Decay estimates of discretized Green's functions for Schrdinger type operators

For a sparse non-singular matrix A, generally A~(-1)is a dense matrix. However, for a class of matrices,A~(-1)can be a matrix with off-diagonal decay properties, i.e., |A_(ij)~(-1)| decays fast to 0 with respect to the increase of a properly defined distance between i and j. Here we consider the off-diagonal decay properties of discretized Green's functions for Schr¨odinger type operators. We provide decay estimates for discretized Green's functions obtained from the finite difference discretization, and from a variant of the pseudo-spectral discretization. The asymptotic decay rate in our estimate is independent of the domain size and of the discretization parameter.We verify the decay estimate with numerical results for one-dimensional Schr¨odinger type operators.     

A criterion for Hill operators to be spectral operators of scalar type

We derive necessary and sufficient conditions for a Hill operator (i.e., a one-dimensional periodic Schrö dinger operator) H = ?d ² /dx ² + V to be a spectral operator of scalar type. The conditions show the remarkable fact that the property of a Hill operator being a spectral operator is independent of smoothness (or even analyticity) properties of the potential V. In the course of our analysis, we also establish a functional model for periodic Schrödinger operators that are spectral operators of scalar type and develop the corresponding eigenfunction expansion.The problem of deciding which Hill operators are spectral operators of scalar type appears to have been open for about 40 years.     

Besov Spaces for the Schrödinger Operator with Barrier Potential

Let H = ?d ²/dx ² + V be a Schrödinger operator on the real line, where \({V=c\chi_{[a,b]}}\) , c > 0. We define the Besov spaces for H by developing the associated Littlewood–Paley theory. This theory depends on the decay estimates of the spectral operator \({{\varphi}_j(H)}\) for the high and low energies. We also prove a Mihlin multiplier theorem on these spaces, including the L ^p boundedness result. Our approach has potential applications to other Schrödinger operators with short-range potentials.     

Two-sided estimates for the trace of the difference of two semigroups

This paper deals with the derivation of two-sided estimates for the trace of the difference of two semigroups generated by two Schrödinger operators in L ₂(?³) with trace class difference of the resolvents. Use is made of a purely operator-theoretic technique. The results are stated in a rather general abstract form. The sharpness of our estimates is confirmed by the fact that they imply the asymptotic behavior of the trace of the difference of the semigroups as t → +0. Our considerations are substantially based on the Krein-Lifshits formula and on the Birman-Solomyak representation for the spectral shift function.     

The Morse and Maslov indices for Schrödinger operators

We study the spectrum of Schrödinger operators with matrixvalued potentials, utilizing tools from infinite-dimensional symplectic geometry. Using the spaces of abstract boundary values, we derive relations between the Morse and Maslov indices for a family of operators on a Hilbert space obtained by perturbing a given self-adjoint operator by a smooth family of bounded self-adjoint operators. The abstract results are applied to the Schrödinger operators with θ-periodic, Dirichlet, and Neumann boundary conditions. In particular, we derive an analogue of the Morse-Smale Index Theorem for multi-dimensional Schrödinger operators with periodic potentials. For quasi-convex domains in Rⁿ, we recast the results, connecting the Morse and Maslov indices using the Dirichlet and Neumann traces on the boundary of the domain.     

On Schrödinger Propagator for the Special Hermite Operator

We establish a Strichartz type estimate for the Schrödinger propagator e ^it? for the special Hermite operator ? on ? ⁿ . Our method relies on a regularization technique. We show that no admissibility condition is required on (q,p) when 1≤q≤2.     

Classes of uniform convergence of spectral expansions for the one-dimensional Schrödinger operator with a distribution potential

For the self-adjoint Schrödinger operator ? defined on ? by the differential operation ?(d/dx)² + q(x) with a distribution potential q(x) uniformly locally belonging to the space W ₂ ^?1, we describe classes of functions whose spectral expansions corresponding to the operator ? absolutely and uniformly converge on the entire line ?. We characterize the sharp convergence rate of the spectral expansion of a function using a two-sided estimate obtained in the paper for its generalized Fourier transforms.     

Heat Kernel Estimates for Schrödinger Operators on Exterior Domains with Robin Boundary Conditions

We study Schrödinger operators with Robin boundary conditions on exterior domains in ? ^d . We prove sharp point-wise estimates for the associated semigroups which show, in particular, how the boundary conditions affect the time decay of the heat kernel in dimensions one and two. Applications to spectral estimates are discussed as well.     

<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-Approximation of the Integrated Density of States for Schrödinger Operators with Finite Local Complexity

We study spectral properties of Schrödinger operators on \({\mathbb R^d}\) . The electromagnetic potential is assumed to be determined locally by a colouring of the lattice points in \({\mathbb Z^d}\) , with the property that frequencies of finite patterns are well defined. We prove that the integrated density of states (spectral distribution function) is approximated by its finite volume analogues, i.e. the normalised eigenvalue counting functions. The convergence holds in the space L ^p (I) where I is any finite energy interval and 1 ≤ p < ∞ is arbitrary.     

Harnack inequality for degenerate and singular operators of <Emphasis Type="Italic">p</Emphasis>-Laplacian type on Riemannian manifolds

We study viscosity solutions to degenerate and singular elliptic equations of p-Laplacian type on Riemannian manifolds. The Krylov–Safonov type Harnack inequality for the p-Laplacian operators with \(1<p<\infty \) is established on the manifolds with Ricci curvature bounded from below based on ABP type estimates. We also prove the Harnack inequality for nonlinear p-Laplacian type operators assuming that a nonlinear perturbation of Ricci curvature is bounded below.     

