Eric Soccorsi

Eigenvalue Asymptotics in a Twisted Waveguide

We consider a twisted quantum wave guide i.e., a domain of the form Ωθ: = r θ ω × ℝ where ω ⊂ ℝ2 is a bounded domain, and r θ = r θ(x 3) is a rotation by the angle θ(x 3) depending on the longitudinal variable x 3. We are interested in the spectral analysis of the Dirichlet Laplacian H acting in L 2(Ωθ). We suppose that the derivative of the rotation angle can be written as (x 3) = β − ϵ(x 3) with a positive constant β and ϵ(x 3) ∼ L|x 3|−α, |x 3| → ∞. We show that if L > 0 and α ∈ (0,2), or if L > L 0 > 0 and α = 2, then there is an infinite sequence of discrete eigenvalues lying below the infimum of the essential spectrum of H, and obtain the main asymptotic term of this sequence.

Stable Determination of Time-Dependent Scalar Potential from Boundary Measurements in a Periodic Quantum Waveguide

We prove logarithmic stability in the determination of the time-dependent scalar potential in a

Inverse boundary value problem for the dynamical heterogeneous Maxwell's system

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Edge currents and eigenvalue estimates for magnetic barrier Schrödinger operators

-periodic quantum cylindrical waveguide, from the boundary measurements of the solution to the dynamic Schrodinger equation.

EDGE CURRENTS FOR QUANTUM HALL SYSTEMS I: ONE-EDGE, UNBOUNDED GEOMETRIES

We consider the inverse problem of determining the isotropic inhomogeneous electromagnetic coefficients of the non-stationary Maxwell equations in a bounded domain of , from a finite number of boundary measurements. Our main result is a Holder stability estimate for the inverse problem, where the measurements are exerted only in some boundary components. For it, we prove a global Carleman estimate for the heterogeneous Maxwell system under boundary conditions.

Stability estimate in an inverse problem for non-autonomous magnetic Schrodinger equations

We study two-dimensional magnetic Schrodinger operators with a magnetic field that is equal to b>0 for x > 0 and (-b) for x < 0. This magnetic Schrodinger operator exhibits a magnetic barrier at x=0. The unperturbed system is invariant with respect to translations in the y-direction. As a result, the Schrodinger operator admits a direct integral decomposition. We analyze the band functions of the fiber operators as functions of the wave number and establish their asymptotic behavior. Because the fiber operators are reflection symmetric, the band functions may be classified as odd or even. The odd band functions have a unique absolute minimum. We calculate the effective mass at the minimum and prove that it is positive. The even band functions are monotone decreasing. We prove that the eigenvalues of an Airy operator, respectively, harmonic oscillator operator, describe the asymptotic behavior of the band functions for large negative, respectively positive, wave numbers. We prove a Mourre estimate for perturbations of the magnetic Schrodinger operator and establish the existence of absolutely continuous spectrum in certain energy intervals. We prove lower bounds on magnetic edge currents for states with energies in the same intervals. We also prove that these lower bounds imply stable lower bounds for the asymptotic currents. We study the perturbation by slowly decaying negative potentials and establish the asymptotic behavior of the eigenvalue counting function for the infinitely-many eigenvalues below the bottom of the essential spectrum.

On the Calderón problem in periodic cylindrical domain with partial Dirichlet and Neumann data

Devices exhibiting the integer quantum Hall effect can be modeled by one-electron Schrodinger operators describing the planar motion of an electron in a perpendicular, constant magnetic field, and under the influence of an electrostatic potential. The electron motion is confined to unbounded subsets of the plane by confining potential barriers. The edges of the confining potential barrier create edge currents. In this, the first of two papers, we prove explicit lower bounds on the edge currents associated with one-edge, unbounded geometries formed by various confining potentials. This work extends some known results that we review. The edge currents are carried by states with energy localized between any two Landau levels. These one-edge geometries describe the electron confined to certain unbounded regions in the plane obtained by deforming half-plane regions. We prove that the currents are stable under various potential perturbations, provided the perturbations are suitably small relative to the magnetic field strength, including perturbations by random potentials. For these cases of one-edge geometries, the existence of, and the estimates on, the edge currents imply that the corresponding Hamiltonian has intervals of absolutely continuous spectrum. In the second paper of this series, we consider the edge currents associated with two-edge geometries describing bounded, cylinder-like regions, and unbounded, strip-like, regions.

ON THE STABILITY OF PERIODICALLY TIME-DEPENDENT QUANTUM SYSTEMS

We consider the inverse problem of determining the time-dependent magnetic field of the Schrödinger equation in a bounded open subset of , , from a finite number of Neumann data, when the boundary measurement is taken on an appropriate open subset of the boundary. We prove the Lipschitz stability of the magnetic potential in the Coulomb gauge class by n times changing initial value suitably.

Edge Currents for Quantum Hall Systems, II. Two-Edge, Bounded and Unbounded Geometries

We consider the Calderon problem in an infinite cylindrical domain, whose cross section is a bounded domain of the plane. We prove log–log stability in the determination of the isotropic periodic conductivity coefficient from partial Dirichlet data and partial Neumann boundary observations of the solution. Copyright

Limiting absorption principle for the magnetic Dirichlet Laplacian in a half-plane

The main motivation of this article is to derive sufficient conditions for dynamical stability of periodically driven quantum systems described by a Hamiltonian H(t), i.e. conditions under which it holds true supt ∈ ℝ|〈ψt, H(t)ψt〉| < ∞ where ψt denotes a trajectory at time t of the quantum system under consideration. We start from an analysis of the domain of the quasi-energy operator. Next, we show, under certain assumptions, that if the spectrum of the monodromy (Floquet) operator U(T, 0) is pure point then there exists a dense subspace of initial conditions for which the mean value of the energy is uniformly bounded in the course of time. Further, we show that if the propagator admits a differentiable Floquet decomposition then ‖H(t)ψt‖ is bounded in time for any initial condition ψ0, and one employs the quantum KAM algorithm to prove the existence of this type of decomposition for a fairly large class of H(t). In addition, we derive bounds uniform in time on transition probabilities between different energy levels, and we also propose an extension of this approach to the case of a higher order of differentiability of the Floquet decomposition. The procedure is demonstrated on a solvable example of the periodically time-dependent harmonic oscillator.