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 .     

Schrödinger operators with highly singular oscillating potentials

We investigate the Feynman-Kac semigroupP _t ^V and its densityp ^V(t,.,.),t>0, associated with the Schrödinger operator ?1/2Δ+V on ?^d\{0}.V will be a highly singular, oscillating potential like $V\left( x \right) = k \cdot \left\| x \right\|^{ - 1} \cdot \sin \left( {\left\| x \right\|^{ - m} } \right)$ with arbitraryk, l, m>0. We derive conditions (onk,l,m) which are sufficientand necessary for the existence of constants α, β, γ, ∈ ? such that for allt, x, y p ^V(t, x, y)≤γ·p(βt, x, y)·e^at. On the other hand, also conditions are derived which imply thatp ^V (t, x, y)≡∞ for allt, x, y. The aim is to see to which extent quick oscillations can lead to annihilations of the singularities ofV. For this purpose, we analyse the above example in great detail. Note that forl≥2 the potential is so singular that none of the usual perturbation techniques applies.     

Essential selfadjointness of singular magnetic Schrödinger operators on Riemannian manifolds

In this paper we extend the well-known Leinfelder–Simader theorem on the essential selfadjointness of singular Schrödinger operators to arbitrary complete Riemannian manifolds. This improves some earlier results of Shubin, Milatovic and others.     

An inverse spectral problem for a fractional Schrödinger operator

Archiv der Mathematik - We establish that the potential appearing in a fractional Schrödinger operator is uniquely determined by an internal spectral data.     

Spectral estimates for Schrödinger and Dirac-type operators on Riemannian manifolds

Supported by funds of M.U.R.S.T. (Italy). The author is grateful to S. Gallot for his encouragement and for helpful discussions and to G. Besson for some interesting remarks     

Inverse scattering theory for one-dimensional Schrödinger operators with steplike finite-gap potentials

We develop direct and inverse scattering theory for one-dimensional Schrödinger operators with steplike potentials which are asymptotically close to different finite-gap potentials on different half-axes. We give a complete characterization of the scattering data, which allows unique solvability of the inverse scattering problem in the class of perturbations with finite second moment.     

Radial Schrödinger equation: The spectral problem

Using the integral transformation method involving the investigation of the Laplace transforms of wave functions, we find the discrete spectra of the radial Schrödinger equation with a confining power-growth potential and with the generalized nuclear Coulomb attracting potential. The problem is reduced to solving a system of linear algebraic equations approximately. We give the results of calculating the discrete spectra of the S-states for the Schrödinger equation with a linearly growing confining potential and the nuclear Yukawa potential.     

On the spectral properties of discrete Schrödinger operators

We use the method of the conjugate operator to prove a limiting absorption principle and the absence of the singular continuous spectrum for discrete Schrödinger operators. We also obtain local decay estimates. Our results apply to a large class of perturbating potentials V decaying arbitrarily slowly to zero at infinity.     

On the inverse spectral problem for the quasi-periodic Schrödinger equation

We study the quasi-periodic Schrödinger equation $$-\psi''(x) + V(x) \psi(x) = E \psi(x), \quad x \in{ \mathbf {R}} $$ in the regime of “small” V. Let $(E_{m}',E''_{m})$ , m∈Z ^ν , be the standard labeled gaps in the spectrum. Our main result says that if $E''_{m} - E'_{m} \le\varepsilon\exp(-\kappa_{0} |m|)$ for all m∈Z ^ν , with ε being small enough, depending on κ ₀>0 and the frequency vector involved, then the Fourier coefficients of V obey $|c(m)| \le \varepsilon^{1/2} \exp(-\frac{\kappa_{0}}{2} |m|)$ for all m∈Z ^ν . On the other hand we prove that if |c(m)|≤εexp(?κ ₀|m|) with ε being small enough, depending on κ ₀>0 and the frequency vector involved, then $E''_{m} - E'_{m} \le2 \varepsilon\exp(-\frac {\kappa_{0}}{2} |m|)$ .     

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.     

Superlinear Schrödinger operators

We find nontrivial and ground state solutions for the nonlinear Schrödinger equation under conditions weaker than those previously assumed.     

Inverse problem of scattering theory for a class one-dimensional Schrödinger equation

Abstract

In this work, direct and inverse scattering problem on the real axis for the Schrödinger equation with piecewise-constant coefficient are studied. Using the new integral representations for solutions, the scattering data is defined, the main integral equations of the inverse scattering problem are obtained, the spectral characteristics of the scattering data are investigated and uniqueness theorem for the solution of inverse problem is proved.     

Dynamical bounds for Sturmian Schrödinger operators

The Fibonacci Hamiltonian, that is a Schrödinger operator associated to a Sturmian potential with respect to the golden number has been investigated intensively in recent years. Damanik and Tcheremchantsev developed a method and found a non-trivial dynamical upper bound for transport exponents for this model. This method can be generalized to obtain results for almost all irrational numbers. As a counter example, we exhibit a pathological irrational number with no possible better bound. Moreover, we establish a global lower bound for the lower box dimension of the spectrum that could be used to obtain a dynamical lower bound for irrational numbers with bounded density. To cite this article: L. Marin, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Bounds on the density of states for Schrödinger operators

We establish bounds on the density of states measure for Schrödinger operators. These are deterministic results that do not require the existence of the density of states measure, or, equivalently, of the integrated density of states. The results are stated in terms of a “density of states outer-measure” that always exists, and provides an upper bound for the density of states measure when it exists. We prove log-Hölder continuity for this density of states outer-measure in one, two, and three dimensions for Schrödinger operators, and in any dimension for discrete Schrödinger operators.     

Multidomain spectral method for Schrödinger equations

A multidomain spectral method with compactified exterior domains combined with stable second and fourth order time integrators is presented for Schrödinger equations. The numerical approach allows high precision numerical studies of solutions on the whole real line. At examples for the linear and cubic nonlinear Schrödinger equation, this code is compared to transparent boundary conditions and perfectly matched layers approaches. The code can deal with asymptotically non vanishing solutions as the Peregrine breather being discussed as a model for rogue waves. It is shown that the Peregrine breather can be numerically propagated with essentially machine precision, and that localized perturbations of this solution can be studied.     

The extraordinary spectral properties of radially periodic Schrödinger operators

Since it became clear that the band structure of the spectrum of periodic Sturm-Liouville operatorst = - (d²/dr²) +q(r) does not survive a spherically symmetric extension to Schrödinger operatorsT =- Δ+ V with V(x) =q(¦x¦) for x ∈ ?^d,d ∈ ? 1, a wealth of detailed information about the spectrum of such operators has been acquired. The observation of eigenvalues embedded in the essential spectrum [μ₀, ∞[ ofT with exponentially decaying eigenfunctions provided evidence for the existence of intervals of dense point spectrum, eventually proved by spherical separation into perturbed Sturm-Liouville operatorst _c = t +(c/r ²). Subsequently, a numerical approach was employed to investigate the distribution of eigenvalues ofT more closely. An eigenvalue was discovered below the essential spectrum in the cased = 2, and it turned out that there are in fact infinitely many, accumulating at μ₀. Moreover, a method based on oscillation theory made it possible to count eigenvalues oft _c contributing to an interval of dense point spectrum ofT. We gained evidence that an asymptotic formula, valid forc → ∞, does in fact produce correct numbers even for small values of the coupling constant, such that a rather precise picture of the spectrum of radially periodic Schrödinger operators has now been obtained.