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 \(R_{j}=\frac {\partial }{\partial x_{j}}L^{-1\slash 2}\) .     

Riesz Transforms Associated with Schrdinger Operators Acting on Weighted Hardy Spaces

Let L =-? + V be a Schrdinger operator acting on L2(Rn), n ≥ 1, where V ≡ 0 is a nonnegative locally integrable function on Rn. In this article, we will intropduce weighted Hardy spaces H L(w) associated with L by means of the square function and then study their atomic decomposition theory. We will also show that the Riesz transform ?L-1/2associated with L is bounded from our new space Hp L(w) to the classical weighted Hardy space Hp(w) when n/(n +1) p 1 and w ∈ A1∩ RH(2/p)′.     

Hardy Spaces H_L~p(R~n) Associated with Higher-Order Schrdinger Type Operators

Let L = L0+V be the higher order Schrdiger type operator where L0 is a homogeneous elliptic operator of order 2m in divergence form with bounded coefficients and V is a real measurable function as multiplication operator(e.g., including(-?)m+V(m∈N) as special examples). In this paper, assume that V satisfies a strongly subcritical form condition associated with L0, the authors attempt to establish a theory of Hardy space Hp L(Rn)(0 p ≤ 1) associated with the higher order Schrdinger type operator L. Specifically, we first define the molecular Hardy space Hp L(Rn) by the so-called( p, q, ε, M) molecule associated to L and then establish its characterizations by the area integral defined by the heat semigroup e-t L.     

Hardy Spaces HpL(Rn) Associated with Higher-Order Schr?dinger Type Operators

Let L=L0+V be the higher order Schr?diger type operator where L0 is a homogeneous elliptic operator of order 2m in divergence form with bounded coeffi-cients and V is a real measurable function as mult...     

Generalized Morrey Spaces Associated to Schrödinger Operators and Applications

We first introduce new weighted Morrey spaces related to certain non-negative potentials satisfying the reverse Hölder inequality. Then we establish the weighted strong-type and weak-type estimates for the Riesz transforms and fractional integrals associated to Schrödinger operators. As an application, we prove the Calderón-Zygmund estimates for solutions to Schrödinger equation on these new spaces. Our results cover a number of known results.     

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.     

Besov and Hardy Spaces Associated with the Schrödinger Operator on the Heisenberg Group

We introduce the Besov space $\dot{B}^{0,L}_{1,1}$ associated with the Schrödinger operator L with a nonnegative potential satisfying a reverse Hölder inequality on the Heisenberg group, and obtain the molecular decomposition. We also develop the Hardy space $H_{L}^{1}$ associated with the Schrödinger operator via the Littlewood–Paley area function and give equivalent characterizations via atoms, molecules, and the maximal function. Moreover, using the molecular decomposition, we prove that $\dot{B}^{0,L}_{1,1}$ is a subspace of $H_{L}^{1}$ .     

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...     

Heat Kernel Estimates of Fractional Schrödinger Operators with Negative Hardy Potential

Potential Analysis - We obtain two-sided estimates for the heat kernel (or the fundamental function) associated with the following fractional Schrödinger operator with negative Hardy potential...     

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.     

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.     

Continuum Schrödinger Operators Associated With Aperiodic Subshifts

We study Schrödinger operators on the real line whose potentials are generated by an underlying ergodic subshift over a finite alphabet and a rule that replaces symbols by compactly supported potential pieces. We first develop the standard theory that shows that the spectrum and the spectral type are almost surely constant and that identifies the almost sure absolutely continuous spectrum with the essential closure of the set of energies with vanishing Lyapunov exponent. Using results of Damanik–Lenz and Klassert–Lenz–Stollmann, we also show that the spectrum is a Cantor set of zero Lebesgue measure if the subshift satisfies the Boshernitzan condition and the potentials are aperiodic and irreducible. We then study the case of the Fibonacci subshift in detail and prove results for the local Hausdorff dimension of the spectrum at a given energy in terms of the value of the associated Fricke–Vogt invariant. These results are elucidated for some simple choices of the local potential pieces, such as piecewise constant ones and local point interactions. In the latter special case, our results explain the occurrence of so-called pseudo bands, which have been pointed out in the physics literature.     

Boundedness of Area Functions Related to Schrödinger Operators and Their Commutators in Weighted Hardy Spaces

In this paper, we consider the area function $S_Q$ related to the Schrödinger operator $\mathcal{L}$ and its commutator $S_{Q,b}$, establish the boundedness of $S_Q$ from $H^p_\rho(w)$ to $L^p(w)$ or $WL^p(w),$ as well as the boundedness of $S_{Q,b}$ from $H^1_\rho(w)$ to $WL^1(w).$     

On Spectral Properties of Translationally Invariant Magnetic Schrödinger Operators

We consider a class of translationally invariant magnetic fields such that the corresponding potential has a constant direction. Our goal is to study basic spectral properties of the Schr?dinger operator with such a potential. In particular, we show that the spectrum of is absolutely continuous and we find its location. Then we study the long-time behaviour of solutions exp of the time dependent Schr?dinger equation. It turns out that a quantum particle remains localized in the plane orthogonal to the direction of the potential. Its propagation in this direction is determined by group velocities. It is to a some extent similar to an evolution of a one-dimensional free particle but “exits” to +∞ and −∞ in the direction of the potential might be essentially different. Submitted: June 7, 2007. Accepted: August 20, 2007.     

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.     

On Weighted Compactness of Commutators Related with Schr?dinger Operators

Let L =-Δ + V be a Schr?dinger operator, where Δ is the Laplacian operator on R^d(d ≥3), while the nonnegative potential V belongs to the reverse H¨older class B_q, q > d/2. In this paper,we study weighted compactness of commutators of some Schr?dinger operators, which include Riesz transforms, standard Calderón–Zygmund operators and Littlewood–Paley functions. These results substantially generalize some well-known results.