Dynamical Localization of Quantum Walks in Random Environments

We study shock statistics in the scalar conservation law ∂ _t u+∂ _x f(u)=0, x∈ℝ, t>0, with a convex flux f and spatially random initial data. We show that the Markov property (in x) is preserved for a large class of random initial data (Markov processes with downward jumps and derivatives of Lévy processes with downward jumps). The kinetics of shock clustering is then described completely by an evolution equation for the generator of the Markov process u(x,t), x∈ℝ. We present four distinct derivations for this evolution equation, and show that it takes the form of a Lax pair. The Lax equation admits a spectral parameter as in Manakov (Funct. Anal. Appl. 10:328–329, 1976), and has remarkable exact solutions for Burgers equation (f(u)=u ²/2). This suggests the kinetic equations of shock clustering are completely integrable.     

Growth of Sobolev Norms in Linear Schrödinger Equations with Quasi-Periodic Potential

In this paper, we consider the following problem. Let iu _t +Δu+V(x,t)u= 0 be a linear Schr?dinger equation ( periodic boundary conditions) where V is a real, bounded, real analytic potential which is periodic in x and quasi periodic in t with diophantine frequency vector λ. Denote S(t) the corresponding flow map. Thus S(t) preserves the L ²-norm and our aim is to study its behaviour on H ^s (T ^D ), s> 0. Our main result is the growth in time is at most logarithmic; thus if φ∈H ^s , then

A Multiplicative Ergodic Theorem and Nonpositively Curved Spaces

We study integrable cocycles u(n,x) over an ergodic measure preserving transformation that take values in a semigroup of nonexpanding maps of a nonpositively curved space Y, e.g. a Cartan–Hadamard space or a uniformly convex Banach space. It is proved that for any y∈Y and almost all x, there exist A≥ 0 and a unique geodesic ray γ (t,x) in Y starting at y such that

Long-Time Asymptotics for Strong Solutions¶of the Thin Film Equation

In this paper we investigate the large-time behavior of strong solutions to the one-dimensional fourth order degenerate parabolic equation u _t =−(u u _xxx ) _x , modeling the evolution of the interface of a spreading droplet. For nonnegative initial values u ₀(x)∈H ¹(ℝ), both compactly supported or of finite second moment, we prove explicit and universal algebraic decay in the L ¹-norm of the strong solution u(x,t) towards the unique (among source type solutions) strong source type solution of the equation with the same mass. The method we use is based on the study of the time decay of the entropy introduced in [13] for the porous medium equation, and uses analogies between the thin film equation and the porous medium equation. Received: 2 February 2001 / Accepted: 7 October 2001     

Propagation of electromagnetic waves generated by moving sources in dispersive media

The propagation of electromagnetic waves issued by modulated moving sources of the form j( t,x ) = a( t )e ^{- iw₀ t} [(x)\dot]₀ ( t )d( x - x₀ ( t ) )j\left( {t,x} \right) = a\left( t \right)e^{ - i\omega _0 t} \dot x_0 \left( t \right)\delta \left( {x - x_0 \left( t \right)} \right) is considered, where j(t, x) stands for the current density vector, x = (x ₁, x ₂, x ₃) ∈ ℝ³ for the space variables, t ∈ ℝ for time, t → x ₀(t) ∈ ℝ³ for the vector function defining the motion of the source, ω ₀ for the eigenfrequency of the source, a(t) for a narrow-band amplitude, and δ for the standard δ function. Suppose that the media under consideration are dispersive. This means that the electric and magnetic permittivity ɛ(ω), μ(ω) depends on the frequency ω. We obtain a representation of electromagnetic fields in the form of time-frequency oscillating integrals whose phase contains a large parameter λ > 0 characterizing the slowness of the change of the amplitude a(t) and the velocity [(x)\dot]₀ ( t )\dot x_0 \left( t \right) and a large distance between positions of the source and the receiver. Applying the two-dimensional stationary phase method to the integrals, we obtain explicit formulas for the electromagnetic field and for the Doppler effects. As an application of our approach, we consider the propagation of electromagnetic waves produced by moving source in a cold nonmagnetized plasma and the Cherenkov radiation in dispersive media.     

