Weakly singular perturbations of a discrete spectrum

We consider weakly singular perturbations ¦x¦^–(0<<2) of an even restoring potential. We compute the matrix elements of the perturbation together with the additional point potential associated with the perturbation. It is shown that even for unperturbed wave functions, the matrix elements exist when 0 < < 3/2. The series for the Rayleigh-Schrödinger coefficients converge in all orders for the same interval in , regardless of the form of the restoring potential. For odd states, the matrix elements of the perturbation exist when 0 < < 3, while estimates for the Rayleigh-Schrödinger coefficients give the boundary = 2.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 14–18, June, 1989.     

Singular perturbations of piecewise monotonic maps of the interval

Let t: [0, 1] [0, 1] be a piecewise monotonic, C₂, and expanding map. In computing an orbit { ⁱ (x ₀)} _i=0 , we model the roundoff error at each iteration by a singular perturbation; i.e.,X _n+1=(X _n )+W , whereW is a random variable taking on discrete values in an interval (-&#x03B5;, ). The main result proves that this process admits an absolutely continuous invariant measure which approaches the absolutely continuous measure invariant under the deterministic map t as the precision of computation 0.     

Some quantum operators with discrete spectrum but classically continuous spectrum

We consider a number of simple quantum Hamiltonians H(?i?,x) with the following property: H(?i?,x) has discrete spectrum even though {(p,q) | H(p,q) <E} has infinite volume.     

Radial wave functions of a discrete spectrum

The system of equations for radial wave functions is written in a form allowing both relativistic and non relativistic wave functions to be obtained. In the case of a discrete spectrum, an asymptotic solution of this system is obtained for a potential which includes not only a Coulomb term but also terms corresponding to dipole and quadrupole polarization. The normalizing factor of the asymptote is determined in the approximation of the relativistic quantum-defect method, which offers the possibility, in principle, of using the functions in semiempirical calculations analogous to nonrelativistic calculations by the Bates — Damgaard — Seaton method. The calculation scheme is illustrated for the example of the calculation of Si IV wave functions.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 12, pp. 39–44, December, 1978.     

Singular continuous Cantor spectrum for magnetic quantum walks

Letters in Mathematical Physics - In this note, we consider a physical system given by a two-dimensional quantum walk in an external magnetic field. In this setup, we show that both the topological...     

Density perturbations due to the inhomogeneous discrete spatial structure of space-time

We demonstrate that if space time admits an inhomogeneous discrete spatial structure due to varying gravitational fields or a foam-like structure of space-time, then thermodynamic reasoning implies that matter-density perturbations will arise in the early universe.     

Boundary effects in local inflation and spectrum of density perturbations

We observe that when a local patch in a radiation filled Robertson–Walker universe inflates by some reason, outside perturbations can enter into the inflating region. Generally, the physical wavelengths of these perturbations become larger than the Hubble radius as they cross into the inflating space and their amplitudes freeze out immediately. It turns out that the corresponding power spectrum is not scale invariant. Although these perturbations cannot reach out to a distance inner observer shielded by a de Sitter horizon, they still indicate a curious boundary effect in local inflationary scenarios.     

Absence of discrete spectrum in highly negative ions

We extend the results of [1] to fermions, i.e., we show that ifH _N is the Hamiltonian forN electrons in the field of a fixed point chargeZ, then there is a constantc such thatH _N has no discrete spectrum forN≧N ₀=cZ ^6/5.     

Absence of discrete spectrum in highly negative ions

LetH _N be the Hamiltonian for the Coulomb system consisting ofN particles of like charge in the field of a fixed point chargeZ. We show that if the particles are bosons, thenH _N has no discrete spectrum whenNN ₀=cZ ² for some constantc. If the particles are fermions, thenH _N is bounded below uniformly inN. These results can be extended to molecules and to other power law potentials.Research supported by the National Science Foundation, MCS78 -20455 USAOn leave from Department of Mathematics, University of Lowell, Lowell, MA O1854 USA     

Strongly nonlinear shear perturbations in discrete and continuous cubic nonlinear systems

