicitly the inverse spectral problem, i.e., recover asymptotic part |x ^?α|V(x) ofB from asymptotics of {µ _k }. ₁ ^∞      

A semi-classical trace formula for Schrödinger operators

LetS _?=??Δ+V, withV smooth. If 0<E ²V(x), the spectrum ofS _? nearE ² consists (for ? small) of finitely-many eigenvalues,λ _j (?). We study the asymptotic distribution of these eigenvalues aboutE ² as ?→0; we obtain semi-classical asymptotics for $$\sum\limits_j {f\left( {\frac{{\sqrt {\lambda _j (\hbar )} - E}}{\hbar }} \right)} $$ with \(\hat f \in C_0^\infty \) , in terms of the periodic classical trajectories on the energy surface \(B_E = \left\{ {\left| \xi \right|^2 + V(x) = E^2 } \right\}\) . This in turn gives Weyl-type estimates for the counting function \(\# \left\{ {j;\left| {\sqrt {\lambda _j (\hbar )} - E} \right| \leqq c\hbar } \right\}\) . We make a detailed analysis of the case when the flow onB _E is periodic.     

On boundary integral operators for diffraction problems on graphs with finitely many exits at infinity

We consider a diffraction problem in a multi-connected domain ?² \ Γ, where Γ is an oriented graph with finitely many edges some of which are infinite. The problem is described by the Helmholtz equation (1) $\mathcal{H}u(x) = \rho (x)\nabla \cdot \rho ^{ - 1} (x)\nabla u(x) + k^2 (x)u(x) = 0,x \in \mathbb{R}^2 \backslash \Gamma ,$ where ρ and k are functions bounded together with all derivatives, and by the transmission conditions (2) $u_ + (t) - u_ - (t) = 0,t \in \Gamma \backslash \mathcal{V},$ (3) $a_ + (t)(\partial u/\partial n_t )_ + (t) - a_ - (t)(\partial u/\partial n_t )_ - (t) + a_0 (t)u(t) = f(t),t \in \Gamma \backslash \mathcal{V},$ where V is the set of vertices, a _± and a ₀ are functions bounded on Γ, slowly oscillating discontinuous at the vertices in V, and slowly oscillating at infinity, and f ∈ L ²(Γ). Using Green’s function for the Helmholtz operator H, we introduce simple- and double-layer potentials and reduce the diffraction problem (1)–(3) to a boundary integral equation. The main objective of the paper is to study the essential spectrum, the Fredholm property, and the index of boundary operators on Γ associated with the problem (1)–(3).     

A rigorous replica trick approach to Anderson localization in one dimension

LetH=?Δ+V onl ²(?), whereV(x),x∈?, are i.i.d.r.v.'s, and letG _L (x,y;E+iη)= 〈x|(H _L?(E+iη))^?1|y〉, whereH _L denotes the operatorH restricted to {?L, ?L+1,...,L} with Dirichlet boundary conditions. We use a supersymmetric replica trick to prove that $$E(|G_L (0,x; E + i\eta )|^2 ) \leqq K\eta ^{ - 2} \exp \{ - m|\log \eta |^{ - \sigma } |x|\} $$ for somem>0, σ>0,K<∞, uniformly inL andE. This estimate, together with the usual necessary estimate on the density of states, implies zero conductivity and gives exponential localization by the Fröhlich, Martinelli, Scoppola, and Spencer method.     

Low temperature phase diagrams for quantum perturbations of classical spin systems

We consider a quantum spin system with Hamiltonian $$H = H^{(0)} + \lambda V,$$ whereH ⁽⁰⁾ is diagonal in a basis ∣s〉=? _x ∣s _x 〉 which may be labeled by the configurationss={s_x} of a suitable classical spin system on ? ^d , $$H^{(0)} |s\rangle = H^{(0)} (s)|s\rangle .$$ We assume thatH ⁽⁰⁾(s) is a finite range Hamiltonian with finitely many ground states and a suitable Peierls condition for excitation, whileV is a finite range or exponentially decaying quantum perturbation. Mapping thed dimensional quantum system onto aclassical contour system on ad+1 dimensional lattice, we use standard Pirogov-Sinai theory to show that the low temperature phase diagram of the quantum spin system is a small perturbation of the zero temperature phase diagram of the classical HamiltonianH ⁽⁰⁾, provided λ is sufficiently small. Our method can be applied to bosonic systems without substantial change. The extension to fermionic systems will be discussed in a subsequent paper.     

