Isospectral Mathieu–Hill Operators

In this paper we prove that the spectra of the Mathieu–Hill Operators with potentials ae^{?i2π x}  +be ^{i2π x} and ce^{?i2π x}  +de ^{i2π x} , where a, b, c and d are complex numbers, are the same if and only if ab = cd. This immediately results in a generalization of the extension of the Harrell–Avron–Simon formula.     

Resonance Tongues and Instability Pockets in the Quasi–Periodic Hill–Schrödinger Equation

This paper concerns Hills equation with a (parametric) forcing that is real analytic and quasi-periodic with frequency vector  ^d and a frequency (or energy) parameter a and a small parameter b. The 1-dimensional Schrödinger equation with quasi-periodic potential occurs as a particular case. In the parameter plane ²={a, b}, for small values of b we show the following. The resonance tongues with rotation number have C -boundary curves. Our arguments are based on reducibility and certain properties of the Schrödinger operator with quasi-periodic potential. Analogous to the case of Hills equation with periodic forcing (i.e., d=1), several further results are obtained with respect to the geometry of the tongues. One result regards transversality of the boundaries at b=0. Another result concerns the generic occurrence of instability pockets in the tongues in a reversible near-Mathieu case, that may depend on several deformation parameters. These pockets describe the generic opening and closing behaviour of spectral gaps of the Schrödinger operator in dependence of the parameter b. This result uses a refined averaging technique. Also consequences are given for the behaviour of the Lyapunov exponent and rotation number in dependence of a for fixed b.     

On the Spectrum of the Schrödinger Operator with Periodic Surface Potential

We consider a discrete Schrödinger operator H=–+V acting in l ²( ^d ), with periodic potential V supported by the subspace surface {0}× ^d ₂. We prove that the spectrum of H is purely absolutely continuous, and that surface waves oscillate in the longitudinal directions to the surface. We also find an explicit formula for the generalized spectral shift function introduced by the author in Helv. Phys. Acta. 72 (1999), 93–122.     

On the Negative Spectrum of the Two-Dimensional Schrödinger Operator with Radial Potential

For a two-dimensional Schrödinger operator H _{α V}  = ?Δ ?αV with the radial potential V(x) = F(|x|), F(r) ≥ 0, we study the behavior of the number N _?(H _{α V} ) of its negative eigenvalues, as the coupling parameter α tends to infinity. We obtain the necessary and sufficient conditions for the semi-classical growth N _?(H _{α V} ) = O(α) and for the validity of the Weyl asymptotic law.     

Hölder Regularity of Integrated Density of States for the Almost Mathieu Operator in a Perturbative Regime

Consider the Almost Mathieu operator H = cos 2(k +)+ on the lattice. It is shown that for large , the integrated density of states is Hölder continuous of exponent < . This result gives a precise version in the perturbative regime of recent work by M. Goldstein and W. Schlag on Hölder regularity of the integrated density of states for 1D quasi-periodic lattice Schrödinger operators, assuming positivity of the Lyapunov exponent (and proven by different means). Our approach provides also a new way to control Green's functions, in the spirit of the author's work in KAM theory. It is by no means restricted to the cosine-potential and extends to band operators.     

On the stability of the shear–free condition

The evolution equation for the shear is reobtained for a spherically symmetric anisotropic, viscous dissipative fluid distribution, which allows us to investigate conditions for the stability of the shear–free condition. The specific case of geodesic fluids is considered in detail, showing that the shear–free condition, in this particular case, may be unstable, the departure from the shear–free condition being controlled by the expansion scalar and a single scalar function defined in terms of the anisotropy of the pressure, the shear viscosity and the Weyl tensor or, alternatively, in terms of the anisotropy of the pressure, the dissipative variables and the energy density inhomogeneity.     

On the Hausdorff Dimension of the Spectrum of the Thue–Morse Hamiltonian

We show that the Hausdorff dimension of the spectrum of the Thue–Morse Hamiltonian has a common positive lower bound for all coupling.     

On the algebraic types of the Bel–Robinson tensor

The Bel–Robinson tensor is analyzed as a linear map on the space of the traceless symmetric tensors. This study leads to an algebraic classification that refines the usual Petrov–Bel classification of the Weyl tensor. The new classes correspond to degenerate type I space-times which have already been introduced in literature from another point of view. The Petrov–Bel types and the additional ones are intrinsically characterized in terms of the sole Bel–Robinson tensor, and an algorithm is proposed that enables the different classes to be distinguished. Results are presented that solve the problem of obtaining the Weyl tensor from the Bel–Robinson tensor in regular cases.     

