Stability estimate for an inverse problem for the magnetic Schrödinger equation from the Dirichlet-to-Neumann map

We consider the problem of stability estimate of the inverse problem of determining the magnetic field entering the magnetic Schrödinger equation in a bounded smooth domain of Rn with input Dirichlet data, from measured Neumann boundary observations. This information is enclosed in the dynamical Dirichlet-to-Neumann map associated to the solutions of the magnetic Schrödinger equation. We prove in dimension n?2 that the knowledge of the Dirichlet-to-Neumann map for the magnetic Schrödinger equation measured on the boundary determines uniquely the magnetic field and we prove a Hölder-type stability in determining the magnetic field induced by the magnetic potential.     

Strichartz estimates for the magnetic Schrödinger equation

We prove global, scale invariant Strichartz estimates for the linear magnetic Schrödinger equation with small time dependent magnetic field. This is done by constructing an appropriate parametrix. As an application, we show a global regularity type result for Schrödinger maps in dimensions n?6.     

Some Properties of Eigenfunctions of the Schrödinger Operator in a Magnetic Field

We study the behavior of eigenfunctions corresponding to a positive point spectrum of the Schrödinger operator with magnetic and electric potentials.     

An asymptotic functional-integral solution for the Schrödinger equation with polynomial potential

A functional integral representation for the weak solution of the Schrödinger equation with a polynomially growing potential is proposed in terms of an analytically continued Wiener integral. The asymptotic expansion in powers of the coupling constant λ of the matrix elements of the Schrödinger group is studied and its Borel summability is proved.     

Absolute Continuity of a Two-Dimensional Magnetic Periodic Schrödinger Operator ith Potentials of the Type of Measure Derivative

A two-dimensional magnetic periodic Schrödinger operator with a variable metric is considered. An electric potential is assumed to be a distribution formally given by an expression , where d is a periodic signed measure with a locally finite variation. We also assume that the perturbation generated by the electric potential is strongly subject (in the sense of forms) to the free operator. Under this natural assumption, we prove that the spectrum of the Schrödinger operator is absolutely continuous. Bibliography: 15 titles.     

Properties of the Schrödinger Operator Spectrum in a Magnetic Field

The properties of the spectrum of the Schrödinger operator with magnetic and electric potentials are investigated. We prove that the operator has no positive eigenvalues and its spectrum is absolutely continuous on the positive semiaxis.     

A remark to a paper of Kato and Ikebe

This paper deals with Schrödinger operators as they were treated by Kato - Ikebe [3]. It is shown that every element of the domain of definition of the adjoint of such an operator has locally square integrable distributional derivatives up to the order 2. For this the potential of the Schrödinger operator must fulfil a local Stummel condition; if the potential is only locally square integrable a somewhat weaker statement is possible for three dimensions (see remark 2 at the end of this paper).     

Smoothing estimates for the Schrödinger equation with unbounded potentials

We prove a local in time smoothing estimate for a magnetic Schrödinger equation with coefficients growing polynomially at spatial infinity. The assumptions on the magnetic field are gauge invariant and involve only the first two derivatives. The proof is based on the multiplier method and no pseudodifferential techniques are required.     

Scattering theory for the Schrödinger equation in some external time-dependent magnetic fields

We study the theory of scattering for a Schrödinger equation in an external time-dependent magnetic field in the Coulomb gauge, in space dimension 3. The magnetic vector potential is assumed to satisfy decay properties in time that are typical of solutions of the free wave equation, and even in some cases to be actually a solution of that equation. That problem appears as an intermediate step in the theory of scattering for the Maxwell-Schrödinger (MS) system. We prove in particular the existence of wave operators and their asymptotic completeness in spaces of relatively low regularity. We also prove their existence or at least asymptotic results going in that direction in spaces of higher regularity. The latter results are relevant for the MS system. As a preliminary step, we study the Cauchy problem for the original equation by energy methods, using as far as possible time derivatives instead of space derivatives.     

Anisotropic subdiffractive solitons

We study solitons in the two-dimensional defocusing nonlinear Schrödinger equation with the spatio-temporal modulation of the external potential. The spatial modulation is due to a square lattice; the resulting macroscopic diffraction is rotationally symmetric in the long-wavelength limit but becomes anisotropic for shorter wavelengths. Anisotropic solitons-solitons with the square (x, y)-geometry - are obtained both in the original nonlinear Schrödinger model and in its averaged amplitude equation.     

