The AKNS Hierarchy and Finite-Gap Schrödinger Potentials

We consider the AKNS hierarchy and find the necessary and sufficient conditions for functions p and q to become solutions of the AKNS hierarchy. Using the functions p and q, we construct finite-gap Schrödinger potentials.     

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.     

Sigma Functions and Lie Algebras of Schrödinger Operators

Functional Analysis and Its Applications - In a 2004 paper by V. M. Buchstaber and D. V. Leikin, published in “Functional Analysis and Its Applications,” for each $$g &gt; 0$$ , a...     

On Isomorphisms of Hardy Spaces Associated with Schrödinger Operators

Let L=?Δ+V is a Schrödinger operator on $\mathbb{R}^{d}$ , d≥3, V≥0. Let $H^{1}_{L}$ denote the Hardy space associated with L. We shall prove that there is an L-harmonic function w, 0<δ≤w(x)≤C, such that the mapping $$H_L^1 \ni f\mapsto wf\in H^1\bigl(\mathbb{R}^d\bigr) $$ is an isomorphism from the Hardy space $H_{L}^{1}$ onto the classical Hardy space $H^{1}(\mathbb{R}^{d})$ if and only if $\Delta^{-1}V(x)=-c_{d}\int_{\mathbb{R}^{d}} |x-y|^{2-d} V(y) dy$ belongs to $L^{\infty}(\mathbb{R}^{d})$ .     

Gauge Optimization and Spectral Properties of Magnetic Schrödinger Operators

We establish new necessary and sufficient conditions for the discreteness of spectrum and strict positivity of magnetic Schrödinger operators with positive scalar potentials. We also derive two-sided estimates for the bottoms of the spectrum and essential spectrum. The main idea is to optimize the gauges of the magnetic field on cubes, thus reducing the quadratic form on the cubes to ones without magnetic field (but with appropriately adjusted scalar potentials).     

A Spectral Property of Discrete Schrödinger Operators with Non-Negative Potentials

In the context of an infinite weighted graph of bounded degree, we give a sufficient condition for the discrete Schrödinger operator with a non-negative potential to have a strictly positive bottom of the spectrum. The main result is a discrete analogue of a theorem of Shen in the setting of complete Riemannian manifolds.     

On the Spectrum of Non-Selfadjoint Schrödinger Operators with Compact Resolvent

We determine the Schatten class for the compact resolvent of Dirichlet realizations, in unbounded domains, of a class of non-selfadjoint differential operators. This class consists of operators that can be obtained via analytic dilation from a Schrödinger operator with magnetic field and a complex electric potential. As an application, we prove, in a variety of examples motivated by physics, that the system of generalized eigenfunctions associated with the operator is complete, or at least the existence of an infinite discrete spectrum.     

On Characterization of Poisson Integrals of Schrödinger Operators with Morrey Traces

Let L be a Schr?dinger operator of the form L =-Δ + V acting on L~2(R~n) where the nonnegative potential V belongs to the reverse H?lder class B_q for some q ≥ n. In this article we will show that a function f ∈ L~(2,λ)(R~n), 0 λ n, is the trace of the solution of L_u =-u_(tt) + L_u =0, u(x, 0) = f(x), where u satisfies a Carleson type condition sup x_B,r_Br_B~(-λ)∫_0~(rB)∫_(B(x_B,r_B))t|u(x,t)|~2dxdt≤C∞.Its proof heavily relies on investigate the intrinsic relationship between the classical Morrey spaces and the new Campanato spaces L_L~(2,λ)(R~n) associated to the operator L, i.e.L_L~(2,λ)(R~n)=L~(2,λ)(R~n).Conversely, this Carleson type condition characterizes all the L-harmonic functions whose traces belong to the space L~(2,λ)(R~n) for all 0 λ n.     

The Dynamics of Zeros of Finite-Gap Solutions of the Schrödinger Equation

We study a system of particles on a Riemann surface with a puncture. This system describes the behavior of zeros of finite-gap solutions of the Schrödinger equation corresponding to a degenerate hyperelliptic curve. We show that this system is Hamiltonian and integrable by constructing action-angle type coordinates.     