Lattice Dislocations in a 1-Dimensional Model

The spectral properties of the Schr?dinger operator T(t)=−d ²/dx ²+q(x,t) in L ²(ℝ) are studied, where the potential q is defined by q=p(x+t), x>0, and q=p(x), x<0; p is a 1-periodic potential and t∈ℝ is the dislocation parameter. For each t the absolutely continuous spectrum σ _ac (T(t))=σ _ac (T(0)) consists of intervals, which are separated by the gaps γ _n (T(t))=γ _n (T(0))=(α _n ⁻,α _n ⁺), n≥1. We prove: in each gap γ _n ≠?, n≥ 1 there exist two unique “states” (an eigenvalue and a resonance) λ _n ^±(t) of the dislocation operator, such that λ _n ^±(0)=α _n ^± and the point λ _n ^±(t) runs clockwise around the gap γ _n changing the energy sheet whenever it hits α _n ^±, making n/2 complete revolutions in unit time. On the first sheet λ _n ^±(t) is an eigenvalue and on the second sheet λ _n ^±(t) is a resonance. In general, these motions are not monotonic. There exists a unique state λ₀(t) in the basic gap γ₀(T(t))=γ₀(T(0))=(−∞ ,α₀ ⁺). The asymptotics of λ _n ^±(t) as n→∞ is determined. Received: 5 April 1999 / Accepted: 3 March 2000     

Inverse problem for the Grad-Shafranov equation with affine right-hand side

For a given domain ω ⋐ ℝ² with boundary γ = ∂ω, we study the cardinality of the set $ \mathfrak{A}_\eta \left( \Phi \right) $ \mathfrak{A}_\eta \left( \Phi \right) of pairs of numbers (a, b) for which there is a function u = u _(a,b): ω → ℝ such that ∇² u(x) = au(x) + b ⩾ 0 for x ∈ ω, u| _γ = 0, and ||∇u(s)| − Φ(s) ⩽ η for s ∈ γ. Here η ⩾ 0 stands for a very small number, Φ(s) = |∇(s)| / ∫ _γ |∇v| d γ, and v is the solution of the problem ∇² v = a ₀ v + 1 ⩾ 0 on ω with v| _γ = 0, where a ₀ is a given number. The fundamental difference between the case η = 0 and the physically meaningful case η > 0 is proved. Namely, for η = 0, the set $ \mathfrak{A}_\eta \left( \Phi \right) $ \mathfrak{A}_\eta \left( \Phi \right) contains only one element (a, b) for a broad class of domains ω, and a = a ₀. On the contrary, for an arbitrarily small η > 0, there is a sequence of pairs (a _j , b _j ) ∈ $ \mathfrak{A}_\eta \left( \Phi \right) $ \mathfrak{A}_\eta \left( \Phi \right) and the corresponding functions u _j such that ‖f _u ^j+1‖ − ‖f _u ^j ‖ > 1, where ‖f _u ^j = max _x∈ω |f _u ^j (x)| and f _u ^j (x) = a _j u _j (x) + b _j . Here the mappings f _u ^j : ω → ℝ necessarily tend as j → ∞ to the δ-function concentrated on γ.     

(GL(<Emphasis Type="Italic">n</Emphasis> + 1, ℝ), GL(<Emphasis Type="Italic">n</Emphasis>, ℝ)) is a generalized Gelfand pair

Denote by G = GL(n + 1, ℝ) the group of invertible (n + 1) × (n + 1) matrices with real entries, acting on ℝ ⁿ⁺¹ in the usual way, and let H ₁ = GL(n, ℝ) be the stabilizer of the first unit vector e ₀. Let H ₀ = GL(1, ℝ) and set H = H ₀ × H ₁. It is known that the pair (G,H) is a generalized Gelfand pair. Define a character χ of H by χ(h) = χ(h ₀ h ₁) = χ0(h ₀) where χ0 is a unitary character of H ₀ (h ₀ ∈ H ₀, h ₁ ∈ H ₁). Let σ be the anti-involution on G given by σ(g) = ^t g. In this note, we show that any distribution T on G satisfying T(h ₁ gh ₂) = χ(h ₁ h ₂) T(g) (g ∈ G; h ₁, h ₂ ∈ H) is invariant under the anti-involution σ. This result implies that (G,H ₁) is a generalized Gelfand pair.     