Systems with constraints, the masses in which move only along guides, can execute strongly nonlinear vibrations. This means that nonlinear phenomena manifest themselves at arbitrary small deviations from equilibrium. The form of vibrations of a single mass is described by elliptical Jacobi functions. The spectrum of these vibrations is found. With an increase in amplitude, the period of vibrations decreases. We deduce equations of strongly nonlinear vibrations of a chain of connected masses. In the continuum limit, we obtain a new nonlinear equation in partial derivatives. We devise transformation of variables leading to linearization of this equation. We implemented a factorization procedure that decreases the order of the equation in partial derivatives from second to first. Exact solutions to the first-order equation describe the slow evolution of the displacement profile in a distributed system. In the absence of preliminary tension of elastic elements in the continued model, traveling waves cannot be achieved; however, time-oscillating solutions like standing waves are possible. We obtain an equation for a field of strongly nonlinear deformations. Its exact solution describes periodic movement in time and space. As well, the period of time oscillations decreases with an increase in amplitude, and the spatial period, in contrast, increases. The product of the vibration frequency multiplied by the spatial period is a constant that depends on the deformation energy. We propose a scheme of the mechanical system producing strongly nonlinear torsional vibrations. We experimentally measured the period of torsional vibrations of a single disc. We show that with an increase in amplitude in the process of vibration attenuation, an increase in the period occurs, which agrees with calculations. We measure the shapes of nonlinear vibrations of a chain of connected discs. A strongly nonlinear behavior of the chain is observed.     

Inflaton field potential producing an exactly flat spectrum of adiabatic perturbations

Presented in this letter is the exact solution of the problem of finding the potential of an inflaton scalar field for which adiabatic perturbations generated during a de Sitter (inflationary) stage in the early Universe have an exactly flat (or, the Harrison-Zeldovich) initial spectrum. This solution lies outside the scope of the slow-roll approximation and higher-order corrections to it. The potential found depends on two arbitrary physical constants, one of which determines the amplitude of the perturbations. For small (zero) values of the other constant, a long (infinite) inflationary stage with slow rolling of the inflaton field exists.     

Black hole spectrum: continuous or discrete?

We formulate a qualitative argument, based on Heisenberg's uncertainty principle, to support the claim that when the effects of matter fields are assumed to overshadow the effects of quantum mechanics of spacetime, the discrete spectrum of black hole radiation, as predicted by Bekenstein's proposal for a discrete black hole area spectrum, reduces to Hawking's black-body spectrum.     

奇异谱分析用于提升双光梳激光测距精度

从含噪数据中提取信号从而提升数据采集系统精度是极为重要的问题.奇异谱分析(singular spectrum analysis,SSA)作为一种无参数频谱估计技术,广泛用于区分系统模型未知情况下的动态系统信号的复杂成分.本文应用SSA方法提取双光梳飞秒激光测距系统中的含噪时间序列的距离信息,数值仿真显示SSA方法可以从含有有色噪声的信号中提取距离信号.实验中,SSA方法成功地从含有量子噪声的测距信号中提取出激光与目标之间的距离信息,提取后的信号有13倍的精度提升.这种方法同样适用于高维信号,如基于飞秒激光测距的高精度、高速率表面形貌测量的图像提取.     

Pure point spectrum under 1-parameter perturbations and instability of Anderson localization

We consider a selfadjoint operator,A, and a selfadjoint rank-one projection,P, onto a vector, , which is cyclic forA. We study the set of all eigenvalues of the operatorA _t =A+tP (t) that belong to its essential spectrum (which does not depend on the parametert). We prove that this set is empty for a dense set of values oft. Then we apply this result or its idea to questions of Anderson localization for 1-dimensional Schrödinger operators (discrete and continuous).     

On the discrete spectrum of spin-orbit Hamiltonians with singular interactions

We give a variational proof of the existence of infinitely many bound states placed below the continuous spectrum for spin-orbit Hamiltonians (including the Rashba and Dresselhaus cases) perturbed by measure potentials, thus extending results of J. Brüning, V. Geyler, K. Pankrashkin, J. Phys. A: Math. Theor. 40 F113–F117 (2007).     

Singular spectrum analysis for enhancing the sensitivity in structural damage detection

Time domain structural condition assessment methods have been studied extensively in the last two decades due to their effectiveness in dealing with limited and short duration measurements from a structure under operational conditions. The sensitivity-based method is revisited in this paper with an enhancement in the sensitivity with respect to local damages via the singular spectrum analysis technique. The measured response of the structure is decomposed and the sensitivity vectors as well as the computed response vectors are projected into the corresponding decomposition subspace. The projected identification equations associate with components which contain the least measurement noise and the most damage information are then used to detect local damages in the structure. The enhanced sensitivity-based method is shown to be capable of yielding more accurate identified results with noisy measurement in a planar truss structure compared to conventional sensitivity-based method. A seven-storey steel frame test is performed in the laboratory, and the proposed method is checked to be able to identify the damage location and extend of this structure with an acceptable accuracy.