Narrow Results

In the present paper we consider Riemannian coverings (X,g) → (M,g) with residually finite covering group Γ and compact base space (M,g). In particular, we give two general procedures resulting in a family of deformed coverings (X,g_ε) → (M,g_ε) such that the spectrum of the Laplacian Δ(X_ε,g_ε) has at least a prescribed finite number of spectral gaps provided ε is small enough.

If Γ has a positive Kadison constant, then we can apply results by Brüning and Sunada to deduce that spec Δ(X,g_ε) has, in addition, band-structure and there is an asymptotic estimate for the number of components of spec Δ(X,g_ε) that intersect the interval [0,λ]. We also present several classes of examples of residually finite groups that fit with our construction and study their interrelations. Finally, we mention several possible applications for our results.

Product vacua with boundary states (PVBS) are cousins of the Heisenberg XXZ spin model and feature $n$ particle species on ${\mathbb{Z}}^d$. The PVBS models were originally introduced as toy models for the classification of ground state phases. A crucial ingredient for this classification is the existence of a spectral gap above the ground state sector. In this work, we derive a spectral gap for PVBS models at arbitrary species number $n$ and in arbitrary dimension $d$ in the perturbative regime of small anisotropy parameters. Instead of using the more common martingale method, the proof verifies a finite-size criterion in the spirit of Knabe.

We consider a family of Schrödinger operators supported by a periodic chain of loops connected either tightly or loosely through connecting links of the length $\ell >0$ with the vertex coupling which is non-invariant with respect to the time reversal. The spectral behavior of the model illustrates that the high-energy behavior of such vertices is determined by the vertex parity. The positive spectrum of the tightly connected chain covers the entire halfline while the one of the loose chain is dominated by gaps. In addition, there is a negative spectrum consisting of an infinitely degenerate eigenvalue in the former case, and of one or two absolutely continuous bands in the latter. Furthermore, we discuss the limit $\ell \to 0$ and show that while the spectrum converges as a set to that of the tight chain, as it should in view of a result by Berkolaiko, Latushkin, and Sukhtaiev, this limit is rather non-uniform.

We present a new method of gap control in two-dimensional periodic systems with the perturbation consisting of a second-order differential operator and a family of narrow potential “walls” separating the period cells in one direction. We show that under appropriate assumptions one can open gaps around points determined by dispersion curves of the associated “waveguide” system, in general any finite number of them, and to control their widths in terms of the perturbation parameter. Moreover, a distinctive feature of those gaps is that their edge values are attained by the corresponding band functions at internal points of the Brillouin zone.

The flow of quantum information in local dephasing channels is analyzed over short and long times in case the structured reservoir of frequency modes exhibits a spectral gap in the density of modes over low frequencies. The presence of the low-frequency gap with upper cut-off frequency ${\omega }_g$ produces an infinite sequence of long-time intervals over which information backflow appears. Though generally irregular, the time intervals exhibit, under certain conditions, upper bounds: the $n$th episode of information backflow has already started at the instant $\pi (2n-1)∕{\omega }_g$, and already ended at the instant $2\pi n∕{\omega }_g$, for every $n=1,2,\dots$. The intervals become regular over long times, tend to the asymptotic length $\pi ∕{\omega }_g$, as the supremum value, and are described analytically in terms of the structure of the spectral density near the upper cut-off frequency of the spectral gap. Consequently, engineering structured reservoirs of frequency modes with a spectral gap over low frequencies produces in local dephasing channels regular sequences of information backflow and recoherence over long times, along with non-Markovian evolution.

The energy of the bosonic bath and the flow of quantum information in local dephasing channels are studied over short and long times in case the distribution of frequency modes of the bosonic bath exhibits a low-frequency gap. The initial conditions consist in special correlations between the qubit and the bosonic bath or are factorized, and involve thermal states of the whole system or of the bath. The low-frequency gap generates damped oscillations of the bath energy around the asymptotic value, for the correlated initial conditions, and induces the open system to alternately loose and gain information, for the factorized initial configurations. The long-time oscillations of the bath energy become regular and the frequency of the oscillations coincides with the upper cut-off frequency of the spectral gap. Regular long-time sequences of intervals are found over which the bath energy increases (decreases), for the correlated initial conditions, and information is lost (gained) by the open system, for the factorized initial configurations, even at different temperatures. This relation is reversed, if compared to the one obtained without the low-frequency gap, and can fail if the spectral density is tailored near the spectral gap according to power laws with odd natural powers.

We consider a periodic magnetic Schrödinger operator on a noncompact Riemannian manifold M such that H¹(M,ℝ) = 0 endowed with a properly discontinuous cocompact isometric action of a discrete group. We assume that there is no electric field and that the magnetic field has a periodic set of compact magnetic wells. We review a general scheme of a proof of existence of an arbitrary large number of gaps in the spectrum of such an operator in the semiclassical limit, which was suggested in our previous paper, and some applications of this scheme. Then we apply these methods to establish similar results in the case when the wells have regular hypersurface pieces.