Endpoint estimates for Riesz transforms of magnetic Schrödinger operators

Let $A=-(\nabla-i\vec{a})^2+VLet be a magnetic Schr?dinger operator acting on L ²(R ⁿ ), n≥1, where and 0≤V∈L ¹ _loc. Following [1], we define, by means of the area integral function, a Hardy space H ¹ _A associated with A. We show that Riesz transforms (∂/∂x _k -i a _k )A ^-1/2 associated with A, , are bounded from the Hardy space H ¹ _A into L ¹. By interpolation, the Riesz transforms are bounded on L ^p for all 1<p≤2.     

Commutators of Riesz transforms of magnetic Schrödinger operators

Let \({A=-(\nabla-i{\vec a})\cdot (\nabla-i{\vec a}) +V}\) be a magnetic Schrödinger operator acting on \({L^2({\mathbb R}^n)}\), n ≥  1, where \({{\vec a}=(a_1, \ldots, a_n)\in L^2_{\rm loc}({\mathbb R}^n, {\mathbb R}^n)}\) and \({0\leq V\in L^1_{\rm loc}({\mathbb R}^n)}\). In this paper, we show that when a function \({b\in {\rm BMO}({\mathbb R}^n)}\), the commutators [b, T _k ]f = T _k (b f) ? b T _k f, k = 1, . . . , n, are bounded on \({L^p({\mathbb R}^n)}\) for all 1 < p < 2, where the operators T _k are Riesz transforms (?/?x _k  ? i a _k )A ^?1/2 associated with A.     

Spectral instability for some Schr?dinger operators

Summary. We show that the condition numbers of isolated eigenvalues of typical non-self-adjoint differential operators such as the harmonic oscillator may be extremely large. We describe a stable procedure for computing the condition numbers for Schr?dinger operators in one dimension, and apply it to the complex resonances of a typical operator with a dilation analytic potential. Received October 9, 1998 / Revised version received September 13, 1999 / Published online 16 March 2000     

Some estimates for commutators of Riesz transform associated with Schrödinger type operators

Let L₁ = ?Δ + V be a Schr:dinger operator and let L² = (?Δ)² + V² be a Schrödinger type operator on ?ⁿ (n ? 5), where V≠ 0 is a nonnegative potential belonging to certain reverse Hölder class B_s for s ? n/2. The Hardy type space \(H_{{L_2}}^1\) is defined in terms of the maximal function with respect to the semigroup \(\left\{ {{e^{ - t{L_2}}}} \right\}\) and it is identical to the Hardy space \(H_{{L_1}}^1\) established by Dziubański and Zienkiewicz. In this article, we prove the L^p-boundedness of the commutator R_b = bRf - R(bf) generated by the Riesz transform \(R = {\nabla ^2}L_2^{ - 1/2}\), where \(b \in BM{O_\theta }(\varrho )\), which is larger than the space BMO(?ⁿ). Moreover, we prove that R_b is bounded from the Hardy space \(H_{\mathcal{L}_1 }^1 \) into weak \(L_{weak}^1 (\mathbb{R}^n )\).     

The BMO L space and Riesz transforms associated with Schrödinger operators

Let L =△ ＋ V be a SchrSdinger operator in Rd, d ≥ 3, where the nonnegative potential V belongs to the reverse HSlder class Sd. We establish the BMOL-boundedness of Riesz transforms З/ЗxiL-1/2, and give the Fefferman-Stein type decomposition of BMOL functions.     

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.     

Maximal inequalities and Riesz transform estimates on L p spaces for magnetic Schrödinger operators II

The paper concerns the magnetic Schrödinger operator ${H({\bf a},V)=\sum_{j=1}^{n} (\frac{1}{i}\frac{\partial}{\partial x_{j}}-a_{j})^{2}+V }$ on ${\mathbb{R}^n}$ . We prove some L ^p estimates on the Riesz transforms of H and we establish some related maximal inequalities. The conditions that we arrive at, are essentially based on the control of the magnetic field by the electric potential.     

Semigroup and Riesz transform for the Dunkl–Schrödinger operators

Semigroup Forum - Let $$L_k=-\Delta _k+V$$ be the Dunkl–Schrödinger operators, where $$\Delta _k=\sum _{j=1}^dT_j^2$$ is the Dunkl Laplace operator associated to the Dunkl operators...     

bmo _ρ(ω) Spaces and Riesz transforms associated to Schrdinger operators

We introduce the BMO-type space bmoρ(ω) and establish the duality between h_ρ~1(ω) and bmo _ρ(ω),where ω∈A_1~(ρ, ∞)(R~n) and ω's locally behave as Muckenhoupt's weights but actually include them. We also give the Fefferman-Stein type decomposition of bmoρ(ω) with respect to Riesz transforms associated to Schrdinger operator L, where L =-? + V is a Schrdinger operator on R~n(n≥3) and V is a non-negative function satisfying the reverse Hlder inequality.     

