The L^p-continuity of the wave operators for the three dimensional Schroedinger operator

On the spectrum of an "even" periodic Schroedinger operator with a rational magnetic flux

We study the Schrödinger operator on $L_2(\mathbb R^3)$ with periodic variable metric, and periodic electric and magnetic fields. It is assumed that the operator is reflection symmetric and the (appropriately defined) flux of the magnetic field is rational. Under these assumptions it is shown that the spectrum of the operator is absolutely continuous. Previously known results on absolute continuity for periodic operators were obtained for the zero magnetic flux.

Magnetic Schroedinger operators with radially symmetric magnetic field and radially symmetric electric potential

The aim of the paper is to derive spectral estimates on the eigenvalue moments of the magnetic Schroedinger operators defined on the two-dimensional disk with a radially symmetric magnetic field and radially symmetric electric potential.

Schroedinger operators on exterior domains with Robin boundary conditions: heat kernel estimates

We study Schroedinger operators with Robin boundary conditions on exterior domains in $\R^d$. We prove sharp point-wise estimates for the associated semi-groups which show, in particular, how the boundary conditions affect the time decay of the heat kernel in dimensions one and two. Applications to spectral estimates are discussed as well.

Resolvent expansion and time decay of the wave functions for two-dimensional magnetic Schroedinger operators

We consider two-dimensional Schroedinger operators with magnetic field. Under certain regularity and decay assumptions on the magnetic and electric field we show that the behavior of the corresponding resolvent at threshold and the behavior of the unitary group for large times are determined by the total flux of the associated magnetic field.

Absence of eigenvalues of two-dimensional magnetic Schroedinger operators

By developing the method of multipliers, we establish sufficient conditions on the electric potential and magnetic field which guarantee that the corresponding two-dimensional Schroedinger operator possesses no point spectrum. The settings of complex-valued electric potentials and singular magnetic potentials of Aharonov-Bohm field are also covered.

Spectral stability of Schroedinger operators with subordinated complex potentials

We prove that the spectrum of Schroedinger operators in three dimensions is purely continuous and coincides with the non-negative semiaxis for all potentials satisfying a form-subordinate smallness condition. By developing the method of multipliers, we also establish the absence of point spectrum for Schroedinger operators in all dimensions under various alternative hypotheses, still allowing complex-valued potentials with critical singularities.

On band spectrum of Schroedinger operator in periodic system of domains coupled by small windows

We consider a periodic system of domains coupled by small windows. In such domain we study the band spectrum of a Schroedinger operator subject to Neumann condition. We show that near each isolated eigenvalue of the similar operator but in the periodicity cell, there are several non-intersecting bands of the spectrum for the perturbed operator. We also discuss the position of the points at which the band functions attain the edges of each band.

Non-trapping magnetic fields and Morrey-Campanato estimates for Schroedinger operators

We prove some uniform in $ε$ a priori estimates for solutions of the equation $$(\nabla-iA)^2u-V(x)u+(λ\pm iε)u=f, λ\geq0, ε\neq0.$$ The estimates are obtained in terms of Morrey-Campanato norms, and can be used to prove absence of zero-resonances, in a suitable sense, for electromagnetic Hamiltonians. Precise conditions on the size of the \textit{trapping component} of the magnetic field and the non repulsive component of the electric field are given.

Dispersive estimates for Schroedinger operators in the presence of a resonance and/or an eigenvalue at zero energy in dimension three: II

We consider non-selfadjoint operators of the kind arising in linearized NLS and prove dispersive bounds for the time-evolution without assuming that the edges of the essential spectrum are regular. Our approach does not depend on any specific properties of NLS. Rather, it is axiomatic on the linear level, and our results are obtained from four assumptions (which are of course motivated by NLS). This work is in three dimensions.

2D- and 3DMagnetic Schroedinger Operators: Short Loops, Pointwise Spectral Asymptotics and Asymptotics of Dirac Energy

We consider 2- and 3-dimensional Schrödinger or generalized Schrödinger-Pauli operators with the non-degenerating magnetic field in the open domain under certain non-degeneracy assumptions we derive pointwise spectral asymptotics. We also consider asymptotics of some related expressions (see below). For all asymptotics loops rather than periodic trajectories play important role.

Pseudomodes for Schroedinger operators with complex potentials

For one-dimensional Schroedinger operators with complex-valued potentials, we construct pseudomodes corresponding to large pseudoeigenvalues. Our (non-semi-classical) approach results in substantial progress in achieving optimal conditions and conclusions as well as in covering a wide class of previously inaccessible potentials, including discontinuous ones.

