Embedded singular continuous spectrum for one-dimensional Schrödinger operators

We investigate one-dimensional Schrödinger operators with sparse potentials (i.e. the potential consists of a sequence of bumps with rapidly growing barrier separations). These examples illuminate various phenomena related to embedded singular continuous spectrum.

     

On the absolutely continuous spectrum of one-dimensional quasi-periodic Schrödinger operators in the adiabatic limit

In this paper we study the spectral properties of families of quasi-periodic Schrödinger operators on the real line in the adiabatic limit in the case when the adiabatic iso-energetic curves are extended along the position direction. We prove that, in energy intervals where this is the case, most of the spectrum is purely absolutely continuous in the adiabatic limit, and that the associated generalized eigenfunctions are Bloch-Floquet solutions.

RÉSUMÉ. Cet article est consacré à l'étude du spectre de certaines familles d'équations de Schrödinger quasi-périodiques sur l'axe réel lorsque les variétés iso-énergetiques adiabatiques sont étendues dans la direction des positions. Nous démontrons que, dans un intervalle d'énergie où ceci est le cas, le spectre est dans sa majeure partie purement absolument continu et que les fonctions propres généralisées correspondantes sont des fonctions de Bloch-Floquet.

     

Bounds on embedded singular spectrum for one-dimensional Schrödinger operators

We show that the solutions of the one-dimensional Schrödinger equation with potential satisfy the WKB asymptotic formulae off a set of energies of Hausdorff dimension . This result gives restrictions on the structure of possible embedded singular spectrum. The proof relies on new norm estimates for the integral transform associated with the WKB method.

     

Bounded eigenfunctions and absolutely continuous spectra for one-dimensional Schrödinger operators

We provide a short proof of that case of the Gilbert-Pearson theorem that is most often used: That all eigenfunctions bounded implies purely a.c. spectrum. Two appendices illuminate Weidmann's result that potentials of bounded variation have strictly a.c. spectrum on a half-axis.

     

Estimates for multiparameter maximal operators of Schrödinger type

Multiparameter maximal estimates are considered for operators of Schrödinger type. Sharp and almost sharp results, that extend work by Rogers and Villarroya, are obtained. We provide new estimates via the integrability of the kernel which naturally appears with a TT^?$T{T}^{?}$ argument and discuss the behavior at the endpoints. We treat in particular the case of global integrability of the maximal operator on finite time for solutions to the linear Schrödinger equation and make some comments on an open problem.     

Local spectral properties of reflectionless Jacobi, CMV, and Schrödinger operators

We prove that Jacobi, CMV, and Schrödinger operators, which are reflectionless on a homogeneous set E (in the sense of Carleson), under the assumption of a Blaschke-type condition on their discrete spectra accumulating at E, have purely absolutely continuous spectrum on E.     

Isometries, Fock spaces, and spectral analysis of Schrödinger operators on trees

We construct conjugate operators for the real part of a completely non-unitary isometry and we give applications to the spectral and scattering theory of a class of operators on (complete) Fock spaces, natural generalizations of the Schrödinger operators on trees. We consider C^*-algebras generated by such Hamiltonians with certain types of anisotropy at infinity, we compute their quotient with respect to the ideal of compact operators, and give formulas for the essential spectrum of these Hamiltonians.     

Estimates for fundamental solutions and spectral bounds for a class of Schrödinger operators

We study a class of Schrödinger operators of the form , where is a nonnegative function singular at 0, that is V(0)=0. Under suitable assumptions on the potential V, we derive sharp lower and upper bounds for the fundamental solution hε. Moreover, we obtain information on the spectrum of the self-adjoint operator defined by Lε in L²(R). In particular, we give a lower bound for the eigenvalues.     

Holomorphic spectral geometry of magnetic Schrödinger operators on Riemann surfaces

We study magnetic Schrödinger operators on line bundles over Riemann surfaces endowed with metrics of constant curvature. We show that for harmonic magnetic fields the spectral geometry of these operators is completely determined by the Bochner Laplacians of the line bundles. Therefore we are led to examine the spectral problem for the Bochner Laplacian ∇^∗∇ of a Hermitian line bundle L with connection ∇ over a Riemann surface S. This spectral problem is analyzed in terms of the natural holomorphic structure on L defined by the Cauchy-Riemann operator associated with ∇. By means of an elliptic chain of line bundles obtained by twisting L with the powers of the canonical bundle we prove that there exists a certain subset of the spectrum σhol(∇^∗∇) such that the eigensections associated with λ∈σhol(∇^∗∇) are given by the holomorphic sections of a certain line bundle of the elliptic chain. For genus p=0,1 we prove that σhol(∇^∗∇) is the whole spectrum, whereas for genus p>1 we get a finite number of eigenvalues.     

Trace asymptotics for fractional Schrödinger operators

This paper proves an analogue of a result of Bañuelos and Sá Barreto [5] on the asymptotic expansion for the trace of Schrödinger operators on R^d${\mathbb{R}}^d$ when the Laplacian −Δ, which is the generator of the Brownian motion, is replaced by the non-local integral operator (−Δ)^α/2${(-\mathrm{\Delta })}^{\alpha /2}$, 0<α<2$0<\alpha <2$, which is the generator of the symmetric stable process of order α. These results also extend recent results of Bañuelos and Yildirim [6] where the first two coefficients for (−Δ)^α/2${(-\mathrm{\Delta })}^{\alpha /2}$ are computed. Some extensions to Schrödinger operators arising from relativistic stable and mixed-stable processes are obtained.     

