Scattering and Wave Operators for One-Dimensional Schrödinger Operators with Slowly Decaying Nonsmooth Potentials

We prove existence of modified wave operators for one-dimensional Schrödinger equations with potential in If in addition the potential is conditionally integrable, then the usual Möller wave operators exist. We also prove asymptotic completeness of these wave operators for some classes of random potentials, and for almost every boundary condition for any given potential.     

Harnack Inequality for Non-Local Schrödinger Operators

Let \(x \in \mathbb {R}^{d}\), d ≥ 3, and \(f: \mathbb {R}^{d} \rightarrow \mathbb {R}\) be a twice differentiable function with all second partial derivatives being continuous. For 1 ≤ i, j ≤ d, let \(a_{ij} : \mathbb {R}^{d} \rightarrow \mathbb {R}\) be a differentiable function with all partial derivatives being continuous and bounded. We shall consider the Schrödinger operator associated to

$$\mathcal{L}f(x) = \frac12 \sum\limits_{i=1}^{d} \sum\limits_{j=1}^{d} \frac{\partial}{\partial x_{i}} \left( a_{ij}(\cdot) \frac{\partial f}{\partial x_{j}}\right)(x) + {\int}_{\mathbb{R}^{d}\setminus{\{0\}}} [f(y) - f(x) ]J(x,y)dy $$

where \(J: \mathbb {R}^{d} \times \mathbb {R}^{d} \rightarrow \mathbb {R}\) is a symmetric measurable function. Let \(q: \mathbb {R}^{d} \rightarrow \mathbb {R}.\) We specify assumptions on a, q, and J so that non-negative bounded solutions to

$$\mathcal{L}f + qf = 0 $$

satisfy a Harnack inequality. As tools we also prove a Carleson estimate, a uniform Boundary Harnack Principle and a 3G inequality for solutions to \(\mathcal {L}f = 0.\)

     

On the Negative Spectrum of One-Dimensional Schrödinger Operators with Point Interactions

We investigate negative spectra of one-dimensional (1D) Schrödinger operators with δ- and δ′-interactions on a discrete set in the framework of a new approach. Namely, using the technique of boundary triplets and the corresponding Weyl functions, we complete and generalize the results of Albeverio and Nizhnik (Lett Math Phys 65:27–35, 2003; Methods Funct Anal Topol 9(4):273–286, 2003). For instance, we propose an algorithm for determining the number of negative squares of the operator with δ-interactions. We also show that the number of negative squares of the operator with δ′-interactions equals the number of negative strengths.     

Anderson Localization for Time Periodic Random Schrödinger Operators

Abstract

We prove that at large disorder, Anderson localization in Z ^d is stable under localized time-periodic perturbations by proving that the associated quasi-energy operator has pure point spectrum. The formulation of this problem is motivated by questions of Anderson localization for non-linear Schrödinger equations.     

Eigenfunction Expansions for Schrödinger Operators on Metric Graphs

We construct an expansion in generalized eigenfunctions for Schr?dinger operators on metric graphs. We require rather minimal assumptions concerning the graph structure and the boundary conditions at the vertices.      

Schrödinger Operators on Zigzag Nanotubes

We consider the Schr?dinger operator with a periodic potential on quasi-1D models of zigzag single-wall carbon nanotubes. The spectrum of this operator consists of an absolutely continuous part (intervals separated by gaps) plus an infinite number of eigenvalues with infinite multiplicity. We describe all compactly supported eigenfunctions with the same eigenvalue. We define a Lyapunov function, which is analytic on some Riemann surface. On each sheet, the Lyapunov function has the same properties as in the scalar case, but it has branch points, which we call resonances. We prove that all resonances are real. We determine the asymptotics of the periodic and antiperiodic spectrum and of the resonances at high energy. We show that there exist two types of gaps: i) stable gaps, where the endpoints are periodic and anti-periodic eigenvalues, ii) unstable (resonance) gaps, where the endpoints are resonances (i.e., real branch points of the Lyapunov function). We describe all finite gap potentials. We show that the mapping: potential all eigenvalues is a real analytic isomorphism for some class of potentials. Submitted: October 5, 2006. Accepted: December 15, 2006.     

Bäcklund Transformations for the One-Dimensional Schrödinger Equation

Journal of Applied and Industrial Mathematics - We study the system of equations which bases on the one-dimensional Schrödinger equation and connects the potential, amplitude, and phase...     

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.     

A-Priori Decay for Eigenfunction of Perturbed Periodic Schrödinger Operators

. Recently Neumayr and Metzner [1] have shown that the connected N-point density-correlation functions of the two-dimensional and the one-dimensional Fermi gas at one-loop order generically (i.e.for nonexceptional energy-momentum configurations) vanish/are regular in the small momentum/small energy-momentum limits. Their result is based on an explicit analysis in the sequel of the results of Feldman et al. [2]. In this note we use Ward identities to give a proof of the same fact - in a considerably shortened and simplified way - for any dimension of space.     

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.     

