The Negative Spectrum of Schrödinger Operators with Fractal Potentials

Let Γ?R;be a regular anisotropic fractal. We discuss the problem of the negative spectrum for the Schr?dinger operators associated with the formal expression H;=id-?+βtr;,β∈R,acting in the anisotropic Sobolev space W;（R;）, where ? is the Dirichlet Laplanian in R;and tr;is a fractal potential（distribution） supported by Γ.

ISOMORPHISMS OF VARIABLE HARDY SPACES ASSOCIATED WITH SCHRÖDINGER OPERATORS

Let L:=-△+V be the Schrodinger operator on R^(n)with n≥3,where V is a non-negative potential satisfying△^(-1)(V)∈L^(∞)(R^(n)).Let w be an L-harmonic function,determined by V,satisfying that there exists a positive constantδsuch that,for any x∈Rn,0<δ≤w(x)≤1.Assume that p(·):R^(n)→(0,1]is a variable exponent satisfying the globally log-Hölder continuous condition.In this article,the authors show that the mappings HL^(p)(·))(R^(n))■f■wf∈H^(p)(·)(R^(n))and HL^(p(·))(R^(n))■f■(-△)^(1/2)L^(-1/2)(f)∈H^(p(·))(R^(n))are isomorphisms between the variable Hardy spaces HL^(p(·))(R^(n)),associated with L,and the variable Hardy spaces H^(p(·))(R^(n)).

Boundedness Estimates for Commutators of Riesz Transforms Related to Schrödinger Operators

Let L =-?+V be a Schr?dinger operator on R^n(n ≥ 3), where the non-negative potential V belongs to reverse H?lder class RH_(q1) for q_1>n/2. Let H_L^p(R^n)be the Hardy space associated with L. In this paper, we consider the commutator[b,T_α], which associated with the Riesz transform T_α= V~α(-?+V)^(-α) with 0 < α ≤ 1,and a locally integrable function b belongs to the new Campanato space Λ_β~θ(ρ). We establish the boundedness of [b,T_α] from L^p(R^n) to L^q(R^n) for 1 < p < q_1/α with 1/q = 1/p-β/n. We also show that [b,T_α] is bounded from H_L^p(R^n) to L^q(R^n) when n/(n+ β) < p ≤ 1,1/q = 1/p-β/n. Moreover, we prove that [b,T_α] maps H_L^(n/n+β)(~Rn)continuously into weak L^1(R^n).

Hardy Type Estimates for Riesz Transforms Associated with Schrdinger Operators on the Heisenberg Group

Abstract. Let H^n be the Heisenberg group and Q = 2n＋2 be its homogeneous dimen- sion. In this paper, we consider the Schr6dinger operator -△H^n ＋V, where △H^n is the sub-Laplacian and V is the nonnegative potential belonging to the reverse H61der class Bql for ql _〉 Q/2. We show that the operators T1 = V（-△H^n-In ＋V）-1 and T2 = V1/2（-△H^n-V）-1/2 are both bounded from 1 n HL^1（H^n ） into L1（H^n）. Our results are also valid on the stratified Lie group.

Note on Gradient Estimate of Heat Kernel for Schrodinger Operators

Let be a Schr?dinger operator on . We show that gradient estimates for the heat kernel of with upper Gaussian bounds imply polynomial decay for the kernels of certain smooth dyadic spectral operators. The latter decay property has been known to play an important role in the Littlewood-Paley theory for and Sobolev spaces. We are able to establish the result by modifying Hebisch and the author’s recent proofs. We give a counterexample in one dimension to show that there exists in the Schwartz class such that the long time gradient heat kernel estimate fails.

A Deep Learning Method for Computing Eigenvalues of the Fractional Schrödinger Operator

The authors present a novel deep learning method for computing eigenvalues of the fractional Schrödinger operator.The proposed approach combines a newly developed loss function with an innovative neural network architecture that incorporates prior knowledge of the problem.These improvements enable the proposed method to handle both high-dimensional problems and problems posed on irregular bounded domains.The authors successfully compute up to the first 30 eigenvalues for various fractional Schrödinger operators.As an application,the authors share a conjecture to the fractional order isospectral problem that has not yet been studied.

