The Dynamics of Zeros of Finite-Gap Solutions of the Schrödinger Equation

We study a system of particles on a Riemann surface with a puncture. This system describes the behavior of zeros of finite-gap solutions of the Schrödinger equation corresponding to a degenerate hyperelliptic curve. We show that this system is Hamiltonian and integrable by constructing action-angle type coordinates.     

Schrödinger Operators with Rapidly Oscillating Potentials

Schr?dinger operators with rapidly oscillating potentials V such as are considered. Such potentials are not relatively compact with respect to the free Schr?dinger operator −Δ. We show that the oscillating potential V do not change the essential spectrum of −Δ. Moreover we derive upper bounds for negative eigenvalue sums of Ĥ.     

Dissipative Schrödinger Operators with Matrix Potentials

Maximal dissipative Schrödinger operators are studied in L ²((–,);E) (dimE=n<) that the extensions of a minimal symmetric operator with defect index (n,n) (in limit-circle case at – and limit point-case at ). We construct a selfadjoint dilation of a dissipative operator, carry out spectral analysis of a dilation, use the Lax–Phillips scattering theory, and find the scattering matrix of a dilation. We construct a functional model of the dissipative operator, determine its characteristic function in terms of the Titchmarsh–Weyl function of selfadjoint operator and investigate its analytic properties. Finally, we prove a theorem on completeness of the eigenvectors and associated vectors of a dissipative Schrödinger operators.     

The Quenched Asymptotics for Nonlocal Schrödinger Operators with Poissonian Potentials

Potential Analysis - We study the quenched long time behaviour of the survival probability up to time t, $mathbf {E}_{x}left [e^{-{{int }_{0}^{t}} V^{omega }(X_{s})mathrm {d}s}right ],$ of a...     

One-Point Commuting Difference Operators of Rank One and Their Relation with Finite-Gap Schrödinger Operators

One-point commuting difference operators of rank one in the case of hyperelliptic spectral curves are studied. A relationship between such operators and one-dimensional finite-gap Schrödinger operators is investigated. In particular, a discretization of finite-gap Lamé operators is obtained.     

The Morse and Maslov indices for Schrödinger operators

We study the spectrum of Schrödinger operators with matrixvalued potentials, utilizing tools from infinite-dimensional symplectic geometry. Using the spaces of abstract boundary values, we derive relations between the Morse and Maslov indices for a family of operators on a Hilbert space obtained by perturbing a given self-adjoint operator by a smooth family of bounded self-adjoint operators. The abstract results are applied to the Schrödinger operators with θ-periodic, Dirichlet, and Neumann boundary conditions. In particular, we derive an analogue of the Morse-Smale Index Theorem for multi-dimensional Schrödinger operators with periodic potentials. For quasi-convex domains in Rⁿ, we recast the results, connecting the Morse and Maslov indices using the Dirichlet and Neumann traces on the boundary of the domain.     

Unique Continuation for Fractional Schrödinger Equations with Rough Potentials

This article deals with the weak and strong unique continuation principle for fractional Schrödinger equations with scaling-critical and rough potentials via Carleman estimates. Our methods extend to “variable coefficient” versions of fractional Schrödinger equations and operators on non-flat domains.     

The Probabilistic Feynman–Kac Formula for an Infinite-Dimensional Schrödinger Equation with Exponential and Singular Potentials

Using the probabilistic Feynman–Kac formula, the existence of solutions of the Schrödinger equation on an infinite dimensional space E is proven. This theorem is valid for a large class of potentials with exponential growth at infinity as well as for singular potentials. The solution of the Schrödinger equation is local with respect to time and space variables. The space E can be a Hilbert space or other more general infinite dimensional spaces, like Banach and locally convex spaces (continuous functions, test functions, distributions). The specific choice of the infinite dimensional space corresponds to the smoothness of the fields to which the Schrödinger equation refers. The results also express an infinite-dimensional Heisenberg uncertainty principle: increasing of the field smoothness implies increasing of divergence of the momentum part of the quantum field Hamiltonian.     

Inverse spectral problem for singular AKNS and Schrödinger operators on [0,1]

We consider an inverse spectral problem for singular Sturm–Liouville equations on the unit interval with explicit singularity $a(a+1)/x^2$, $a\in \mathbb{N}$. This problem arises by splitting of the Schrödinger operator with radial potential acting on the unit ball of ${\mathbb{R}}^3$. Our goal is the global parametrization of potentials by spectral data noted by ${\lambda }^a$, and some norming constants noted by ${\kappa }^a$. For $a=0$ and 1, ${\lambda }^a\times {\kappa }^a$ was already known to be a global coordinate system on $L_{\mathbb{R}}^2(0,1)$. We extend this to any non-negative integer a. Similar result is obtained for singular AKNS operator. To cite this article: F. Serier, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

The Negative Spectrum of Schr?dinger Operators with Fractal Potentials

Let Γ?R~2 be a regular anisotropic fractal. We discuss the problem of the negative spectrum for the Schr?dinger operators associated with the formal expression H_β=id-?+βtr_b~Γ,β∈R,acting in the anisotropic Sobolev space W_2~(1,α)(R~2), where ? is the Dirichlet Laplanian in R~2 and tr_b~Γ is a fractal potential(distribution) supported by Γ.     

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.     

The Spectral Function and Principal Eigenvalues for Schrödinger Operators

Let m , 0 m₊ in Kato's class. We investigate the spectral function s( + m) where s( + m) denotes the upper bound of the spectrum of the Schrödinger operator + m. In particular, we determine its derivative at 0. If m_- is sufficiently large, we show that there exists a unique ₁ > 0 such that s( + ₁m) = 0. Under suitable conditions on m⁺ it follows that 0 is an eigenvalue of + ₁m with positive eigenfunction.     

The spectral bound of Schrödinger operators

Let V: R ^N [0, ] be a measurable function, and >0 be a parameter. We consider the behaviour of the spectral bound of the operator 1/2–V as a function of . In particular, we give a formula for the limiting value as , in terms of the integrals of V over subsets of R ^N on which the Laplacian with Dirichlet boundary conditions has prescribed values. We also consider the question whether this limiting value is attained for finite .     

Scattering and Wave Operators for One-Dimensional Schrödinger Operators with Slowly Decaying Nonsmooth Potentials

We prove existence of modified wave operators for one-dimensional Schrödinger equations with potential in If in addition the potential is conditionally integrable, then the usual Möller wave operators exist. We also prove asymptotic completeness of these wave operators for some classes of random potentials, and for almost every boundary condition for any given potential.     

Rotation Numbers of Linear Schrödinger Equations with Almost Periodic Potentials and Phase Transmissions

In this paper we study the linear Schrödinger equation with an almost periodic potential and phase transmission. Based on the extended unique ergodic theorem by Johnson and Moser, we will show for such an equation the existence of the rotation number. This extends the work of Johnson and Moser (in Commun Math Phys 84:403–438, 1982; Erratum Commun Math Phys 90:317–318, 1983) where no phase transmission is considered. The continuous dependence of rotation numbers on potentials and transmissions will be proved.     

Lifshits Tails for Random Schrödinger Operators with Nonsign Definite Potentials

This paper is devoted to the study of Lifshits tails for random Schr?dinger operator acting on of the form , where H ₀ is a -periodic Schr?dinger operator, λ is a positive coupling constant, are i.i.d and bounded random variables and V is the single site potential with changing sign. We prove that, in the weak disorder regime, at an open band edge, a true Lifshits tail for the random Schr?dinger operator occurs under a certain set of conditions on H ₀ and on V. Submitted: April 17, 2007. Accepted: December 13, 2007.