Spectral Asymptotics for Magnetic Schroedinger Operator

In this article we obtain eigenvalue asymptotics for 2D and 3D-Schroedinger, Schroedinger-Pauli and Dirac operators in the situations in which the role of the magnetic field is important. These operators are essentially different and there is a significant difference between 2D and 3D-operators.

Exact solutions to the Navier-Stokes equation for an incompressible flow from the interpretation of the Schroedinger wave function

The existence of the velocity potential is a direct consequence from the derivation of the continuity equation from the Schroedinger equation. This implies that the Cole-Hopf transformation is applicable to the Navier-Stokes equation for an incompressible flow and allows reducing the Navier-Stokes equation to the Einstein-Kolmogorov equation, in which the reaction term depends on the pressure. The solution to the resulting equation, and to the Navier-Stokes equation as well, can then be written in terms of the Green's function of the heat equation and is given in the form of an integral mappin…

Sharp Spectral Asymptotics for Operators with Irregular Coefficients. III. Schroedinger operator with a strong magnetic field

Sharp spectral asymptotics for 2- and 3-dimensional Schroedinger operators with strong magnetic field are derived under rather weak smoothness conditions. In comparison with version 1 of 2005 new results are added and minor errors corrected.

Schroedinger Operator with Strong Magnetic Field near Boundary

We consider 2-dimensional Schroedinger operator with the non-degenerating magnetic field in the domain with the boundary and under certain non-degeneracy assumptions we derive spectral asymptotics with the remainder estimate better than $O(h^{-1})$, up to $O(μ^{-1}h^{-1})$ and the principal part $\asymp h^{-2}$ where $h\ll 1$ is Planck constant and $μ\gg 1$ is the intensity of the magnetic field; $μh \le 1$. We also consider generalized Schrödinger-Pauli operator in the same framework albeit with $μh\ge 1$ and derive spectral asymptotics with the remainder estimate up to O(1) and with the prin…

Banach空间中线性算子的虚能级与虚态及其在薛定谔算子中的应用

Virtual levels and virtual states of linear operators in Banach spaces. Applications to Schroedinger operators

本文提出Banach空间中虚能级的统一刻画方法，证明其等价于存在虚态、缺乏极限吸收原理及小扰动产生特征值，并应用于任意维非自伴非局部势薛定谔算子的阈值预解估计。

Schroedinger Operator: Heat Kernel and Its Applications

In this paper, we study the geometry associated with Schroedinger operator via Hamiltonian and Lagrangian formalism. Making use of a multiplier technique, we construct the heat kernel with the coefficient matrices of the operator both diagonal and non-diagonal. For applications, we compute the heat kernel of a Schroedinger operator with terms of lower order, and obtain a globally closed solution to a matrix Riccati equations as a by-product. Besides, we finally recover and generalise several classical results on some celebrated operators.

Nodal sets for the groundstate of the Schroedinger operator with zero magnetic field in a non simply connected domain

We investigate nodal sets of magnetic Schroedinger operators with zero magnetic field, acting on a non simply connected domain in $\r^2$. For the case of circulation 1/2 of the magnetic vector potential around each hole in the region, we obtain a charactisation of the nodal set, and use this to obtain bounds on the multiplicity of the groundstate. For the case of one hole and a fixed electric potential, we show that the first eigenvalue takes its highest value for circulation 1/2.

Determining a magnetic Schroedinger operator from partial Cauchy data

In this paper we show, in dimension n >=3, that knowledge of the Cauchy data for the Schroedinger equation in the presence of a magnetic potential, measured on possibly very small subsets of the boundary, determines uniquely the magnetic field and the electric potential.

Maximal regularity for non-autonomous Schroedinger type equations

In this paper we study the maximal regularity property for non-autonomous evolution equations $\partial_t u(t)+A(t)u(t)=f(t), u(0)=0.$ If the equation is considered on a Hilbert space $H$ and the operators $A(t)$ are defined by sesquilinear forms $ a(t,.,.)$ we prove the maximal regularity under a Holder continuity assumption of $t \to a(t,.,.)$. In the non-Hilbert space situation we focus on Schrodinger type operators $A(t):= -Δ+ m(t, .)$ and prove $L^p-L^q$ estimates for a wide class of time and space dependent potentials $m$.

Sharp Spectral Asymptotics for Dirac Energy. II. Magnetic Schroedinger operator

I derive sharp semiclassical asymptotics of \int |e_h(x,y,0)|^2ω(x,y)dxdy where e_h(x,y,τ) is the Schwartz kernel of the spectral projector of Magnetic Schroedinger operator and ω(x,y) is singular as x=y. I also consider asymptotics of more general expressions.