On the similarity solution of evolution equation u t =H(x,t, u,u x,u xx , ...)

Let $$\begin{gathered} u^* = u + \in \eta (x,{\text{ }}t,{\text{ }}u), \hfill \\ \hfill \\ \hfill \\ x^* = x + \in \xi (x, t, u{\text{),}} \hfill \\ \hfill \\ \hfill \\ {\text{t}}^{\text{*}} = {\text{ }}t + \in \tau {\text{(}}x,{\text{ }}t,{\text{ }}u), \hfill \\ \end{gathered}$$ be an infinitesimal invariant transformation of the evolution equation u _t =H(x,t,u,?u/?x,...,? ⁿ :u/?x ⁿ . In this paper we give an explicit expression for \(\eta ^{X^i }\) in the ‘determining equation’ $$\eta ^T = \sum\limits_{i = 1}^n {{\text{ }}\eta ^{X^i } {\text{ }}\frac{{\partial H}}{{\partial u_i }} + \eta \frac{{\partial H}}{{\partial u_{} }} + \xi \frac{{\partial H}}{{\partial x}} + \tau } \frac{{\partial H}}{{\partial t}},$$ where u _i =? ⁱ u/?x ⁱ . By using this expression we derive a set of equations with η, ξ, τ as unknown functions and discuss in detail the cases of heat and KdV equations.     

On the field dependence of random walks in the presence of random fields

Numerical simulations and scaling arguments are used to study the field dependence of a random walk in a one-dimensional system with a bias field on each site. The bias is taken randomly with equal probability to be +E or ?E. The probability density¯P(x, t) is found to scale asymptotically as $$\left\{ {[A(E)]^{\beta /2} /\ln ^2 t} \right\}\exp \left( { - \left\{ {x[A(E)]^{\beta /2} /\ln ^2 t} \right\}^\alpha } \right)$$ withA(E)=ln[(1+E)/(1-E)],β=4.25, and α=1.25. The mean square displacement scales as \(\langle x^2 \rangle \sim [A(E)]^{ - \beta } F[tA^\beta (E)]\) , where F(u)～ln⁴ u asymptotically.     

Localization in the ground state of a disordered array of quantum rotators

We consider the zero-temperature behavior of a disordered array of quantum rotators given by the finite-volume Hamiltonian: $$H_\Lambda = - \mathop \Sigma \limits_{x \in \Lambda } \frac{{h(x)}}{2}\frac{{\partial ^2 }}{{\partial \varphi (x)^2 }} - J\mathop \Sigma \limits_{\left\langle {x,y} \right\rangle \in \Lambda } \cos (\varphi (x) - \varphi (y))$$ , wherex,y∈Z ^d , 〈,〉 denotes nearest neighbors inZ ^d ;J>0 andh={h(x)>0,x∈Z ^d } are independent identically distributed random variables with common distributiondμ(h), satisfying ∫h ^?δ dμ(h)<∞ for some δ>0. We prove that for anym>0 it is possible to chooseJ(m) sufficiently small such that, if 0<J<J(m), for almost every choice ofh and everyx∈Z ^d the ground state correlation function satisfies $$\left\langle {\cos (\varphi (x) - \varphi (y))} \right\rangle \leqq C_{x,h,J} e^{ - m\left| {x - y} \right|} $$ for ally∈Z ^d withC _x,h,J <∞.     

Boundary values of analytic functions

It is known that a complex — valued continuous functionS(x) as well as a Schwartz distribution on the real axis can be extended in the complex plane minus the support ofS to an analytic function?(z). In the case of a continuous function the jump of?(z) on the real axis represents exactlyS(x): $$\mathop {\lim }\limits_{\varepsilon \to 0 + } [\hat S(x + i\varepsilon ) - \hat S(x - i\varepsilon )] = S(x)$$ . We call regular a pointx on the support ofS such that \(\mathop {\lim }\limits_{\varepsilon \to 0 + } [\hat S(x + i\varepsilon ) - \hat S(x - i\varepsilon )]\) exists. Conditions are found for the existence of regular points on the support of a distribution. It is possible to call this limit (if this exists) the valueS(x) of the distributionS in the pointx. Properties of this type occur in the theory of dispersion relations.