Local exponential estimates for h-pseudodifferential operators and tunneling for Schrödinger, Dirac, and square root Klein-Gordon operators

We consider the class of matrix h-pseudodifferential operators Op _h (a) with symbols a = (a _ij ) _i,j=1 ^N , where the coefficients a _ij ∈ C ^∞(? _x ⁿ × ? _ξ ⁿ ? C(0, 1] satisfy the estimates |? _x ^β g6 _ξ ^α α _ij (x, ξ, h)| ? C _αβ 〈ξ〉 ^m and 〈ξ〉 = (1 + |ξ|²)^1/2 for every multi-indices α, β. We also assume that a _ij (x, ξ) is analytically continued with respect to ξ to a tube domain ? ⁿ + i $ \mathcal{B} We consider the class of matrix h-pseudodifferential operators Op _h (a) with symbols a = (a _ij ) _i,j=1^N, where the coefficients a _ij ∈ C ^∞(ℝ _x ⁿ × ℝ _ξ ⁿ ⊗ C(0, 1] satisfy the estimates |ϖ _x ^β g6 _ξ ^α α _ij (x, ξ, h)| ⩽ C _αβ 〈ξ〉 ^m and 〈ξ〉 = (1 + |ξ|²)^1/2 for every multi-indices α, β. We also assume that a _ij (x, ξ) is analytically continued with respect to ξ to a tube domain ℝ ⁿ + i , where is a bounded domain in ℝ ⁿ containing the origin. The main results of the paper are the local estimates for solutions of h-pseudodifferential equations. Let H _h ^s (ℝ ⁿ , ℂ ^N ) be the space of distributions with values in ℂ ^N which is equipped with the norm , let Ω ⊂ ℝ ⁿ be a bounded open set, let v ∈ C ^∞(ℝ ⁿ ), let ▿v(x) ∈ for any x ∈ Ω, and let . Let u _h (∈ H _h ^s (ℝ ⁿ ,‒ ^N )) be a solution of the equation Op _h (α)u = 0. In this case, for every ϕ ∈ C ₀^∞ (Ω) such that ϕ(x) = 1 on Supp v and for a sufficiently small h ₀ > 0, there exists a constant C > 0 such that the following estimate holds for every h ∈ (0, h ₀]:

((1))

We apply estimate (1) to local tunnel exponential estimates for the behavior as h → 0 of the eigenfunctions of matrix Schr?dinger, Dirac, and square-root Klein-Gordon operators. To the memory of Professor V. A. Borovikov     

Non-linear Stability of Modulated Fronts¶for the Swift–Hohenberg Equation

We consider front solutions of the Swift–Hohenberg equation ∂ _t u= -(1+ ∂ _x ²)² u + ɛ² u -u ³. These are traveling waves which leave in their wake a periodic pattern in the laboratory frame. Using renormalization techniques and a decomposition into Bloch waves, we show the non-linear stability of these solutions. It turns out that this problem is closely related to the question of stability of the trivial solution for the model problem ∂ _t u(x,t) = ∂ _x ² u (x,t)+(1+tanh(x-ct))u(x,t)+u(x,t) ^p with p>3. In particular, we show that the instability of the perturbation ahead of the front is entirely compensated by a diffusive stabilization which sets in once the perturbation has hit the bulk behind the front. Received: 23 February 2001 / Accepted: 27 August 2001     

Spectral theory of elliptic operators in exterior domains

Diverse closed (and selfadjoint) realizations of elliptic differential expressions A = Σ_{0⩽|α|,|β|⩽m} (−1) ^α D ^α a _α,β (x)D ^β , a _α,β (·) ∈ C ^∞($ \bar \Omega $ \bar \Omega ) on smooth (bounded or unbounded) domains Ω in ℝ ⁿ with compact boundary ∂Ω are considered. Trace-ideal properties of powers of resolvent differences for these closed realizations of A are proved by using the concept of boundary triples and operator-valued Weyl-Titchmarsh functions, and estimates for negative eigenvalues of certain selfadjoint extensions of the nonnegative minimal operator are derived. Our results extend classical theorems due to Vishik, Povzner, Birman, and Grubb.     

