Spectral analysis of percolation Hamiltonians

We study the family of Hamiltonians which corresponds to the adjacency operators on a percolation graph. We characterise the set of energies which are almost surely eigenvalues with finitely supported eigenfunctions. This set of energies is a dense subset of the algebraic integers. The integrated density of states has discontinuities precisely at this set of energies. We show that the convergence of the integrated densities of states of finite box Hamiltonians to the one on the whole space holds even at the points of discontinuity. For this we use an equicontinuity-from-the-right argument. The same statements hold for the restriction of the Hamiltonian to the infinite cluster. In this case we prove that the integrated density of states can be constructed using local data only. Finally we study some mixed Anderson-Quantum percolation models and establish results in the spirit of Wegner, and Delyon and Souillard.Mathematics Subject Classification (2000): 35J10,81Q10,82B43     

Poisson-Type Limit Theorems for Eigenvalues of Finite-Volume Anderson Hamiltonians

We consider the spectral problem for the random Schrödinger operator on the multidimensional lattice torus increasing to the whole of lattice, with an i.i.d. potential (Anderson Hamiltonian). We prove complete Poisson-type limit theorems for the (normalized) eigenvalues and their locations, provided that the upper tails of the distribution of potential decay at infinity slower than the double exponential tails. For the fractional-exponential tails, the strong influence of the parameters of the model on a specification of the normalizing constants is described.     

Spectral theory of one-channel operators and application to absolutely continuous spectrum for Anderson type models

A one-channel operator is a self-adjoint operator on ${?}^2(\mathbb{G})$ for some countable set $\mathbb{G}$ with a rank 1 transition structure along the sets of a quasi-spherical partition of $\mathbb{G}$. Jacobi operators are a very special case. In essence, there is only one channel through which waves can travel across the shells to infinity. This channel can be described with transfer matrices which include scattering terms within the shells and connections to neighboring shells. Not all of the transfer matrices are defined for some countable set of energies. Still, many theorems from the world of Jacobi operators are translated to this setup. The results are then used to show absolutely continuous spectrum for the Anderson model on certain finite dimensional graphs with a one-channel structure. This result generalizes some previously obtained results on antitrees.     

Spectral localization in the hierarchical Anderson model

We prove that a large class of hierarchical Anderson models with spectral dimension has only pure point spectrum.

     

Spectral localization for quantum Hamiltonians with weak random delta interaction

We consider a negative Laplacian in multi-dimensional Euclidean space (or a multi-dimensional layer) with a weak disorder random perturbation. The perturbation consists of a sum of lattice translates of a delta interaction supported on a compact manifold of co-dimension one and modulated by coupling constants, which are independent identically distributed random variables times a small disorder parameter. We establish that the spectrum of the considered operator is almost surely a fixed set, characterize its minimum, give an initial length scale estimate and the Wegner estimate, and conclude that there is a small zone of a pure point spectrum containing the almost sure spectral bottom. The length of this zone is proportional to the small disorder parameter.     

Lower Bounds on the Lowest Spectral Gap of Singular Potential Hamiltonians

We analyze Schr?dinger operators whose potential is given by a singular interaction supported on a sub-manifold of the ambient space. Under the assumption that the operator has at least two eigenvalues below its essential spectrum we derive estimates on the lowest spectral gap. In the case where the sub-manifold is a finite curve in two dimensional Euclidean space the size of the gap depends only on the following parameters: the length, diameter and maximal curvature of the curve, a certain parameter measuring the injectivity of the curve embedding, and a compact sub-interval of the open, negative energy half-axis which contains the two lowest eigenvalues. Dedicated to Krešimir Veselić on the occasion of his 65^th birthday. Submitted: February 20, 2006; Accepted: May 8, 2006     

Spectra of Anderson type models with decaying randomness

In this paper we consider some Anderson type models, with free parts having long range tails and with the random perturbations decaying at different rates in different directions and prove that there is a.c. spectrum in the model which is pure. In addition, we show that there is pure point spectrum outside some interval. Our models include potentials decaying in all directions in which case absence of singular continuous spectrum is also shown.     

Spectral properties of Hamiltonians with a magnetic field at a fixed pseudomomentum. III

The general results (see previous Part II) on the structure of the discrete spectra of energy operators of neutral systems in a homogeneous magnetic field at a fixed pseudomomentum are proved to be applicable to Hamiltonians of arbitrary atoms. Asymptotic expressions for the discrete spectra of Hamiltonians in the presence of a homogeneous magnetic field are found for arbitrary atoms. This paper completes the investigation of the spectral properties of Hamiltonians of neutral systems in a homogeneous magnetic field at a fixed pseudomomentum. The essential and discrete parts of the spectrum for such systems were found previously; however, whether the theorems in Part II were valid for actual n-particle systems remained an open question for the case n>3. Translated from Teoreticheskaya i Mathematicheskaya Fizika, Vol. 120, No. 2, pp. 291–308, August, 1999.     

Spectral properties of Hamiltonians with a magnetic field at a fixed pseudomoment. II

The discrete spectrum of multiparticle Hamiltonians H₀ of neutral systems in a homogeneous magnetic field is studied at a fixed pseudomoment. A general theorem is proved, which describes the discrete spectrum of H₀ under certain conditions in terms of constructed effective one-dimensional differential operators with a known spectrum structure. Based on this theorem, the conditions for a finite or infinite spectrum and the spectral asymptotic forms of the operator H₀ are obtained. The results can be applied to Hamiltonians of any atoms. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 118, No. 1, pp. 15–39, January, 1999.     

