Spectral localization for quantum Hamiltonians with weak random delta interaction

Localizations at infinity and essential spectrum of quantum Hamiltonians: I. General theory

We isolate a large class of self-adjoint operators H whose essential spectrum is determined by their behavior at large x and we give a canonical representation of their essential spectrum in terms of spectra of limits at infinity of translations of H. The configuration space is an arbitrary abelian locally compact not compact group.

Spectral asymptotics of a broken delta interaction

This paper is concerned with the spectral analysis of a Hamiltonian with a $δ$-interaction supported along a broken line with angle $θ$. The bound states with energy slightly below the threshold of the essential spectrum are estimated in the semiclassical regime $θ\to 0$.

One-dimensional Schroedinger operators with delta-prime-interactions on Cantor-type sets

We introduce a novel approach for defining a $δ'$-interaction on a subset of the real line of Lebesgue measure zero which is based on Sturm-Liouville differential expression with measure coefficients. This enables us to establish basic spectral properties (e.g., self-adjointness, lower semiboundedness and spectral asymptotics) of Hamiltonians with $δ'$-interactions concentrated on sets of complicated structures.

Eigenvalues of Šeba billiards with localization of low-energy eigenfunctions

We study the localization of eigenfunctions produced by a point scatterer on a thin rectangle. We find an explicit set of eigenfunctions localized to part of the rectangle by showing that the one-dimensional Schrödinger operator with a Dirac delta potential asymptotically governs the spectral properties of the two-dimensional point scatterer. We also find the rate of localization in terms of the aspect ratio of the rectangle. In addition, we present numerical results regarding the asymptotic behavior of the localization.

Spectral properties of dynamical localization for Schrödinger operators

We investigate the equivalence between dynamical localization and localization properties of eigenfunctions of Schrödinger Hamiltonians. We introduce three classes of equivalent properties and study the relationships between them. These relationships are shown to be optimal thanks to counter examples.

具有纯点谱的弱正则Floquet Hamilton算子

Weakly regular Floquet Hamiltonians with pure point spectrum

我们研究了Floquet哈密顿量：-i ω d/dt + H + V（t）随参数ω的变化.我们假设H的谱是离散的，{h_m（m = 1.. infinity）}，其中h_m的重数为M_m。V是厄米特算子，在t中是2 pi周期的。设J > 0，设Ω_0 = [8 J/9，9 J/8]。假设对于某个sigma > 0：sum_{m,n such that h_m > h_n} mu_{mn}（h_m-h_n）^（-sigma）<无穷大其中mu_{mn} = sqrt（min（M_m，M_n））M_m M_n.我们证明了在这种情况下，存在一个合适的范数来度量V的正则性，记为ω，和正的常数，δ_* & delta_*，使得：如果ω <δ_*，则存在一个可测子集|Omega_infinity| > |Omega_0| - delta_*，并且Floquet哈密顿算子对Ω_infinity中的所有Ω都有纯点谱。