Inverse Problems,Trace Formulae for Discrete Schrödinger Operators

We study discrete Schrödinger operators with compactly supported potentials on Z ^d . Constructing spectral representations and representing S-matrices by the generalized eigenfunctions, we show that the potential is uniquely reconstructed from the S-matrix of all energies. We also study the spectral shift function \({\xi(\lambda)}\) for the trace class potentials, and estimate the discrete spectrum in terms of the moments of \({\xi(\lambda)}\) and the potential.     

Decay of the Fourier transform of surfaces with vanishing curvature

We prove L ^p -bounds on the Fourier transform of measures μ supported on two dimensional surfaces. Our method allows to consider surfaces whose Gauss curvature vanishes on a one-dimensional submanifold. Under a certain non-degeneracy condition, we prove that \({\hat{\mu}\in L^{4+\beta}}\) , β > 0, and we give a logarithmically divergent bound on the L ⁴-norm. We use this latter bound to estimate almost singular integrals involving the dispersion relation, \({e(p)= \sum_1^3 [1-\cos p_j]}\) , of the discrete Laplace operator on the cubic lattice. We briefly explain our motivation for this bound originating in the theory of random Schrödinger operators.     

Restriction Theorems on M\'{e}tiver Groups Associated to Joint Functional Calculus

The authors get on Métivier groups the spectral resolution of a class of operators Open image in new window , the joint functional calculus of the sub-Laplacian and Laplacian on the centre, and then give some restriction theorems together with their asymptotic estimates, asserting the mix-norm boundedness of the spectral projection operators \(\mathcal{P}_\mu^m\) for two classes of functions m(a, b) = (a^α + b^β)^γ or (1 + a^α + b^β)^γ, with α, β > 0, γ ≠ 0.     

Uniform,on the entire axis,convergence of the spectral expansion for Schrödinger operator with a potential-distribution

A uniform, on ?, estimate for the increment of the spectral function θ(λ; x, y) at x = y is proved for the self-adjoint Schrödinger operator A defined on the entire axis ? by the differential operation (?d/dx)² + q(x) with a potential-distribution q(x) that uniformly locally belongs to the space W ₂ ^?1. As a consequence, it is shown that for any function f(x) from the domain of power Aα of the operator with α > 1/4, the spectral expansion of the function that corresponds to the operator A is convergent absolutely and uniformly on the entire axis ?.     

Linear natural operators lifting p-vectors to tensors of type (q, 0) on Weil bundles

We give a classification of all linear natural operators transforming p-vectors (i.e., skew-symmetric tensor fields of type (p, 0)) on n-dimensional manifolds M to tensor fields of type (q, 0) on T^AM, where T^A is a Weil bundle, under the condition that p ≥ 1, n ≥ p and n ≥ q. The main result of the paper states that, roughly speaking, each linear natural operator lifting p-vectors to tensor fields of type (q, 0) on T^A is a sum of operators obtained by permuting the indices of the tensor products of linear natural operators lifting p-vectors to tensor fields of type (p, 0) on T^A and canonical tensor fields of type (q ? p, 0) on T^A.     

Intrinsic Ultracontractivity for Schrödinger Operators Based on Fractional Laplacians

We study the Feynman-Kac semigroup generated by the Schrödinger operator based on the fractional Laplacian ??(???Δ)^α/2???q in R ^d , for q?≥?0, α?∈?(0,2). We obtain sharp estimates of the first eigenfunction φ ₁ of the Schrödinger operator and conditions equivalent to intrinsic ultracontractivity of the Feynman-Kac semigroup. For potentials q such that lim_{|x| →?∞?} q(x)?=?∞ and comparable on unit balls we obtain that φ ₁(x) is comparable to (|x|?+?1)^???d???α (q(x)?+?1)^???1 and intrinsic ultracontractivity holds iff lim_{|x| →?∞?} q(x)/log|x|?=?∞. Proofs are based on uniform estimates of q-harmonic functions.     

<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> estimates for the Schrödinger type operators

Let L _k = (?Δ) ^k + V ^k be a Schrödinger type operator, where k ≥ 1 is a positive integer and V is a nonnegative polynomial. We obtain the L ^p estimates for the operators ?^2k L _k ^?1 and ? ^k L _k ^?1/2 .     

Riesz Transforms Associated with Higher-Order Schrödinger Type Operators

Let L = L ₀ + V be a Schrödinger type operator, where L ₀ is a higher order elliptic operator with bounded complex coefficients in divergence form and V is a signed measurable function. Under the strongly subcritical assumption on V, we study the L ^q boundedness of Riesz transform ? ^m L ^?1/2 for q ≤ 2 based on the off-diagonal estimates of semigroup e ^{?t L} . Furthermore, the authors impose extra regularity assumptions on V to obtain the L ^q boundedness of Riesz transform ? ^m L ^?1/2 for some q > 2. In particular, these results are applied to the more interesting Schrödinger operators L = P(D) + V, where P(D) is any homogeneous positive elliptic operator with constant coefficients.     

On infinite-rank singular perturbations of the Schrödinger operator

Schrödinger operators with infinite-rank singular potentials V=Σ _i,j=1 ^∞ b _ij〈φ_j,·〉φ_i are studied under the condition that the singular elements ψ _j are ξ _j(t)-invariant with respect to scaling transformationsin ?³.