Cantor singular continuous spectrum for operators along interval exchange transformations

It is shown that Schrödinger operators, with potentials along the shift embedding of Lebesgue almost every interval exchange transformations, have Cantor spectrum of measure zero and pure singular continuous for Lebesgue almost all points of the interval.

     

The first instability interval for Hill equations with symmetric single well potentials

For Hill equations with symmetric single well (or symmetric single barrier) potentials, the first instability interval is absent when and only when the potential is constant.

     

Parametrization of the isospectral set for the vector-valued Sturm-Liouville problem

We obtain a parametrization of the isospectral set of matrix-valued potentials for the vector-valued Sturm-Liouville problem on a finite interval.     

Local Remez-type inequalities for exponentials of a potential on a piecewise analytic ARC

Based on techniques from potential theory and geometric function theory, we give a new proof of the pointwise Remez-type inequality for exponentials of logarithmic potentials on the unit interval [−1, 1]. This inequality was first proved by Erdélyi, Li and Saff [8, Theorem 2.2.]. We generalize our result to the case of a piecewise analytic arc.     

Sparse potentials with fractional Hausdorff dimension

We construct non-random bounded discrete half-line Schrödinger operators which have purely singular continuous spectral measures with fractional Hausdorff dimension (in some interval of energies). To do this we use suitable sparse potentials. Our results also apply to whole line operators, as well as to certain random operators. In the latter case we prove and compute an exact dimension of the spectral measures.     

Krein Systems and Canonical Systems on a Finite Interval: Accelerants with a Jump Discontinuity at the Origin and Continuous Potentials

This paper is devoted to connections between accelerants and potentials of Krein systems and of canonical systems of Dirac type, both on a finite interval. It is shown that a continuous potential is always generated by an accelerant, provided the latter is continuous with a possible jump discontinuity at the origin. Moreover, the generating accelerant is uniquely determined by the potential. The results are illustrated on pseudo-exponential potentials. The paper is a continuation of the earlier paper of the authors (Alpay et al. in Modern Analysis and Applications. The Mark Krein Centenary Conference, vol. 2, pp. 19–36, OT 191. Birkhäuser, Basel, 2009) dealing with the direct problem for Krein systems.     

Numerical solution of inverse spectral problems for Sturm-Liouville operators with discontinuous potentials

We consider Sturm-Liouville differential operators on a finite interval with discontinuous potentials having one jump. As the main result we obtain a procedure of recovering the location of the discontinuity and the height of the jump. Using our result, we apply a generalized Rundell-Sacks algorithm of Rafler and Böckmann for a more effective reconstruction of the potential and present some numerical examples.     

Formula for the number of eigenvalues of a three-particle Schrödinger operator on a lattice

We consider a system of three arbitrary quantum particles on a three-dimensional lattice that interact via short-range attractive potentials. We obtain a formula for the number of eigenvalues in an arbitrary interval outside the essential spectrum of the three-particle discrete Schrödinger operator and find a sufficient condition for the discrete spectrum to be finite. We give an example of an application of our results.     

Boundary conditions of Sturm-Liouville operators with mixed spectra

We study bounds on averages of spectral functions corresponding to Sturm-Liouville operators on the half line for different boundary conditions. As a consequence constraints are obtained which imply existence of singular spectrum embedded in a.c. spectrum for sets of boundary conditions with positive measure and potentials vanishing in an interval [0,N]. These constraints are related to estimates on the measure of sets where the spectral density is positive.     

Some inverse scattering problems on star-shaped graphs

Having in mind applications to the fault-detection/diagnosis of lossless electrical networks, here we consider some inverse scattering problems for Schrödinger operators over star-shaped graphs. We restrict ourselves to the case of minimal experimental setup consisting in measuring, at most, two reflection coefficients when an infinite homogeneous (potential-less) branch is added to the central node. First, by studying the asymptotic behavior of only one reflection coefficient in the high-frequency limit, we prove the identifiability of the geometry of this star-shaped graph: the number of edges and their lengths. Next, we study the potential identification problem by inverse scattering, noting that the potentials represent the inhomogeneities due to the soft faults in the network wirings (potentials with bounded H¹-norms). The main result states that, under some assumptions on the geometry of the graph, the measurement of two reflection coefficients, associated to two different sets of boundary conditions at the external vertices of the tree, determines uniquely the potentials; it can be seen as a generalization of the theorem of the two boundary spectra on an interval.     

