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Milivoje Lukic

Publications and source records attributed to Milivoje Lukic.

24 records · Page 2Linked to original sources
arXiv2013

Wigner-von Neumann type perturbations of periodic Schrödinger Operators

We consider decaying oscillatory perturbations of periodic Schrödinger operators on the half line. More precisely, the perturbations we study satisfy a generalized bounded variation condition at infinity and an $L^p$ decay condition. We show that the absolutely continuous spectrum is preserved, and give bounds on the Hausdorff dimension of the singular part of the resulting perturbed measure. Under additional assumptions, we instead show that the singular part embedded in the essential spectrum is contained in an explicit countable set. Finally, we demonstrate that this explicit countable set is optimal. That is, for every point in this set there is an open and dense class of periodic Schrödinger operators for which an appropriate perturbation will result in the spectrum having an embedded eigenvalue at that point.

math.SP↗
arXiv2012

Schrodinger operators with slowly decaying Wigner--von Neumann type potentials

We consider Schrödinger operators with potentials satisfying a generalized bounded variation condition at infinity and an $L^p$ decay condition. This class of potentials includes slowly decaying Wigner--von Neumann type potentials $\sin(ax)/x^b$ with $b>0$. We prove absence of singular continuous spectrum and show that embedded eigenvalues in the continuous spectrum can only take values from an explicit finite set. Conversely, we construct examples where such embedded eigenvalues are present, with exact asymptotics for the corresponding eigensolutions.

math.SP↗
arXiv2012

A class of Schrodinger operators with decaying oscillatory potentials

We discuss Schrödinger operators on a half-line with decaying oscillatory potentials, such as products of an almost periodic function and a decaying function. We provide sufficient conditions for preservation of absolutely continuous spectrum and give bounds on the Hausdorff dimension of the singular part of the spectral measure. We also discuss the analogs for orthogonal polynomials on the real line and unit circle.

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arXiv2011

Derivatives of L^p eigenfunctions of Schrodinger operators

Assuming the negative part of the potential is uniformly locally $L^1$, we prove a pointwise $L^p$ estimate on derivatives of eigenfunctions of one-dimensional Schrodinger operators. In particular, if an eigenfunction is in $L^p$, then so is its derivative, for $1\le p\le \infty$.

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arXiv2010

Orthogonal Polynomials with Recursion Coefficients of Generalized Bounded Variation

We consider probability measures on the real line or unit circle with Jacobi or Verblunsky coefficients satisfying an $\ell^p$ condition and a generalized bounded variation condition. This latter condition requires that a sequence can be expressed as a sum of sequences $β^{(l)}$, each of which has rotated bounded variation, i.e., $\sum_{n=0}^\infty | e^{iϕ_l} β_{n+1}^{(l)} - β_n^{(l)} |$ is finite for some $ϕ_l$. This includes discrete Schrödinger operators on a half-line or line with finite linear combinations of Wigner--von Neumann type potentials. For the real line, we prove that in the Lebesgue decomposition $dμ=f dm + dμ_s$ of such measures, the intersection of (-2,2) with the support of $dμ_s$ is contained in an explicit finite set S (thus, $dμ$ has no singular continuous part), and f is continuous and non-vanishing on $(-2,2) \setminus S$. The results for the unit circle are analogous, with (-2,2) replaced by the unit circle.

math.SP↗