Estimates of the Root Functions of a One-Dimensional Schrödinger Operator with a Strong Boundary Singularity

For any operator defined by the differential operation Lu = ?u″ + q(x)u on the interval G = (0, 1) with complex-valued potential q(x) locally integrable on G and satisfying the inequalities \(\int_{{x_1}}^{{x_2}} {\zeta |(q(\zeta ))|d\zeta \leqslant ln({x_1}/{x_2})} \) and \(\int_{{x_1}}^{{x_2}} {\zeta |(q(1 - \zeta ))|d\zeta \leqslant \gamma ln({x_1}/{x_2})} \) with some constant γ for all sufficiently small 0 < x₁ < x₂, we estimate the norms of root functions in the Lebesgue spaces L _p (G), 1 ≤ p < ∞. We show that for sufficiently small γ these norms satisfy the same estimates asymptotic in the spectral parameter as in the unperturbed case.     

Stability Estimates for the Lowest Eigenvalue of a Schrödinger Operator

There is a family of potentials that minimize the lowest eigenvalue of a Schrödinger operator under the constraint of a given L ^p norm of the potential. We give effective estimates for the amount by which the eigenvalue increases when the potential is not one of these optimal potentials. Our results are analogous to those for the isoperimetric problem and the Sobolev inequality. We also prove a stability estimate for Hölder’s inequality, which we believe to be new.     

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...     

Carleman Estimates for the Schrödinger Operator. Applications to Quantitative Uniqueness

On a closed manifold, we give a quantitative Carleman estimate on the

Schrödinger operator. We then deduce quantitative uniqueness results for solutions to the Schrödinger equation using doubling estimates.

Finally we investigate the sharpness of this results with respect to the electric potential.     

Schrödinger Equations with Fractional Laplacians

It is shown that the unique solution of } can be represented as { } where X=(X _t , t≥ 0) is a stable process whose generator is (-Δ) ^α/2 with X ₀ =0 . Accepted 24 July 2000. Online publication 13 November 2000.     

Resolvent Estimates and Matrix-Valued Schrödinger Operator with Eigenvalue Crossings; Application to Strichartz Estimates

We consider a semi-classical Schrödinger operator with a matrix-valued potential presenting eigenvalue crossings on isolated points. We obtain estimates for the boundary values of the resolvent under a generalized non-trapping assumption. As a consequence, we prove the smoothing effect of this operator, derive Strichartz type estimate for the propagator and get an existence theorem for a system of non-linear Schrödinger equations.     

Dispersive Estimates of Solutions to the Schrödinger Equation

We prove time decay L¹ → L^∞ estimates for the Schr?dinger group e^{it(−Δ + V)} for real-valued potentials satisfying V (x) = O (|x|^−δ), |x| ≫ 1, with δ > 5/2. Communicated by Bernard Helffer submitted 27/11/04, accepted 29/04/05     

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...     

Fractional Schrödinger equation; solvability and connection with classical Schrödinger equation

We consider the Dirichlet boundary problem for semilinear fractional Schrödinger equation with subcritical nonlinear term. Local and global in time solvability and regularity properties of solutions are discussed. But our main task is to describe the connections of the fractional equation with the classical nonlinear Schrödinger equation, including convergence of the linear semigroups and continuity of the nonlinear semigroups when the fractional exponent α approaches 1.     

Dispersion Estimates for Spherical Schrödinger Equations

We derive a dispersion estimate for one-dimensional perturbed radial Schrödinger operators. We also derive several new estimates for solutions of the underlying differential equation and investigate the behavior of the Jost function near the edge of the continuous spectrum.     

Absolute Continuity of the Spectrum of a Periodic Schrödinger Operator

We prove the absolute continuity of the spectrum of the Schrödinger operator in , , with periodic (with a common period lattice ) scalar and vector potentials for which either , , or the Fourier series of the vector potential converges absolutely, , where is an elementary cell of the lattice , for , and for , and the value of is sufficiently small, where and otherwise, , and .     