Some estimates for the Schrödinger type operators with nonnegative potentials

In this paper we consider the Schrödinger operator ?Δ + V on \({\mathbb R^d}\), where the nonnegative potential V belongs to the reverse Hölder class \({B_{q_{_1}}}\) for some \({q_{_1}\geq \frac{d}{2}}\) with d ≥ 3. Let \({H^1_L(\mathbb R^d)}\) denote the Hardy space related to the Schrödinger operator L = ?Δ + V and \({BMO_L(\mathbb R^d)}\) be the dual space of \({H^1_L(\mathbb R^d)}\). We show that the Schrödinger type operator \({\nabla(-\Delta +V)^{-\beta}}\) is bounded from \({H^1_L(\mathbb R^d)}\) into \({L^p(\mathbb R^d)}\) for \({p=\frac{d}{d-(2\beta-1)}}\) with \({ \frac{1}{2}<\beta<\frac{3}{2} }\) and that it is also bounded from \({L^p(\mathbb R^d)}\) into \({BMO_L(\mathbb R^d)}\) for \({p=\frac{d}{2\beta-1}}\) with \({ \frac{1}{2}<\beta< 2}\).     

Spectral estimates for Schrödinger and Dirac-type operators on Riemannian manifolds

Supported by funds of M.U.R.S.T. (Italy). The author is grateful to S. Gallot for his encouragement and for helpful discussions and to G. Besson for some interesting remarks     

Spectral estimates for Schrödinger operators with sparse potentials on graphs

A construction of “sparse potentials,” suggested by the authors for the lattice \mathbbZ^d {\mathbb{Z}^d} , d > 2, is extended to a large class of combinatorial and metric graphs whose global dimension is a number D > 2. For the Schr?dinger operator − Δ − αV on such graphs, with a sparse potential V, we study the behavior (as α → ∞) of the number N_(−Δ − αV) of negative eigenvalues of − Δ − αV. We show that by means of sparse potentials one can realize any prescribed asymptotic behavior of N_(−Δ − αV) under very mild regularity assumptions. A similar construction works also for the lattice \mathbbZ² {\mathbb{Z}^2} , where D = 2. Bibliography: 13 titles.     

Global estimates of fundamental solutions for higher-order Schr?dinger equations

In this paper we first establish global pointwise time-space estimates of the fundamental solution for Schr?dinger equations, where the symbol of the spatial operator is a real non-degenerate elliptic polynomial. Then we use such estimates to establish related L ^p ?CL ^q estimates on the Schr?dinger solution. These estimates extend known results from the literature and are sharp. This result was lately already generalized to a degenerate case (cf. [4]).     

L~p estimates for Riesz transform and their commutators associated with Schr?dinger type operator

Let H_2 =(-△)~2 +V~2 be the Schrodinger type operator,where V satisfies reverse Holder inequality.In this paper,we establish the L~P boundedness for V~2H_2~(-1),H_2~(-1)V~2,VH_2~(-1/2)and H_2~(-1/2)V,and that of their commutators.We also prove that H_2~(-1)V~2,H_2~(-1/2)V are bounded from BMO_L to BMO_L.     

Some estimates of Schrödinger type operators on the Heisenberg group

In this paper, we consider the Schrödinger type operator ${H = (-\Delta _{\mathbb {H}}^n)^2 +V ^{2}}In this paper, we consider the Schr?dinger type operator H = (-D_{\mathbb H}ⁿ)² +V ²{H = (-\Delta _{\mathbb {H}}^n)^2 +V ^{2}}, where the nonnegative potential V belongs to the reverse H?lder class B_q1 for q₁ 3 \frac Q 2,Q 3 6{B_{{q}_{1}}\, {\rm for}\, q_{1}\geq {\frac {Q}{ 2}},Q \geq 6}, and D_{\mathbb Hⁿ}{\Delta_{\mathbb {H}^n}} is the sublaplacian on the Heisenberg group \mathbb Hⁿ{\mathbb {H}^n}. An L ^p estimate and a weak type L ¹ estimate for the operator ?⁴_{\mathbb Hⁿ} H^-1{\nabla^4_{\mathbb {H}^n} H^{-1}} when V ? B_q1{V \in B_{{q}_{1}}} for 1 < p ￡ \fracq₁2{1 < p \leq \frac{q_{1}}{2}} are obtained.