环面上薛定谔算子的控制

Control for Schroedinger operators on tori

本文证明Jaffard关于环面上自由定态和动态薛定谔方程L2控制的结果在存在光滑时间无关势时仍然成立，并推测该结果对任意有界时间相关势也成立。

On space-time estimates for the Schroedinger operator

We prove mixed norm space-time estimates for solutions of the Schroedinger equation, with initial data in $L^p$ Sobolev or Besov spaces, and clarify the relation with adjoint restriction.

A general approximation of quantum graph vertex couplings by scaled Schroedinger operators on thin branched manifolds

We demonstrate that any self-adjoint coupling in a quantum graph vertex can be approximated by a family of magnetic Schroedinger operators on a tubular network built over the graph. If such a manifold has a boundary, Neumann conditions are imposed at it. The procedure involves a local change of graph topology in the vicinity of the vertex; the approximation scheme constructed on the graph is subsequently `lifted' to the manifold. For the corresponding operator a norm-resolvent convergence is proved, with the natural identification map, as the tube diameters tend to zero.

Spectral multipliers for Schroedinger operators with Poeschl-Teller potential

We prove a sharp Mihlin-Hormander multiplier theorem for Schroedinger operators $H$ on $\R^n$. The method, which allows us to deal with general potentials, improves Hebisch's method relying on heat kernel estimates for positive potentials. Our result applies to, in particular, the negative Poeschl-Teller potential $V(x)= -ν(ν+1) \sech^2 x $, $ν\in \N$, for which $H$ has a resonance at zero.

A dispersive bound for three-dimensional Schroedinger operators with zero energy eigenvalues

We prove a dispersive estimate for the evolution of Schroedinger operators $H = -Δ+ V(x)$ in ${\mathbb R}^3$. The potential is allowed to be a complex-valued function belonging to $L^p(\R^3)\cap L^q(\R^3)$, $p < \frac32 < q$, so that $H$ need not be self-adjoint or even symmetric. Some additional spectral conditions are imposed, namely that no resonances of $H$ exist anywhere within the interval $[0,\infty)$ and that eigenfunctions at zero (including generalized eigenfunctions) decay rapidly enough to be integrable.

Magnetic Schroedinger operators and the d-bar equation

In this paper we characterize compactness of the canonical solution operator to d-bar on weigthed $L^2$ spaces on $\mathbb C.$ For this purpose we consider certain Schrödinger operators with magnetic fields and use a condition which is equivalent to the property that these operators have compact resolvents. We also point out what are the obstructions in the case of several complex variables.

Sharp Spectral Asymptotics for Operators with Irregular Coefficients. IV. Multidimensional Schroedinger operator with a strong magnetic field. Full-rank case

With derive sharp spectral asymptotics (with the remainder estimate $O(μ^{-1}h^{1-d}+μ^{\frac{d} {2}-1}h^{1-\frac{d}{2}})$ for $d$-dimensional Schrödinger operator with a strong magnetic field; here $h$ and $μ$ are Plank and binding constants respectively and magnetic intensity matrix has full rank at each point. In comparison with version 1 of 4.5 year ago this version contains more results (we also study some degenerations), improvements and some minor corrections.

Strichartz and Smoothing Estimates for Schroedinger Operators with Large Magnetic Potentials in R^3

We show that the time evolution of the operator $H = -Δ+ i(A \cdot \nabla + \nabla \cdot A) + V$ in R^3 satisfies Strichartz and smoothing estimates under suitable smoothness and decay assumptions on A and V but without any smallness assumptions. We require that zero energy is neither an eigenvalue nor a resonance.

Universal bounds and semiclassical estimates for eigenvalues of abstract Schroedinger operators

We prove trace inequalities for a self-adjoint operator on an abstract Hilbert space. These inequalities lead to universal bounds on spectral gaps and on moments of eigenvalues lambda_k that are analogous to those known for Schroedinger operators and the Dirichlet Laplacian, on which the operators of interest are modeled. In addition we produce inequalities that are new even in the model case. These include a family of differential inequalities for generalized Riesz means and theorems stating that arithmetic means of lambda_k^p for p <= 3 are universally bounded from above by multiples of the …

Unique continuation properties for Schroedinger operators in Hilbert spaces

Here, the Morgan type uncertainty principle and unique continuation properties of abstract Schredinger equations with time dependent potentials are obtained in Hilbert space valued function classes. The equations include linear operator in abstract Hilbert spaces H dependent on space variables. So, by selecting appropriate spaces H and operators, we derive unique continuation properties for numerous classes of Schrödinger type equations and its systems, which occur in a wide variety of physical systems.