The BMO_L Space and Riesz Transforms Associated with Schrdinger 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.

Dirichlet-to-Neumann Map for a Hyperbolic Equation

In this paper, we provide an explicit expression for the full Dirichlet-to-Neumann map corresponding to a radial potential for a hyperbolic differential equation in 3-dimensional. We show that the Dirichlet-Neumann operators corresponding to a potential radial have the same properties for hyperbolic differential equations as for elliptic differential equations. We numerically implement the coefficients of the explicit formulas. Moreover, a Lipschitz type stability is established near the edge of the domain by an estimation constant. That is necessary for the reconstruction of the potential from Dirichlet-to-Neumann map in the inverse problem for a hyperbolic differential equation.

On Weighted Compactness of Commutators Related with Schrodinger Operators

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

Holder Continuity of Spectral Measures for the Finitely Differentiable Quasi-Periodic Schrodinger Operators

In the present paper,we prove the 1/2-Holder continuity of spectral measures for the C^k Schrodinger operators.This result is based on the quantitative almost reducibility and an estimate for the growth of the Schrodinger cocycles in[5].

Boundedness of High Order Commutators of Riesz Transforms Associated with Schrodinger Type Operators

Let L2=(-△)^2+ V^2 be the Schrödinger type operator, where V≠0 is a nonnegative potential and belongs to the reverse Holder class RHq1 for q1> n/2, n ≥5. The higher Riesz transform associated with L2 is denoted by R=△^2L2^(-1/2)and its dual is denoted by R^*=L2^(-1/2)△^2. In this paper, we consider the m-order commutators [b^m, R] and [bm, R^*], and establish the(L^p, L^q)-boundedness of these commutators when b belongs to the new Campanato space Λβ^θ(ρ) and 1/q = 1/p-mβ/n.

The Weighted Estimates of the Schrodinger Operators on the Nilpotent Lie Group

In this paper we consider the SchrSdinger operator -△G ＋ W on the nilpotent Lie group G where the nonnegative potential W belongs to the reverse H51der class Bq1 for some q1 ≥ D and D is the dimension at infinity of G. The weighted L^p -L6q estimates for the operators W^a（-△G ＋ W）^-β and W^a△G（-△G ＋ W）^-β are obtained.

Refined Scattering and Hermitian Spectral Theory for Linear Higher-Order Schrōdinger Equations

The Cauchy problem for a linear 2mth-order Schrōdinger equation ut=-i（-△）^mu, in R^N×R＋,u｜t=0=u0;m≥1 is an integer,is studied, for initial data uo in the weighted space L^2ρ（R^N）,withρ^＊（x）=e｜x｜^a and a=2m/2m-1∈（1,2].The following five problems are studied： （I） A sharp asymptotic behaviour of solutions as t → ＋∞ is governed by a discrete spectrum and a countable set Ф of the eigenfunctions of the linear rescaled operator B=-i（-△）^m＋1/2my·↓△＋N/2mI,with the spectrum σ（B）={λβ=-｜β｜≥0}. （Ⅱ） Finite-time blow-up local structures of nodal sets of solutions as t → 0^- and a formation of ＂multiple zeros＂ are described by the eigenfunctions, being generalized Hermite polynomials, of the ＂adjoint＂ operator B=-i（-△）^m-1/2my·↓△,with the same spectrum σ（B^＊）=σ（B）.Applications of these spectral results also include： （Ⅲ） a unique continuation theorem, and （IV） boundary characteristic point regularity issues. Some applications are discussed for more general linear PDEs and for the nonlinear Schr6dinger equations in the focusing （＂＋＂） and defocusing （＂-＂） cases ut=-（-△）^mu±i｜u｜^p-1u,in R^N×R＋,where P〉1,as well as for： （V） the quasilinear Schr6dinger equation of a ＂porous medium type＂ ut=-（-△）^m（｜u｜^nu）,in R^N×R＋,where n〉0.For the latter one, the main idea towards countable families of nonlinear eigenfunctions is to perform a homotopic path n → 0^＋ and to use spectral theory of the pair {B,B^＊}.