Intrinsic Ultracontractivity for Schrödinger Operators with Mixed Boundary Conditions

Let be a domain in , . Let be a divergence form uniformly elliptic operator with Dirichlet boundary conditions on and Neumann boundary conditions on , where is a closed subset of . We prove intrinsic ultracontractivity for the semigroup associated to the Schrödinger operator , where is a potential in the Kato class, provided that is locally Lipschitz and is given by the boundary of either a Hölder domain of order or a uniformly Hölder domain of order , . Our results extend to the mixed boundary case the results of Bañuelos, Bass and Burdzy, Bass and Hsu, and Davies and Simon.     

The Spectral Function and Principal Eigenvalues for Schrödinger Operators

Let m , 0 m₊ in Kato's class. We investigate the spectral function s( + m) where s( + m) denotes the upper bound of the spectrum of the Schrödinger operator + m. In particular, we determine its derivative at 0. If m_- is sufficiently large, we show that there exists a unique ₁ > 0 such that s( + ₁m) = 0. Under suitable conditions on m⁺ it follows that 0 is an eigenvalue of + ₁m with positive eigenfunction.     

Semiclassical Low Energy Scattering for One-Dimensional Schr?dinger Operators with Exponentially Decaying Potentials

We consider semiclassical Schr?dinger operators on the real line of the form $$H(\hbar)=-\hbar^2 \frac{{\rm d}^2}{{\rm d}x^2}+V(\cdot;\hbar)$$ with ${\hbar >0 }$ small. The potential V is assumed to be smooth, positive and exponentially decaying towards infinity. We establish semiclassical global representations of Jost solutions ${f_\pm(\cdot,E;\hbar)}$ with error terms that are uniformly controlled for small E and ${\hbar}$ , and construct the scattering matrix as well as the semiclassical spectral measure associated with ${H(\hbar)}$ . This is crucial in order to obtain decay bounds for the corresponding wave and Schr?dinger flows. As an application we consider the wave equation on a Schwarzschild background for large angular momenta ? where the role of the small parameter ${\hbar}$ is played by ? ^?1. It follows from the results in this paper and Donninger et al. (Commun Math Phys 2009, arXiv:0911.3179), that the decay bounds obtained in Donninger et al. (Adv Math 226(1):484–540, 2011) and Donninger and Wilhelm (Int Math Res Not IMRN 22:4276–4300, 2010) for individual angular momenta ? can be summed to yield the sharp t ^?3 decay for data without symmetry assumptions.     

Schrödinger Operators with Rapidly Oscillating Potentials

Schr?dinger operators with rapidly oscillating potentials V such as are considered. Such potentials are not relatively compact with respect to the free Schr?dinger operator −Δ. We show that the oscillating potential V do not change the essential spectrum of −Δ. Moreover we derive upper bounds for negative eigenvalue sums of Ĥ.     

Dissipative Schrödinger Operators with Matrix Potentials

Maximal dissipative Schrödinger operators are studied in L ²((–,);E) (dimE=n<) that the extensions of a minimal symmetric operator with defect index (n,n) (in limit-circle case at – and limit point-case at ). We construct a selfadjoint dilation of a dissipative operator, carry out spectral analysis of a dilation, use the Lax–Phillips scattering theory, and find the scattering matrix of a dilation. We construct a functional model of the dissipative operator, determine its characteristic function in terms of the Titchmarsh–Weyl function of selfadjoint operator and investigate its analytic properties. Finally, we prove a theorem on completeness of the eigenvectors and associated vectors of a dissipative Schrödinger operators.     

Global Solutions of Modified One-Dimensional Schrödinger Equation

In this paper, we consider the modified one-dimensional Schrödinger equation:$(D_t-F(D))u=λ|u|^2u,$where F(ξ) is a second order constant coefficients classical elliptic symbol, and with smooth initial datum of size $ε≪1$. We prove that the solution is global-in-time, combining the vector fields method and a semiclassical analysis method introduced by Delort. Moreover, we get a one term asymptotic expansion for $u$when $t→+∞$.     

A One-Dimensional Schrödinger Operator with Square-Integrable Potential

We study the spectral properties of a one-dimensional Schrödinger operator with squareintegrable potential whose domain is defined by the Dirichlet boundary conditions. The main results are concerned with the asymptotics of the eigenvalues, the asymptotic behavior of the operator semigroup generated by the negative of the differential operator under consideration. Moreover, we derive deviation estimates for the spectral projections and estimates for the equiconvergence of the spectral decompositions. Our asymptotic formulas for eigenvalues refine the well-known ones.     

Lifshits Tails for Random Schrödinger Operators with Nonsign Definite Potentials

This paper is devoted to the study of Lifshits tails for random Schr?dinger operator acting on of the form , where H ₀ is a -periodic Schr?dinger operator, λ is a positive coupling constant, are i.i.d and bounded random variables and V is the single site potential with changing sign. We prove that, in the weak disorder regime, at an open band edge, a true Lifshits tail for the random Schr?dinger operator occurs under a certain set of conditions on H ₀ and on V. Submitted: April 17, 2007. Accepted: December 13, 2007.     

Locality of Quadratic Forms for Point Perturbations of Schrödinger Operators

In this paper we study point perturbations of the Schrödinger operators within the framework of Krein's theory of self-adjoint extensions. A locality criterion for quadratic forms is proved for such perturbations.