Multidimensional discrete Schr�dinger equation with limit periodic potential

2007

J Anal Math

Abstract. We consider a polyharmonic operator H = (−∆) l +V (x) in dimension two with l ≥ 6 and a limit-periodic potential V (x). We prove that the spectrum of H contains a semiaxis and there is a family of generalized eigenfunctions at every point of this semiaxis with the following properties. First, the eigenfunctions are close to plane waves e i k, x at the high energy region. Second, the isoenergetic curves in the space of momenta k corresponding to these eigenfunctions have a form of slightly distorted circles with holes (Cantor type structure).

Section: )mentioning

confidence: 99%

Section: Contracted Set Omentioning

confidence: 99%

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Spectral properties of polyharmonic operators with limit-periodic potential in dimension two

2007

J Anal Math

“…It is shown that the spectrum is a Cantor set of positive Lebesgue measure and purely absolutely continuous for a dense set of sampling functions, and it is a Cantor set of zero Lebesgue measure and purely singular continuous for a dense G δ set of sampling functions. Properties of eigenfunctions of discrete multidimensional limit-periodic Schrödinger operator are studied in [16]. As to the continuum multidimensional case, it is proven in [14] that the integrated density of states for (1.1) is the limit of densities of states for periodic operators.…”

Section: Introductionmentioning

confidence: 99%

Spectral properties of a limit-periodic Schrödinger operator in dimension two

2013

JAMA

Abstract. We study Schrödinger operator H = −∆ + V (x) in dimension two, V (x) being a limit-periodic potential. We prove that the spectrum of H contains a semiaxis and there is a family of generalized eigenfunctions at every point of this semiaxis with the following properties. First, the eigenfunctions are close to plane waves e i k, x at the high energy region. Second, the isoenergetic curves in the space of momenta k corresponding to these eigenfunctions have a form of slightly distorted circles with holes (Cantor type structure). Third, the spectrum corresponding to the eigenfunctions (the semiaxis) is absolutely continuous.

“…Integrated density of states is investigated in [11]- [14]. Properties of eigenfunctions of a discrete multidimensional limit-periodic Schrödinger operator are studied in [15]. As to the continuum multidimensional case, it is known that the integrated density of states for (1) is the limit of densities of states for periodic operators [14].…”

Section: Yulia Karpeshina and Young-ran Lee (Communicated By Svetlanamentioning

confidence: 99%

On polyharmonic operators with limit-periodic potential in dimension two

Electron. Res. Announc. Amer. Math. Soc.

2006

This is an announcement of the following results. We consider a polyharmonic operator H = ( − Δ ) l + V ( x ) H=(-\Delta )^l+V(x) in dimension two with l ≥ 6 l\geq 6 and V ( x ) V(x) being a limit-periodic potential. We prove that the spectrum of H H contains a semiaxis and there is a family of generalized eigenfunctions at every point of this semiaxis with the following properties. First, the eigenfunctions are close to plane waves at the high-energy region. Second, the isoenergetic curves in the space of momenta corresponding to these eigenfunctions have the form of slightly distorted circles with holes (Cantor-type structure). Third, the spectrum corresponding to the eigenfunctions (the semiaxis) is absolutely continuous.