Inverse problems for Sturm–Liouville operators on a star-shaped graph with mixed spectral data

The partial inverse spectral problem for Sturm–Liouville operators on a star-shaped graph was studied. The authors showed that if the potentials but one were known a priori, then the unknown potential on the whole interval can be uniquely determined by part of information of the potential and part of eigenvalues. The methods employed rest on the Weyl's m-function and theory concerning densities of zeros of entire functions.     

Equilibrium states,pressure and escape for multimodal maps with holes

For a class of non-uniformly hyperbolic interval maps, we study rates of escape with respect to conformal measures associated with a family of geometric potentials. We establish the existence of physically relevant conditionally invariant measures and equilibrium states and prove a relation between the rate of escape and pressure with respect to these potentials. As a consequence, we obtain a Bowen formula: we express the Hausdorff dimension of the set of points which never exit through the hole in terms of the relevant pressure function. Finally, we obtain an expression for the derivative of the escape rate in the zero-hole limit.     

Ambarzumian’s Theorem for a Sturm-Liouville Boundary Value Problem on a Star-Shaped Graph

Ambarzumian’s theorem describes the exceptional case in which the spectrum of a single Sturm-Liouville problem on a finite interval uniquely determines the potential. In this paper, an analog of Ambarzumian’s theorem is proved for the case of a Sturm-Liouville problem on a compact star-shaped graph. This case is also exceptional and corresponds to the Neumann boundary conditions at the pendant vertices and zero potentials on the edges.__________Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 39, No. 2, pp. 78–81, 2005Original Russian Text Copyright © by V. N. Pivovarchik     

Remarks on a nonlinear transport problem

A nonlinear transport problem of hyperbolic–elliptic type is studied. Estimates of potentials over varying domains and the method of characteristics enable one to treat the initial value problem for Hölder continuous data as an abstract evolution equation via Picard–Lindelöf theorem. In addition, existence for all times is proved. Similar techniques yield the existence of shock front solutions with smooth interfaces at least for a small time interval. By a priori estimates of approximating solutions, the results extend to the case of only bounded initial values. A modification of the system applies to the construction of a diffeomorphism with prescribed Jacobian determinant.     

On a two-point boundary value problem for the Sturm-Liouville operator with a nonclassical asymptotics of the spectrum

We consider a spectral problem generated by a Sturm-Liouville equation on the interval (0, π) with degenerate boundary conditions. We prove the existence of potentials q(x) ∈ L ₂(0, π) such that the multiplicities of the eigenvalues λ _n monotonically tend to infinity as n → ∞.     

Lifshitz tails for Schrödinger operators with random breather potential

We prove a Lifshitz tail bound on the integrated density of states of random breather Schrödinger operators. The potential is composed of translated single-site potentials. The single-site potential is an indicator function of the set tA where t is from the unit interval and A is a measurable set contained in the unit cell. The challenges of this model are that, since A is not assumed to be star-shaped, the dependence of the potential on the parameter t is not monotone. It is also non-linear and not differentiable.     

On the Reconstruction of the Sturm-Liouville Problems with Spectral Parameter in the Discontinuity Conditions

In this paper, we are concerned with the problem of recovering the Sturm–Liouville problem under the circumstance of the discontinuity conditions involved spectral parameter at finite interior points of a finite interval. We provide procedures for constructing their potentials and boundary conditions either from the Weyl function, or from spectral data, or from two spectra in terms of the method of spectral mappings.     

One‐sided operators in Lp (x) spaces

Boundedness of one‐sided maximal functions, singular integrals and potentials is established in L(I) spaces, where I is an interval in R . (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Half‐inverse problems for the quadratic pencil of the Sturm–Liouville equations with impulse

In this paper, we consider the inverse spectral problem for the impulsive Sturm–Liouville differential pencils on [0, π] with the Robin boundary conditions and the jump conditions at the point . We prove that two potentials functions on the whole interval and the parameters in the boundary and jump conditions can be determined from a set of eigenvalues for two cases: (i) the potentials given on and (ii) the potentials given on , where 0 < α < 1 , respectively. Inverse spectral problems, Sturm–Liouville operator, spectrum, uniqueness.     

Spectra of Anderson type models with decaying randomness

In this paper we consider some Anderson type models, with free parts having long range tails and with the random perturbations decaying at different rates in different directions and prove that there is a.c. spectrum in the model which is pure. In addition, we show that there is pure point spectrum outside some interval. Our models include potentials decaying in all directions in which case absence of singular continuous spectrum is also shown.