Function spaces associated with Schroedinger operators: the Poeschl-Teller potential

We address the function space theory associated with the Schroedinger operator H. The discussion is featured with the Poeschl-Teller potential in quantum physics. Using biorthogonal dyadic system, we introduce Besov spaces and Triebel-Lizorkin spaces (including Sobolev spaces) associated with H. We then use interpolation method to identify these spaces with the classical ones for a certain range of p,q> 1. A physical implication is that the corresponding wave function $ψ(t,x)=e^{-i t H }f(x)$ admits appropriate time decay in the Besov space scale.

Littlewood-Paley theorem for Schroedinger operators

Let $H$ be a Schrödinger operator on $\R^n$. Under a polynomial decay condition for the kernel of its spectral operator, we show that the Besov spaces and Triebel-Lizorkin spaces associated with $H$ are well defined. We further give a Littlewood-Paley characterization of $L_p$ spaces as well as Sobolev spaces in terms of dyadic functions of $H$. This generalizes and strengthens the previous result when the heat kernel of $H$ satisfies certain upper Gaussian bound.

Schroedinger and elliptic operators with distributional coefficients on a bounded domain

We study the operator $L=-Δ+q$ on a bounded domain $Ω\subset\mathbb R^n$, where $q(x)$ is a distributional potential. We find sufficient conditions for $q(x)$ which guarantee that $L$ is well--defined with Dirichlet and generalized Neumann boundary conditions. The asymptotics of the eigenvalues and the basis properties of the eigen- and associated functions of such operators are studied. The results are generalized for strongly elliptic operator of order $2m$ with Dirichlet boundary conditions.

Schroedinger Operators on Regular Metric Trees with Long Range Potentials: Weak Coupling Behavior

Consider a regular $d$-dimensional metric tree $Γ$ with root $o$. Define the Schroedinger operator $-Δ- V$, where $V$ is a non-negative, symmetric potential, on $Γ$, with Neumann boundary conditions at $o$. Provided that $V$ decays like $x^{-γ}$ at infinity, where $1 < γ\leq d \leq 2, γ\neq 2$, we will determine the weak coupling behavior of the bottom of the spectrum of $-Δ- V$. In other words, we will describe the asymptotical behavior of $\inf σ(-Δ- αV)$ as $α\to 0+$

Sharp Spectral Asymptotics for four-dimensional Schroedinger operator with a strong magnetic field. II

I consider 4-dimensional Schrödinger operator with the generic non-degenerating magnetic field and for a generic potential I derive spectral asymptotics with the remainder estimate $O(μ^{-1}h^{-3})$ and the principal part $\asymp h^{-4}$ where $h\ll 1$ is Planck constant and $μ\gg 1$ is the intensity of the magnetic field. For general potentials remainder estimate $O(μ^{-1}h^{-3}+μ^2h^{-2})$ is achieved.

Rough controls for Schroedinger operators on 2-tori

The purpose of this note is to use the results and methods of our previous work with Bourgain to obtain control and observability by rough functions and sets on rectangular 2-tori. We show that any Lebesgue measurable set of positive measure can be used for observability for the Schroedinger equation. This leads to controllability with rough localization and control functions. For non-empty open sets this follows from the results of Haraux '89 and Jaffard '89 while for square tori and sufficiently long times this can be deduced from the results of Jakobson '97.

Schroedinger Operator with Strong Magnetic Field: Propagation of singularities and sharper asymptotics

We consider 2-dimensional Schrödinger operator with the non-degenerating magnetic field and we discuss spectral asymptotics with the remainder estimate $o(μ^{-1}h^{-1})$ or better. We also consider 3-dimensional Schrödinger operator with the non-degenerating magnetic field and we discuss spectral asymptotics with the remainder estimate $o(μ^{-1}h^{-1})$ or better. We also consider generalized Schrödinger-Pauli operator in the same framework albeit with $μh\ge 1$ and derive spectral asymptotics with the remainder estimate up to $O(h^{-1}|\log h|)$ and with the principal part $\asymp μh^{-2}$.

On Schroedinger type operators with unbounded coefficients: Generation and heat kernel estimates

We consider the Schrödinger type operator ${\mathcal A}=(1+|x|^α)Δ-|x|^β$, for $α\in [0,2]$ and $β\ge 0$. We prove that, for any $p\in (1,\infty)$, the minimal realization of operator ${\mathcal A}$ in $L^p(\R^N)$ generates a strongly continuous analytic semigroup $(T_p(t))_{t\ge 0}$. For $α\in [0,2)$ and $β\ge 2$, we then prove some upper estimates for the heat kernel $k$ associated to the semigroup $(T_p(t))_{t\ge 0}$. As a consequence we obtain an estimate for large $|x|$ of the eigenfunctions of ${\mathcal A}$. Finally, we extend such estimates to a class of divergence type elliptic operat…