Andrzej Radosz

Some properties of double-Morse potentials

In the framework of various exact and approximate methods, the properties of the quasi-exactly solvable double-Morse potential are discussed. The relation between the hidden algebraic structure of the model and its geometrical properties is examined. The potential is used as a testing model for a semiclassical approximation which proves to work well both in the symmetric and asymmetric cases. The relation between semiclassical energy-level splitting and perturbation theory is underlined, leading to conclusions on the interpretation of the semiclassical results.

A CERTAIN DOUBLE-WELL POTENTIAL RELATED TO SU(2) SYMMETRY

In the present paper we analyse a class of partially exactly solvable one-dimensional double-well potentials. We show how the underlying SU(2) dynamical symmetry makes it possible analytically to find several energy levels.

Dynamics of a hydrogen-bonded linear chain with a new type of one-particle potential

The linear hydrogen-bonded chain with the new type of one-particle potential is considered in the continuum limit. This double Morse potential might be a single- or double-well, symmetric or asymmetric one, depending on the parameters A and B. The solutions of the equations of motion in the form of oscillatory and solitary waves are derived and discussed. It is shown that in the asymmetric case there are no stable solitary waves.

The Doppler shift in a Schwarzschild spacetime

We demonstrate that in the case of Schwarzschild spacetime the Doppler shift is partially factorized into terms representing relativistic, kinematical and the gravitational contributions. The condition for the complete factorization is derived. Application of these results to the simplest cases and possible implementation in the framework of GPS is briefly discussed.

A pseudo-goldstone mode in the exactly solvable model of phase transitions

Abstract It is shown that a phase transition within the model proposed by Schneider, Stoll and Beck is associated with a spontaneously broken discrete symmetry; so the gapless mode in the ordered phase is not the Golstone-type excitation. On the basis of the results for T =0, it is suggested that this excitation (called the pseudo-Goldstone mode) exhibits an unusual behaviour at q =0 for all T T c . Such a phenomenon should be common for some class of the exactly solvable models of phase transitions.

Touching ghosts: observing free fall from an infalling frame of reference into a Schwarzschild black hole

The problem of communication between observers in the vicinity of a black hole in a Schwarzschild metric is considered. The classic example of an infalling observer Alice and a static distant mother station (MS) is extended to include a second infalling observer Bob, who follows Alice in falling towards the event horizon. Kruskal coordinates are introduced to analyse this situation, and the pedagogic value of introducing this alternative coordinate system is demonstrated by their ability to provide a graphically based solution to and illustration of this problem. The counterintuitive result that is obtained, which will be of interest to those studying general relativity at an introductory level at final year undergraduate or at graduate level, is that Bob appears to reach a ghostly image of Alice at the event horizon.

Soft mode in the class of exactly soluble models of phase transitions

The temperature dependence of susceptibility of a class of exactly soluble models of phase transitions is discussed. Instead of a Goldstone mode-type behaviour found in earlier treatments it is shown that this function reveals a quite rich structure. Inverse susceptibility vanishes in the ordered phase for d⩽dc=4 and has a singularity at T=0, but is a continuous function of temperature for d > 4 with a classical critical index, γ′=1. Such a characteristic behaviour is probably associated with a non-ergodic behaviour in the former case (d⩽dc) and an ergodic behaviour in the latter case (d >dc); a similar singularity of the susceptibility function (d⩽dc) is found for a spherical model.

Comparative analysis of the isovolume calibration method for non-invasive respiratory monitoring techniques based on area transduction versus circumference transduction using the connected cylinders model

An analytical formalism developed previously to examine the robustness of the isovolume calibration technique for non-invasive respiratory monitoring devices based on measurements of torso circumference (e.g. fibre-optic respiratory plethysmography) is extended here to techniques based on area measurement (e.g. respiratory inductive plethysmography), and the results are compared. The earlier perturbation approach is adopted, and an exact method is also presented. It is demonstrated that the area-based techniques have less dependence on the cylindrical compartmental parameters of radius and height, and are independent of compartmental volume if height variations are negligible, in contrast to circumference-based techniques. It is also demonstrated that both the area- and the circumference-based techniques provide similar inferences of volume when calibrated using the isovolume method under reasonable assumptions for the dimensions of the compartments that constitute a model of the torso.

Doppler effect in Schwarzschild and Kerr geometries

Calculation of the Doppler shift in general relativity involves contributions of gravitational and kinematical origins and for most metrics or trajectories these contributions are coupled. The exact expression for this Doppler shift may simplify for particular symmetries. Here the specific case for a light signal emitted by a distant inertial observer and received by an in-falling observer in a Schwarzschild geometry is discussed. The resulting expression the Doppler shift is composed of simple factors that can be clearly identified with contributions arising from classical kinematical, special relativistic and general relativistic origins. This result turns out to be more general and it holds for a case of an arbitrary radial in-fall in Schwarzschild geometry and for a particular type of in-fall in the case of a Kerr metric.

A useful expression for relativistic energy conservation of a point mass in an isotropic static gravitational field

By using the symmetry and time-independence properties of Schwarzschild spacetime it is demonstrated that an energy conservation law may be expressed in terms of local velocity. From this form three important results may be derived very concisely. This highlights analogies and differences between relativistic and classical approaches to mechanics and serves as an illustration of the power that the use of particular expressions may provide in solving physical problems.