On the Non-Integrability of the Stark–Zeeman Hamiltonian System

In this paper we present a proof of the non-integrability in the Liouvillian sense of the Stark–Zeeman Hamiltonian. In particular, we generalize the result of Kummer and Saenz about the non-integrability of the pure Zeeman Hamiltonian. The proof we give is an application of the theorem of Morales and Ramis (1998) about non-integrability, based on differential Galois theory. Received: 30 March 1999 / Accepted: 16 May 1999     

On the Diffuse Structure of the Toluene–Water Interface

The electric density profile along the normal to the phase interface between aromatic hydrocarbon toluene and water has been studied by X-ray reflectometry using synchrotron radiation. According to the experimental data, the width of the interface under normal conditions is (3.9 ± 0.1) Å. This value is much larger than a theoretical value of (5.7 ± 0.2) Å predicted by the theory of capillary waves with an interphase tension of (36.0 ± 0.1) mN/m. The observed broadening of the interface is attributed to its own diffuse near-surface structure with a width no less than Å, which is about the value previously discussed for (high-molecular-weight saturated hydrocarbon–water) and (1,2-dichloroethane–water) interfaces.     

On the Eigenstates of the Elliptic Calogero–Moser Model

It is known that the trigonometric Calogero–Sutherland model is obtained by the trigonometric limit (–1) of the elliptic Calogero–Moser model, where (1, ) is a basic period of the elliptic function. We show that for all square-integrable eigenstates and eigenvalues of the Hamiltonian of the Calogero–Sutherland model, if exp(2–1) is small enough then there exist square-integrable eigenstates and eigenvalues of the Hamiltonian of the elliptic Calogero–Moser model which converge to the ones of the Calogero–Sutherland model for the 2-particle and the coupling constant l is positive integer cases and the 3-particle and l=1 case. In other words, we justify the regular perturbation with respect to the parameter exp(2–1). With some assumptions, we show analogous results for N-particle and l is positive integer cases.     

On an Essential Spectrum of the Random p-Adic Schrödinger-type Operator in the Anderson Model

The random p-adic Schrödinger-type operators are considered. The p-adic analogue of the Anderson model is defined for these operators and the spectral properties of this model are investigated.     

On the Effect of Stochastic Fluctuations in the Dynamics of the Lifshitz–Slyozov–Wagner Model

In this paper the dynamics of a system of spherical particles that fill a small volume fraction of the space and that evolves in a concentration field is discussed. Corrections to the Lifshitz–Slyozov–Wagner (LSW) model that take into account the stochastic character of the problem are computed. It is proved, under suitable smallness assumptions for the volume fraction filled by the particles, that the effect of these corrections does not modify much the dynamics of the self-similar solutions of the LSW system of equations.     

Explosive Development of the Kelvin–Helmholtz Quantum Instability on the He-II Free Surface

Journal of Experimental and Theoretical Physics - We analyze nonlinear dynamics of the Kelvin–Helmholtz quantum instability of the He-II free surface, which evolves during counterpropagation...     

On the Orthocomplementation of State–Property–Systems of Contextual Systems

We adopt an operational approach to quantum mechanics in which a physical system is defined by the mathematical structure of its set of states and properties. We present a model in which the maximal change of state of the system due to interaction with the measurement context is controlled by a parameter which corresponds with the number N of possible outcomes in an experiment. In the case N=2 the system reduces to a model for the spin measurements on a quantum spin-1/2 particle. In the limit N→∞ the system is classical, i.e. the experiments are deterministic and its set of properties is a Boolean lattice. For intermediate situations the change of state due to measurement is neither ‘maximal’ (i.e. quantum) nor ‘zero’ (i.e. classical). We show that two of the axioms used in Piron’s representation theorem for quantum mechanics are violated, namely the covering law and weak modularity. Next, we discuss a modified version of the model for which it is even impossible to define an orthocomplementation on the set of properties. Another interesting feature for the intermediate situations of this model is that the probability of a state transition in general not only depends on the two states involved, but also on the measurement context which induces the state transition.     

On the structural properties of B–C–N nanotubes

We apply a first-principles method, based on the density functional theory, to calculate the structural stability of B–C–N armchair nanotubes, comparing such results with the ones obtained for zigzag configuration. Analysis of the corresponding strain energies confirm that the stability of BC₂N nanotubes is independent of their chirality and demonstrate that such nanostructures have lower strain than BCN and carbon nanotubes. The results show that the formation energy decreases with the tube diameter and indicate that the most stable nanotubes have the maximum number of B–N and C–C bonds. Therefore, from the experimental point of view, larger diameter BC₂N model-I nanotubes should be more probable to be synthesized.