Behaviour of single-particle states and their eigenvalues near zero energy

For a state in a single-particle potential we derive simple expressions that the eigenvalue behavior and normalization integral properties near threshold. For the square-well, the various parameters in these expressions can be derived analytically. For the more realistic Saxon-Woods well-shape, we have made numerical calculations for s- and p-states and obtained values of these parameters and give rules for their evaluation. We show that those parameters determining the negative energy eigenvalues are also involved in determining the positive energy behavior, and the normalization properties. These studies have direct relevance to recent theories of some types of (d, p) and (n, γ) reactions which critically involve threshold behavior.     

Spectral density of mixtures of random density matrices for qubits

We derive the spectral density of the equiprobable mixture of two random density matrices of a two-level quantum system. We also work out the spectral density of mixture under the so-called quantum addition rule. We use the spectral densities to calculate the average entropy of mixtures of random density matrices, and show that the average entropy of the arithmetic-mean-state of n qubit density matrices randomly chosen from the Hilbert–Schmidt ensemble is never decreasing with the number n. We also get the exact value of the average squared fidelity. Some conjectures and open problems related to von Neumann entropy are also proposed.     

Singularity of the density of states for one-dimensional chains with random couplings

We prove that the density of states for the tight-binding model with off-diagonal disorder under general conditions diverges forR0 at least as . This result is established through the study of the recurrence properties of an associated Markov chain.Partial financial support by GNAFA (CNR)Partial financial support by CNPq, grant n.303795-77FA     

On the particle density of mass zero bosons and energy density of mass zero fermions

It is shown that even nonlocal generalizations of the usual expressions for

1. the particle probability density of mass zero, spin zero (one) bosons;
2. the energy density of mass zero spin one half fermions;

can become negative, indicating that the method of obtaining positive definite expressions for massive particles^2,4 for the above quantities does not work here.     

A dispersion relation for the density of states with application to the Casimir effect

The trace of a function of a Schrödinger operator minus the same for the Laplacian can be expressed in terms of the determinant of its scattering matrix. The naive formula for this determinant is divergent. Using a dispersion relation, we find another expression for it which is convergent, but needs one piece of information beyond the scattering matrix: the spatial integral of the potential. Except for this ‘anomaly’, we can express the Casimir energy of a compact body in terms of its optical scattering matrix, without assuming any rotational symmetry for its shape.     

Efficient, multiple-range random walk algorithm to calculate the density of states

We present a new Monte Carlo algorithm that produces results of high accuracy with reduced simulational effort. Independent random walks are performed (concurrently or serially) in different, restricted ranges of energy, and the resultant density of states is modified continuously to produce locally flat histograms. This method permits us to directly access the free energy and entropy, is independent of temperature, and is efficient for the study of both 1st order and 2nd order phase transitions. It should also be useful for the study of complex systems with a rough energy landscape.     

Landau Hamiltonians with random potentials: Localization and the density of states

We prove the existence of localized states at the edges of the bands for the two-dimensional Landau Hamiltonian with a random potential, of arbitrary disorder, provided that the magnetic field is sufficiently large. The corresponding eigenfunctions decay exponentially with the magnetic field and distance. We also prove that the integrated density of states is Lipschitz continuous away from the Landau energies. The proof relies on a Wegner estimate for the finite-area magnetic Hamiltonians with random potentials and exponential decay estimates for the finitearea Green's functions. The proof of the decay estimates for the Green's functions uses fundamental results from two-dimensional bond percolation theory.Supported in part by CNRS.Supported in part by NSF grants INT 90-15895 and DMS 93-07438.Unité Propre de Recherche 7061.     

On the zero temperature statistics for the density of gap states of hydrogenated amorphous silicon

It is shown that one may obtain much larger value of density-of-states in the gap of hydrogenated amorphous silicon than real value, if the zero temperature statistics is used in studying that material. So care must be taken when one employs zero temperature statistics in studying the density-of-states of amorphous materials.     

A new renormalization group with possible application to the bound state problem

The mass chosen for basis vectors in the experiment of a general state is arbitrary. Exploiting this fact leads to a new type of renormalization group, the basis renormalization group, which may be useful in describing bound states.     

Lyapunov exponents of large,sparse random matrices and the problem of directed polymers with complex random weights

We present results on two different problems: the Lyapunov exponent of large, sparse random matrices and the problem of polymers on a Cayley tree with random complex weights. We give an analytic expression for the largest Lyapunov exponent of products of random sparse matrices, with random elements located at random positions in the matrix. This expression is obtained through an analogy with the problem of random directed polymers on a Cayley tree (i.e., in the mean field limit), which itself can be solved using its relationship with random energy models (REM and GREM). For the random polymer problem with complex weights we find that, in addition to the high- and the low-temperature phases which were already known in the case of positive weights, the mean field theory predicts a new phase (phase III) which is dominated by interference effects.     

Probing the photonic local density of states with electron energy loss spectroscopy

Electron energy loss spectroscopy performed in transmission electron microscopes is shown to directly render the photonic local density of states with unprecedented spatial resolution, currently below the nanometer. Two special cases are discussed in detail: (i) 2D photonic structures with the electrons moving along the translational axis of symmetry and (ii) quasiplanar plasmonic structures under normal incidence. Nanophotonics in general and plasmonics, in particular, should benefit from these results connecting the unmatched spatial resolution of electron energy loss spectroscopy with its ability to probe basic optical properties such as the photonic local density of states.     

Asymptotic expansion for the density of states of the magnetic Schrödinger operator with a random potential

We study the asymptotics for the density of states of the magnetic Schrödinger operator with a random potential. By using the methods of effective Hamiltonian, complex dilation and complex translation, we obtain in the large magnetic field limit, the asymptotic expansion for the density of states measure considered as a distribution.     

ε-expansion for the density of states of a disordered system near an Anderson transition

The density of states for the Schrödinger equation with a Gaussian random potential is calculated in a space of dimension d=4?ε in the entire energy range, including the vicinity of an Anderson transition.     

On the Lifschitz singularity and the tailing in the density of states for random lattice systems

We prove rigorously the existence of a Lifschitz singularity in the density of states at zero energy in some random lattice systems of noninteracting bosons and fermions in any numberv of dimensions. The basic tool is a simple modification of the method of Fukushima to yield the correct upper and lower bounds for allv. We also comment on the mathematical difference between the models treated and the system of phonons with mass disorder in the harmonic approximation, whose behavior is known to be of Debye form, not Lifschitz, at low temperatures.Supported by the Swiss National Science Foundation.On leave of absence from the Institute de Fisica, University of São Paulo, Brazil.     

Tunneling in a superconductor with an energy dependent electronic density of states

Three generalized Eliashberg equations for the gap, renormalization and chemical potential shift have been solved in order to study the effects of energy dependence in the electronic density of states (EDOS) on the tunneling characteristics. Inversion of the tunneling characteristics within the context of the usual Eliashberg theory (with flat EDOS) shows that the effects of a nonconstant EDOS cannot always be described by some effective values of the usual parameters in the standard theory.     

Electronic density of states for the linear random binary alloy in the three-site coherent potential approximation

Morita and Chen's scheme of generalizing the coherent potential approximation in which the diagonal and off-diagonal randomness are included, is applied to the calculation of the electronic density of states for the linear random binary alloy. The result obtained in the three-site coherent potential approximation is reported.