New data for the 197Au(γ, nX) and 197Au(γ, 2nX) reaction cross sections

The results from experimental and theoretical studies of the total and partial cross sections of photoneutron reactions on the ¹⁹⁷Au isotope were analyzed. The cross sections for reactions σ(γ, nX) = σ(γ, n) + σ(γ, np) + … + σ(γ, 2nX) = σ(γ, 2n) + σ(γ, 2np) + … were evaluated in the energy range 7 ≤ E _γ ≤ 30 MeV using an approach free of the shortcomings of experimental photoneutron multiplicity sorting methods. The total photoneutron reaction cross sections σ^exp(γ, xn) = σ^exp(γ, nX) + 2σ^exp(γ, 2nX) + … = σ^exp(γ, n) + σ^exp(γ, np) + 2σ^exp(γ, 2n) + 2σ^exp(γ, 2np) + … were used as the initial experimental data. The contributions from the cross sections σ(γ, nX) and σ(γ, 2nX) to the cross sections σ^exp(γ, xn) were separated using the multiplicity transition functions F ₁ ^theor = σ^theor(γ, 1nX)/σ^theor(γ, xn) and F ₂ ^theor = σ^theor(γ, 2nX)/σ^theor(γ, xn), calculated within an updated version of the pre-equilibrium model of photonuclear reactions. New evaluated data for both partial reaction cross sections, i.e., σ^eval (γ, 1nX) = F ₁ ^theorσ^exp(γ, xn) and σ^eval(γ, 2nX) = F ₂ ^theorσ^exp(γ, xn), were obtained. The cross sections σeval(γ, nX) and σ^eval.(γ, 2nX) evaluated using the theoretically calculated functions F _1,2^theor are consistent with the Livermore data, but substantially contradict the Saclay data.     

Distribution of a Particle’s Position in the ASEP with the Alternating Initial Condition

In this paper we give the distribution of the position of a particle in the asymmetric simple exclusion process (ASEP) with the alternating initial condition. That is, we find ℙ(X _m (t)≤x) where X _m (t) is the position of the particle at time t which was at m=2k−1, k∈ℤ at t=0. As in the ASEP with step initial condition, there arises a new combinatorial identity for the alternating initial condition, and this identity relates the integrand of the integral formula for ℙ(X _m (t)≤x) to a determinantal form together with an extra product.     

One Dimensional Behavior of Singular N Dimensional Solutions of Semilinear Heat Equations

We consider u(x,t) a solution of u _t =Δu+|u| ^{p − 1} u that blows up at time T, where u:ℝ ^N ×[0, T)→ℝ, p>1, (N−2)p<N+2 and either u(0)≥ 0 or (3N−4)p<3N+8. We are concerned with the behavior of the solution near a non isolated blow-up point, as T−t→ 0. Under a non-degeneracy condition and assuming that the blow-up set is locally continuous and N−1 dimensional, we escape logarithmic scales of the variable T−t and give a sharper expansion of the solution with the much smaller error term (T−t)^{1, 1/2−η} for any η>0. In particular, if in addition p>3, then the solution is very close to a superposition of one dimensional solutions as functions of the distance to the blow-up set. Finally, we prove that the mere hypothesis that the blow-up set is continuous implies that it is C ^{1, 1/2−η} for any η>0. Received: 20 June 2001 / Accepted: 6 October 2001     

Evaluated cross sections of the σ(γ, nX) and σ(γ, 2nX) reactions on 112, 114, 116, 117, 118, 119, 120, 122, 124Sn isotopes