Band structure of the spectra of Hamiltonians of regular polynucleotide duplexes

We calculate the band structure of the spectra of Hamiltonians of regular DNA duplexes and show that in single-stranded periodic polynucleotides whose period is determined by the number m of nucleotides in an elementary cell, the spectrum consists of m nonintersecting energy bands. In DNA duplexes, the number of energy bands is equal to 2m, and the bands can intersect. Discrete energy levels can be present in forbidden bands in the case of (semi)bounded chains or duplexes.     

Correlation structure of intermittency in the parabolic Anderson model

Consider the Cauchy problem ∂u(x, t)/∂t = ℋu(x, t) (x∈ℤ^d, t≥ 0) with initial condition u(x, 0) ≡ 1 and with ℋ the Anderson Hamiltonian ℋ = κΔ + ξ. Here Δ is the discrete Laplacian, κ∈ (0, ∞) is a diffusion constant, and ξ = {ξ(x): x∈ℤ ^d } is an i.i.d.random field taking values in ℝ. G?rtner and Molchanov (1990) have shown that if the law of ξ(0) is nondegenerate, then the solution u is asymptotically intermittent. In the present paper we study the structure of the intermittent peaks for the special case where the law of ξ(0) is (in the vicinity of) the double exponential Prob(ξ(0) > s) = exp[−e ^{s /θ}] (s∈ℝ). Here θ∈ (0, ∞) is a parameter that can be thought of as measuring the degree of disorder in the ξ-field. Our main result is that, for fixed x, y∈ℤ ^d and t→∈, the correlation coefficient of u(x, t) and u(y, t) converges to ∥w _ρ∥⁻² _ℓ2Σ_{z ∈ℤd} w _ρ(x+z)w _ρ(y+z). In this expression, ρ = θ/κ while w _ρ:ℤ^d→ℝ⁺ is given by w _ρ = (v _ρ)^{⊗ d} with v _ρ: ℤ→ℝ⁺ the unique centered ground state (i.e., the solution in ℓ²(ℤ) with minimal l ²-norm) of the 1-dimensional nonlinear equation Δv + 2ρv log v = 0. The uniqueness of the ground state is actually proved only for large ρ, but is conjectured to hold for any ρ∈ (0, ∞). empty It turns out that if the right tail of the law of ξ(0) is thicker (or thinner) than the double exponential, then the correlation coefficient of u(x, t) and u(y, t) converges to δ _{x, y} (resp.the constant function 1). Thus, the double exponential family is the critical class exhibiting a nondegenerate correlation structure. Received: 5 March 1997 / Revised version: 21 September 1998     

Hamiltonians and conjugate Hamiltonians of some fourth-order nonlinear ODEs

We first derive the Lagrangians of the reduced fourth-order ordinary differential equations studied by Kudryashov under the assumption that they satisfy the conditions stated by Fels [M.E. Fels, The inverse problem of the calculus of variations for scalar fourth-order ordinary differential equations, Trans. Amer. Math. Soc. 348, 1996, 5007-5029], using Jacobi’s last multiplier technique. In addition we derive the Hamiltonians of these equations using the Jacobi-Ostrogradski theory. Next, we derive the conjugate Hamiltonian equations for such fourth-order equations passing the Painlevé test. Finally, we investigate the conjugate Hamiltonian formulation of certain additional equations belonging to this family.     

A Time-Independent Approach for the Study of the Spectral Shift Function and an Application to Stark Hamiltonians

In this paper we develop a time-independent approach for the study of the spectral shift function (SSF for short). We apply this method for the perturbed Stark Hamiltonian. We obtain a weak and a Weyl-type asymptotics with optimal remainder estimate of the SSF of the operator pair (P = P₀ + V(x), P₀ = ? h²Δ +x₁), x = (x₁,…, x_n) where V(x) ∈ 𝒞^∞(?ⁿ, ?) decays sufficiently fast at infinity, and h is a small positive parameter. Near a non-trapping energy λ, we give a pointwise asymptotic expansions in powers of h of the derivative of the SSF, and we compute explicitly the two leading terms.     

The Hamiltonian structure of equations for quantum averages in systems with matrix Hamiltonians

An infinite system of ordinary differential equations for¯x, ¯p, and for averages of a set of operators is derived for quantum-mechanical problems with a (K×K) matrix HamiltonianH(x,p), x ^N . The set of operators is chosen to be basis in the space Mat _K U(W _N), whereU(W _N) is the universal enveloping algebra of the Heisenberg-Weyl algebraW _N, generated by the time-dependent operatorsÎ, x–¯x(t) · Î, andP–¯p(t) · Î, whereÎ is the identity operator and¯x and¯p are the averages of the position and momentum operators. The system in question can be written in Hamiltonian form; the corresponding Poisson bracket is degenerate and is equal to the sum of the standard bracket on ^2N with respect to the variables (x, p) and the generalized Dirac bracket with respect to the other variables. The possibility of obtaining finite-dimensional approximations to the infinite-dimensional system in the semiclassical limit0 is investigated.Translated fromMatematicheskie Zametki, Vol. 58, No. 6, pp. 803–817, December, 1995.This research was supported by the International Science Foundation under grant No. 78000.     

Spectral synthesis for Euler type operators

A=(a _ij) _i ^j=1 — k-o ,a _ij . :

Adiabatic charge transport and the Kubo formula for Landau‐type Hamiltonians

The adiabatic charge transport is investigated in a two‐dimensional Landau model perturbed by a bounded potential at zero temperature. We show that if the Fermi level lies in a spectral gap, then in the adiabatic limit the accumulated excess Hall charge is given by the linear response Kubo‐?treda formula. The proof relies on the expansion of Nenciu, some generalized phase space estimates, and a bound on the speed of propagation. © 2004 Wiley Periodicals, Inc.