Anderson localization for the discrete one-dimensional quasi-periodic Schrödinger operator with potential defined by a Gevrey-class function

In this paper we consider the discrete one-dimensional Schrödinger operator with quasi-periodic potential v_n=λv(x+nω). We assume that the frequency ω satisfies a strong Diophantine condition and that the function v belongs to a Gevrey class, and it satisfies a transversality condition. Under these assumptions we prove—in the perturbative regime—that for large disorder λ and for most frequencies ω the operator satisfies Anderson localization. Moreover, we show that the associated Lyapunov exponent is positive for all energies, and that the Lyapunov exponent and the integrated density of states are continuous functions with a certain modulus of continuity. We also prove a partial nonperturbative result assuming that the function v belongs to some particular Gevrey classes.     

Essential spectrum and -solutions of one-dimensional Schrödinger operators

In 1949, Hartman and Wintner showed that if the eigenvalue equations of a one-dimensional Schrödinger operator possess square integrable solutions, then the essential spectrum is nowhere dense. Furthermore, they conjectured that this statement could be improved and that under this condition the essential spectrum might always be void. This is shown to be false. It is proved that, on the contrary, every closed, nowhere dense set does occur as the essential spectrum of Schrödinger operators which satisfy the condition of existence of -solutions. The proof of this theorem is based on inverse spectral theory.

     

Schrödinger operators and de Branges spaces

We present an approach to de Branges's theory of Hilbert spaces of entire functions that emphasizes the connections to the spectral theory of differential operators. The theory is used to discuss the spectral representation of one-dimensional Schrödinger operators and to solve the inverse spectral problem.     

Uniqueness theorems in inverse spectral theory for one-dimensional Schrödinger operators

New unique characterization results for the potential in connection with Schrödinger operators on and on the half-line are proven in terms of appropriate Krein spectral shift functions. Particular results obtained include a generalization of a well-known uniqueness theorem of Borg and Marchenko for Schrödinger operators on the half-line with purely discrete spectra to arbitrary spectral types and a new uniqueness result for Schrödinger operators with confining potentials on the entire real line.

     

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.     

Variable coefficient Schrödinger flows for ultrahyperbolic operators

We consider Schrödinger flows which are given by a real symmetric non-degenerate matrix of variable coefficient second order differential operators. After establishing the local smoothing effect we treat non-linear perturbations for first and zero order terms. A fundamental step is the construction of an integrating factor using some non-standard symbols.     

“Localized” self-adjointness of Schrödinger type operators on Riemannian manifolds

We prove self-adjointness of the Schrödinger type operator , where ∇ is a Hermitian connection on a Hermitian vector bundle E over a complete Riemannian manifold M with positive smooth measure dμ which is fixed independently of the metric, and V∈L_loc¹(EndE) is a Hermitian bundle endomorphism. Self-adjointness of H_V is deduced from the self-adjointness of the corresponding “localized” operator. This is an extension of a result by Cycon. The proof uses the scheme of Cycon, but requires a refined integration by parts technique as well as the use of a family of cut-off functions which are constructed by a non-trivial smoothing procedure due to Karcher.     

Semi-classical spectral estimates for Schrödinger operators at a critical level. Case of a degenerate maximum of the potential

We study the semi-classical trace formula at a critical energy level for a Schrödinger operator on Rn. We assume here that the potential has a totally degenerate critical point associated to a local maximum. The main result, which establishes the contribution of the associated equilibrium in the trace formula, is valid for all time in a compact subset of R and includes the singularity in t=0. For these new contributions the asymptotic expansion involves the logarithm of the parameter h. Depending on an explicit arithmetic condition on the dimension and the order of the critical point, this logarithmic contribution can appear in the leading term.     

Absolutely continuous spectrum for one-dimensional Schrödinger operators with slowly decaying potentials: Some optimal results

The absolutely continuous spectrum of one-dimensional Schrödinger operators is proved to be stable under perturbation by potentials satisfying mild decay conditions. In particular, the absolutely continuous spectra of free and periodic Schrödinger operators are preserved under all perturbations satisfying This result is optimal in the power scale. Slightly more general classes of perturbing potentials are also treated. A general criterion for stability of the absolutely continuous spectrum of one-dimensional Schrödinger operators is established. In all cases analyzed, the main term of the asymptotic behavior of the generalized eigenfunctions is shown to have WKB form for almost all energies. The proofs rely on maximal function and norm estimates, and on almost everywhere convergence results for certain multilinear integral operators.

     

Dirac and magnetic Schrödinger operators on fractals

In this paper we define (local) Dirac operators and magnetic Schrödinger Hamiltonians on fractals and prove their (essential) self-adjointness. To do so we use the concept of 1-forms and derivations associated with Dirichlet forms as introduced by Cipriani and Sauvageot, and further studied by the authors jointly with Röckner, Ionescu and Rogers. For simplicity our definitions and results are formulated for the Sierpinski gasket with its standard self-similar energy form. We point out how they may be generalized to other spaces, such as the classical Sierpinski carpet.