A combined analysis of experimental data on total and partial photoneutron reaction cross sections, obtained using bremsstrahlung γ-radiation and quasi-monoenergetic annihilation photon beams, was performed for nine Sn isotopes. The partial reactions σ^eval(γ, nX) and σ^eval(γ, 2nX) cross sections were evaluated using an approach free of the shortcomings of experimental neutron multiplicity sorting methods. This approach involves calculations within the photonuclear reaction model, based on Fermi gas densities and considering the effects of nucleus deformation, the isospin splitting of its giant dipole resonance (GDR), and experimental data on the total photoneutron cross sections σ^exp(γ, xn) = σ^exp(γ, nX) + 2σ^exp(γ, 2nX) = σ^exp(γ, n) + σ^exp(γ, np) + … + 2σ^exp(γ, 2n) + 2σ^exp(γ, 2np) + …. The evaluated σ^eval(γ, nX) and σ^eval(γ, 2nX) reactions cross sections were obtained using the introduced transition multiplicity functions F ^theor = σ^theor(γ, 2nX)/σ^theor(γ, xn) = σ^theor(γ, 2nX)/[σ^theor(γ, nX) + 2σ^theor(γ, 2nX) + …]; and σ^eval.(γ, 2nX) = F ^theor.σ^exp.(γ, xn) = σ^eval(γ, nX) = (1 − 2F ^theor)σ^exp(γ, xn). The evaluated partial reaction cross sections were used to assess the total photoneutron reaction cross sections σ^eval(γ, sn) = σ^eval(γ, nX) + σ^eval(γ, 2nX) + … as functions of the mass number A. The GDR features of ^{112, 114, 116, 117, 118, 119, 120, 122, 124}Sn isotopes were studied and are discussed here.     

Length of the Instability Intervals for Hill's Equation

The length of instability intervals is investigated for the Hill equation y′′+ω(ω− 2∈p(x)y = 0, where p(x) has an infinite Fourier series of coefficients c _n. For any small ∈ it is shown that these lengths are completely characterized by the c _n's.     

Robin boundary condition and shock problem for the focusing nonlinear Schrödinger equation

We consider the initial boundary value (IBV) problem for the focusing nonlinear Schrödinger equation in the quarter plane x>0, t >0 in the case of periodic initial data, u(x,0) = α exp(?2iβx) (or asymptotically periodic, u(x, 0) =α exp(?2iβx)→0 as x→∞), and a Robin boundary condition at x = 0: u_x(0, t)+qu(0, t) = 0, q ≠ 0. Our approach is based on the unified transform (the Fokas method) combined with symmetry considerations for the corresponding Riemann-Hilbert (RH) problems. We present the representation of the solution of the IBV problem in terms of the solution of an associated RH problem. This representation also allows us to determine an initial value (IV) problem, of a shock type, a solution of which being restricted to the half-line x > 0 is the solution of the original IBV problem. In the case β < 0, the large-time asymptotics of the solution of the IBV problem is presented in the “rarefaction” sector, demonstrating, in particular, an oscillatory behavior of the boundary values in the case q > 0, contrary to the decay to 0 in the case q < 0.     

Existence of solutions for Schrödinger evolution equations

We study the existence, uniqueness and regularity of the solution of the initial value problem for the time dependent Schrödinger equationiu/t=(–1/2)u+V(t,x)u,u(0)=u ₀. We provide sufficient conditions onV(t,x) such that the equation generates a unique unitary propagatorU(t,s) and such thatU(t,s)u ₀C ¹(,L ²) C ⁰(H ²( ⁿ )) foru ₀H ²( ⁿ ). The conditions are general enough to accommodate moving singularities of type x^–2+(n4) or x^–n/2+(n3).     

An invariant measure for the equationu tt −u xx +u 3=0

Numerical studies of the initial boundary-value problem of the semilinear wave equationu _tt –u _xx +u ³=0 subject to periodic boundary conditionsu(t, 0)=u(t, 2),u _t (t, 0)=u _t (t, 2) and initial conditionsu(0,x)=u ₀(x),u _t(0,x)=v ₀(x), whereu ₀(x) andv ₀(x) satisfy the same periodic conditions, suggest that solutions ultimately return to a neighborhood of the initial stateu ₀(x),v ₀(x) after undergoing a possibly chaotic evolution. In this paper an appropriate abstract space is considered. In this space a finite measure is constructed. This measure is invariant under the flow generated by the Hamiltonian system which corresponds to the original equation. This enables one to verify the above returning property.     

Statistics of a Flux in Burgers Turbulence¶with One-Sided Brownian Initial Data

We study the statistics of the flux of particles crossing the origin, which is induced by the dynamics of ballistic aggregation in dimension 1, under certain random initial conditions for the system. More precisely, we consider the cases when particles are uniformly distributed on ℝ at the initial time, and if u(x,t) denotes the velocity of the particle located at x at time t, then u(x,0)= 0 for x<0 and (u(x,0), x≥ 0) is either a white noise or a Brownian motion. Received: 18 April 2001 / Accepted: 16 July 2001