Spectral properties for perturbations of unitary operators

AbstractConsider a unitary operator U0 acting on a complex separable Hilbert space H. In this paper we study spectral properties for perturbations of U0 of the type,Uβ=U0eiKβ, with K a compact self-adjoint operator acting on H and β a real parameter. We apply the commutator theory developed for unitary operators in Astaburuaga et al. (2006) [1] to prove the absence of singular continuous spectrum for Uβ. Moreover, we study the eigenvalue problem for Uβ when the unperturbed operator U0 does not have any. A typical example of this situation corresponds to the case when U0 is purely absolutely continuous. Conditions on the eigenvalues of K are given to produce eigenvalues for Uβ for both cases finite and infinite rank of K, and we give an example where the results can be applied

On the energy growth of some periodically driven quantum systems with shrinking gaps in the spectrum

We consider quantum Hamiltonians of the form H(t)=H+V(t) where the spectrum of H is semibounded and discrete, and the eigenvalues behave as E_n~n^\alpha, with 0<\alpha<1. In particular, the gaps between successive eigenvalues decay as n^{\alpha-1}. V(t) is supposed to be periodic, bounded, continuously differentiable in the strong sense and such that the matrix entries with respect to the spectral decomposition of H obey the estimate |V(t)_{m,n}|0, p>=1 and \gamma=(1-\alpha)/2. We show that the energy diffusion exponent can be arbitrarily small provided p is sufficiently large and \epsilon is small enough. More precisely, for any initial condition \Psi\in Dom(H^{1/2}), the diffusion of energy is bounded from above as _\Psi(t)=O(t^\sigma) where \sigma=\alpha/(2\ceil{p-1}\gamma-1/2). As an application we consider the Hamiltonian H(t)=|p|^\alpha+\epsilon*v(\theta,t) on L^2(S^1,d\theta) which was discussed earlier in the literature by Howland

Global attractor and finite dimensionality for a class of dissipative equations of BBM's type

In this work we study the Cauchy problem for a class of nonlinear dissipative equations of Benjamin-Bona-Mahony's type. We discuss the existence of a global attractor and estimate its Hausdorff and fractal dimensions

Short-Term Changes in Algometry, Inclinometry, Stabilometry, and Urinary pH Analysis After a Thoracolumbar Junction Manipulation in Patients with Kidney Stones

[eng] Objectives: To determine the efficacy of a high-velocity low-amplitude manipulation of the thoracolumbar junction in different urologic and musculoskeletal parameters in subjects suffering from renal lithiasis. Design: Randomized controlled blinded clinical study. Settings/location: The Nephrology Departments of 2 hospitals and one private consultancy of physiotherapy in Valencia (Spain). Subjects: Fourty-six patients suffering from renal lithiasis. Interventions: The experimental group (EG, n=23) received a spinal manipulation of the thoracolumbar junction, and the control group (CG, n=23) received a sham procedure. Outcome measures: Pressure pain thresholds (PPTs) of both quadratus lumborum and spinous processes from T10 to L1, lumbar flexion range of motion, stabilometry and urinary pH were measured before and immediately after the intervention. A comparison between pre and postintervention phases was performed and an analysis of variance for repeated measures using time (pre- and postintervention) as intrasubject variable and group (CG or EG) as intersubject variable. Results: Intragroup comparison showed a significative improvement for the EG in the lumbar flexion range of motion (p <0.001) and in all the PPT (p<0.001 in all cases). Between-groups comparison showed significant changes in PPT in quadratus lumborum (p<0.001) as well as in the spinous processes of all of the evaluated levels (p<0.05). No changes in urinary pH were observed (p=0.419). Conclusion: Spinal manipulation of the thoracolumbar junction seems to be effective in short term to improve pain sensitivity, as well as to increase the lumbar spine flexion