An Explicit Formula of the Dirichlet-to-Neumann Map for a Radial Potential in Dimension 3

In this paper, we provide an explicit expression for the full Dirichlet-to-Neumann map corresponding to a radial potential for the Schr&#246;dinger equation in 3-dimensional. We numerically implement the coefficients of the explicit formulas. In this work, Lipschitz type stability is established near the edge of the domain with giving estimation constant. That is necessary for the reconstruction of the potential from Dirichlet-to-Neuman map.

Hardy spaces H_L^p(R^n) associated with operators satisfying k-Davies-Gaffney estimates

Let L be a one-to-one operator of type w having a bounded H∞ functional calculus and satisfying the k-Davies-Gaffney estimates with k C N. In this paper, the authors introduce the Hardy space HPL（Rn） with p ∈（0, 1] associated with L in terms of square functions defined via {e-t2kL}t〉O and establish their molecular and generalized square function characterizations. Typical examples of such operators include the 2k-order divergence form homogeneous elliptic operator L1 with complex bounded measurable coefficients and the 2k-order Schr6dinger type operator L2 ：= （-△）k ＋ Vk, where A is the Laplacian and 0≤V C Llkoc（Rn）. Moreover, as an application, for i E {1, 2}, the authors prove that the associated Riesz transform Vk（Li-1/2） p n HP（Rn） for @ （n/（n ＋ k）, 1] and establish the Riesz transform characterizations is bounded from HLI（IR ） to p of HPL1（]Rn） for p C （rn/（n ＋ kr）, 1] if {e-tL1 }t〉o satisfies the Lr - L2 k-off-diagonal estimates with r C （1, 2]. These results when k ：= I and L ：= L1 are known.

Accretivity of the General Second Order Linear Differential Operator

For the general second order linear differential operator ■ with complex-valued distributional coefficients a_(jk), b_j, and c in an open set Ω ? R^n(n ≥ 1), we present conditions which ensure that-L^0 is accretive, i.e., Re<-L_0φ, φ >≥ 0 for all φ∈ C_0~∞(Ω).

Spectral Aspects of the Skew-Shift Operator:A Numerical Perspective

In this paper we perform a numerical study of the spectra,eigenstates,and Lyapunov exponents of the skew-shift counterpart to Harper’s equation.This study is motivated by various conjectures on the spectral theory of these’pseudo-random’models,which are reviewed in detail in the initial sections of the paper.The numerics carried out at different scales are within agreement with the conjectures and show a striking difference compared with the spectral features of the Almost Mathieu model.In particular our numerics establish a small upper bound on the gaps in the spectrum(conjectured to be absent).

Some mathematical aspects of Anderson localization:boundary effect,multimodality,and bifurcation

Anderson localization is a famous wave phenomenon that describes the absence of diffusion of waves in a disordered medium.Here we generalize the landscape theory of Anderson localization to general elliptic operators and complex boundary conditions using a probabilistic approach,and further investigate some mathematical aspects of Anderson localization that are rarely discussed before.First,we observe that under the Neumann boundary condition,the low energy quantum states are localized on the boundary of the domain with high probability.We provide a detailed explanation of this phenomenon using the concept of extended subregions and obtain an analytical expression of this probability in the one-dimensional case.Second,we find that the quantum states may be localized in multiple different subregions with high probability in the one-dimensional case and we derive an explicit expression of this probability for various boundary conditions.Finally,we examine a bifurcation phenomenon of the localization subregion as the strength of disorder varies.The critical threshold of bifurcation is analytically computed based on a toy model and the dependence of the critical threshold on model parameters is analyzed.