A Characterization of Hardy Spaces Associated with Certain Schrödinger Operators

Let {K _t } _t>0 be the semigroup of linear operators generated by a Schrödinger operator ?L = Δ ? V (x) on ? ^d , d ≥ 3, where V (x) ≥ 0 satisfies Δ ^?1 V ∈ L ^∞. We say that an L ¹-function f belongs to the Hardy space \({H^{1}_{L}}\) if the maximal function ? _L f (x) = sup _t>0 |K _t f (x)| belongs to L ¹ (? ^d ). We prove that the operator (?Δ)^1/2 L ^?1/2 is an isomorphism of the space \({H^{1}_{L}}\) with the classical Hardy space H ¹(? ^d ) whose inverse is L ^1/2(?Δ)^?1/2. As a corollary we obtain that the space \({H^{1}_{L}}\) is characterized by the Riesz transforms \(R_{j}=\frac {\partial }{\partial x_{j}}L^{-1\slash 2}\) .     

Fall-Off of Eigenfunctions for Non-Local Schrödinger Operators with Decaying Potentials

We study the spatial decay of eigenfunctions of non-local Schrödinger operators whose kinetic terms are generators of symmetric jump-paring Lévy processes with Kato-class potentials decaying at infinity. This class of processes has the property that the intensity of single large jumps dominates the intensity of all multiple large jumps. We find that the decay rates of eigenfunctions depend on the process via specific preference rates in particular jump scenarios, and depend on the potential through the distance of the corresponding eigenvalue from the edge of the continuous spectrum. We prove that the conditions of the jump-paring class imply that for all eigenvalues the corresponding positive eigenfunctions decay at most as rapidly as the Lévy intensity. This condition is sharp in the sense that if the jump-paring property fails to hold, then eigenfunction decay becomes slower than the decay of the Lévy intensity. We furthermore prove that under reasonable conditions the Lévy intensity also governs the upper bounds of eigenfunctions, and ground states are comparable with it, i.e., two-sided bounds hold. As an interesting consequence, we identify a sharp regime change in the decay of eigenfunctions as the Lévy intensity is varied from sub-exponential to exponential order, and dependent on the location of the eigenvalue, in the sense that through the transition Lévy intensity-driven decay becomes slower than the rate of decay of the Lévy intensity. Our approach is based on path integration and probabilistic potential theory techniques, and all results are also illustrated by specific examples.     

Spectral Theory of Semibounded Schrödinger Operators with δ′-Interactions

We study spectral properties of Hamiltonians H _X,β,q with δ′-point interactions on a discrete set ${X = \{x_k\}_{k=1}^\infty \subset (0, +\infty)}$ . Using the form approach, we establish analogs of some classical results on operators H _q =  ?d²/dx ² + q with locally integrable potentials ${q \in L^1_{\rm loc}[0, +\infty)}$ . In particular, we establish the analogues of the Glazman–Povzner–Wienholtz theorem, the Molchanov discreteness criterion, and the Birman theorem on stability of an essential spectrum. It turns out that in contrast to the case of Hamiltonians with δ-interactions, spectral properties of operators H _X,β,q are closely connected with those of ${{\rm H}_{X,q}^N = \oplus_{k}{\rm H}_{q,k}^N}$ , where ${{\rm H}_{q,k}^N}$ is the Neumann realization of ?d²/dx ² + q in L ²(x _k-1,x _k ).     

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.     

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 Spectral Problems of Discrete Schrödinger Operators

Applications of Mathematics - A special type of Jacobi matrices, discrete Schrödinger operators, is found to play an important role in quantum physics. In this paper, we show that given the...     

Self-Adjoint Extensions of Discrete Magnetic Schrödinger Operators

Using the concept of an intrinsic metric on a locally finite weighted graph, we give sufficient conditions for the magnetic Schrödinger operator to be essentially self-adjoint. The present paper is an extension of some recent results proven in the context of graphs of bounded degree.     

Mixed Norm Estimates of Schrödinger Waves and Their Applications

In this paper we establish mixed norm estimates of interactive Schrödinger waves and apply them to study smoothing properties and global well-posedness of the nonlinear Schrödinger equations with mass critical nonlinearity.