Schrödinger operators and associated hyperbolic pencils

For a large class of Schrödinger operators, we introduce the hyperbolic quadratic pencils by making the coupling constant dependent on the energy in the very special way. For these pencils, many problems of scattering theory are significantly easier to study. Then, we give some applications to the original Schrödinger operators including one-dimensional Schrödinger operators with L²-operator-valued potentials, multidimensional Schrödinger operators with slowly decaying potentials.     

Convergence of quantum systems on grids

We discuss here the convergence of quantum systems on grids embedded in Rd and generalize the earlier results found for scalar-valued potentials to the case of matrix-valued potentials. We also discuss the essential self-adjointness of Schrödinger operators for a large class of matrix potentials and give a Feynman-Kac formula for their associated imaginary time Schrödinger semigroups when the matrix potential is positive and continuous. Furthermore, we establish an operator kernel estimate for the semigroups.     

Asymptotic behavior of the discrete spectrum of the Schrödinger operator in electric and homogeneous magnetic fields

The asymptotic behavior of bound states is considered to the left of the boundary of the essential spectrum of the Schrödinger operator in homogeneous magnetic and decreasing electric fields. The electric potential is not considered nonpositive. It is assumed that the integral of the potential along the direction of the magnetic field has a powerlike behavior at infinity. It is shown that the asymptotic behavior of the bound states is powerlike and its leading term is computed.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol.182, pp. 131–141, 1990.     

Discrete Spectrum of a Three-Particle Schrodinger Operator with a Homogeneous Magnetic Field

The discrete spectrum of the Schrödinger operator is studiedfor a system of three identical particles with short-range interactionsin a homogeneous magnetic field. All the two-particle subsystemsare supposed to be unstable. Finiteness of the discrete spectrumis established under some assumptions about the solutions ofthe corresponding two-particle Schrödinger equation.     

The integrated density of states in strong magnetic fields

We consider three-dimensional Schrödinger operators with constant magnetic fields and ergodic electric potentials. We study the strong magnetic field asymptotic behaviour of the integrated density of states, distinguishing between the asymptotics far from the Landau levels, and the asymptotics near a given Landau level.     

Continuous integral kernels for unbounded Schrödinger semigroups and their spectral projections

By suitably extending a Feynman-Kac formula of Simon (Canad. Math. Soc. Conf. Proc. 28 (2000) 317), we study one-parameter semigroups generated by (the negative of) rather general Schrödinger operators, which may be unbounded from below and include a magnetic vector potential. In particular, a common domain of essential self-adjointness for such a semigroup is specified. Moreover, each member of the semigroup is proven to be a maximal Carleman operator with a continuous integral kernel given by a Brownian-bridge expectation. The results are used to show that the spectral projections of the generating Schrödinger operator also act as Carleman operators with continuous integral kernels. Applications to Schrödinger operators with rather general random scalar potentials include a rigorous justification of an integral-kernel representation of their integrated density of states—a relation frequently used in the physics literature on disordered solids.     

Eigenvalue Asymptotics for Locally Perturbed Second-Order Differential Operators

We consider the appearance of discrete spectrum in spectralgaps of magnetic Schrödinger operators with electric backgroundfield under strong, localised perturbations. We show that forcompactly supported perturbations the asymptotics of the countingfunction of the occurring eigenvalues in the limit of a strongperturbation does not depend on the magnetic field nor on thebackground field.     

Spectral properties of a class of reflectionless Schrödinger operators

We prove that one-dimensional reflectionless Schrödinger operators with spectrum a homogeneous set in the sense of Carleson, belonging to the class introduced by Sodin and Yuditskii, have purely absolutely continuous spectra. This class includes all earlier examples of reflectionless almost periodic Schrödinger operators. In addition, we construct examples of reflectionless Schrödinger operators with more general types of spectra, given by the complement of a Denjoy-Widom-type domain in C, which exhibit a singular component.     

Riesz Lp summability of spectral expansions related to the Schrödinger operator with constant magnetic field

For the spectral expansions related to the Schrödinger operator with constant magnetic field we establish Riesz-Bochner summability in L_p.     

Boundary controllability for the semilinear Schrödinger equations on Riemannian manifolds

We study the boundary exact controllability for the semilinear Schrödinger equation defined on an open, bounded, connected set Ω of a complete, n-dimensional, Riemannian manifold M with metric g. We prove the locally exact controllability around the equilibria under some checkable geometrical conditions. Our results show that exact controllability is geometrical characters of a Riemannian metric, given by the coefficients and equilibria of the semilinear Schrödinger equation. We then establish the globally exact controllability in such a way that the state of the semilinear Schrödinger equation moves from an equilibrium in one location to an equilibrium in another location.