Local Dispersive and Strichartz Estimates for the Schrödinger Operator on the Heisenberg Group

It was proved by Bahouri et al. [9] that the Schrödinger equation on the Heisenberg group $\mathbb{H}^d,$ involving the sublaplacian, is an example of a totally non-dispersive evolution equation: for this reason global dispersive estimates cannot hold. This paper aims at establishing local dispersive estimates on $\mathbb{H}^d$ for the linear Schrödinger equation, by a refined study of the Schrödinger kernel $S_t$ on $\mathbb{H}^d.$ The sharpness of these estimates is discussed through several examples. Our approach, based on the explicit formula of the heat kernel on $\mathbb{H}^d$ derived by Gaveau [19], is achieved by combining complex analysis and Fourier-Heisenberg tools. As a by-product of our results we establish local Strichartz estimates and prove that the kernel $S_t$ concentrates on quantized horizontal hyperplanes of $\mathbb{H}^d.$     

Virtual Eigenvalues of the High Order Schrödinger Operator II

We consider the Schr?dinger operator H_γ = ( − Δ)^l + γ V(x)· acting in the space where 2l ≥ d, V (x) ≥ 0, V (x) is continuous and is not identically zero, and We study the asymptotic behavior as of the non-bottom negative eigenvalues of H_γ, which are born at the moment γ = 0 from the lower bound λ = 0 of the spectrum σ(H₀) of the unperturbed operator H₀ = ( − Δ)^l (virtual eigenvalues). To this end we use the Puiseux-Newton diagram for a power expansion of eigenvalues of some class of polynomial matrix functions. For the groups of virtual eigenvalues, having the same rate of decay, we obtain asymptotic estimates of Lieb-Thirring type.     

Stability of the Magnetic Schrödinger Operator in a Waveguide

Abstract

The spectrum of the Schrödinger operator in a quantum waveguide is known to be unstable in two and three dimensions. Any local enlargement of the waveguide produces eigenvalues beneath the continuous spectrum. Also, if the waveguide is bent, eigenvalues will arise below the continuous spectrum. In this paper a magnetic field is added into the system. The spectrum of the magnetic Schrödinger operator is proved to be stable under small local deformations and also under small bending of the waveguide. The proof includes a magnetic Hardy-type inequality in the waveguide, which is interesting in its own right.     

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.     

Approximation of the Fractional Schrödinger Propagator on Compact Manifolds

Let L be a second order positive, elliptic differential operator that is self-adjoint with respect to some C~∞ density dx on a compact connected manifold M. We proved that if 0 α 1,α/2 s α and f ∈ H~s(M) then the fractional Schr?dinger propagator e~(it Lα/2) on M satisfies e~(it Lα/2) f(x)-f(x) = o(t~(s/α-ε)) almost everywhere as t → 0~+, for any ε 0.     

A One-Dimensional Schrödinger Operator with Square-Integrable Potential

We study the spectral properties of a one-dimensional Schrödinger operator with squareintegrable potential whose domain is defined by the Dirichlet boundary conditions. The main results are concerned with the asymptotics of the eigenvalues, the asymptotic behavior of the operator semigroup generated by the negative of the differential operator under consideration. Moreover, we derive deviation estimates for the spectral projections and estimates for the equiconvergence of the spectral decompositions. Our asymptotic formulas for eigenvalues refine the well-known ones.     

A Fast Conservative Scheme for the Space Fractional Nonlinear Schrödinger Equation with Wave Operator

A new efficient compact difference scheme is proposed for solving a space fractional nonlinear Schrödinger equation with wave operator. The scheme is proved to conserve the total mass and total energy in a discrete sense. Using the energy method, the proposed scheme is proved to be unconditionally stable and its convergence order is shown to be of $ \mathcal{O}( h^6 + \tau^2) $ in the discrete $ L_2 $ norm with mesh size $ h $ and the time step $ \tau $. Moreover, a fast difference solver is developed to speed up the numerical computation of the scheme. Numerical experiments are given to support the theoretical analysis and to verify the efficiency, accuracy, and discrete conservation laws.     

A Poisson Formula for the Schrödinger Operator

We prove a Poisson type formula for the Schrödinger group. Such a formula had been derived in a previous article by the authors, as a consequence of the study of the asymptotic behavior of nonlinear wave operators for small data. In this note, we propose a direct proof, and extend the range allowed for the power of the nonlinearity to the set of all short range nonlinearities. Moreover, H ¹-critical nonlinearities are allowed.     

Maximum Principle for the Regularized Schrödinger Operator

In this paper we present analogues of the maximum principle and of some parabolic inequalities for the regularized time-dependent Schrödinger operator on open manifolds using Günter derivatives. Moreover, we study the uniqueness of bounded solutions for the regularized Schrödinger–Günter problem and obtain the corresponding fundamental solution. Furthermore, we present a regularized Schrödinger kernel and prove some convergence results. Finally, we present an explicit construction for the fundamental solution to the Schrödinger–Günter problem on a class of conformally flat cylinders and tori.