Nonlinear Impurity in a Lattice: Dispersion Effects

We examine the bound state(s) associated with a single cubic nonlinear impurity, in a one-dimensional tight-binding lattice, where hopping to first--and--second nearest neighbors is allowed. The model is solved in closed form {\em v\`{\i}a} the use of the appropriate lattice Green function and a phase diagram is obtained showing the number of bound states as a function of nonlinearity strength and the ratio of second to first nearest--neighbor hopping parameters. Surprisingly, a finite amount of hopping to second nearest neighbors helps the formation of a bound state at smaller (even vanishingly small) nonlinearity values. As a consequence, the selftrapping transition can also be tuned to occur at relatively small nonlinearity strength, by this increase in the lattice dispersion.Comment: 24 pages, 10 figure

Ideal Gas in a Finite Container

The thermodynamics of an ideal gas enclosed in a box of volume a1 x a2 x a3 at temperature T is considered. The canonical partition function of the system is expressed in terms of complete elliptic integrals of the first kind, whose argument obeys a transcendental equation. For high and low temperatures we derive explicitly the main finite-volume corrections to the standard thermodynamic quantities.Comment: 13 pages total (Latex source), including one table and one ps figur

New DNLS Equations for Anharmonic Vibrational Impurities

We examine some new DNLS-like equations that arise when considering strongly-coupled electron-vibration systems, where the local oscillator potential is anharmonic. In particular, we focus on a single, rather general nonlinear vibrational impurity and determine its bound state(s) and its dynamical selftrapping properties.Comment: 16 pages, 5 figure

The attractive nonlinear delta-function potential

We solve the continuous one-dimensional Schr\"{o}dinger equation for the case of an inverted {\em nonlinear} delta-function potential located at the origin, obtaining the bound state in closed form as a function of the nonlinear exponent. The bound state probability profile decays exponentially away from the origin, with a profile width that increases monotonically with the nonlinear exponent, becoming an almost completely extended state when this approaches two. At an exponent value of two, the bound state suffers a discontinuous change to a delta-like profile. Further increase of the exponent increases again the width of the probability profile, although the bound state is proven to be stable only for exponents below two. The transmission of plane waves across the nonlinear delta potential increases monotonically with the nonlinearity exponent and is insensitive to the sign of its opacity.Comment: submitted to Am. J. of Phys., sixteen pages, three figure

Exponential versus linear amplitude decay in damped oscillators

We comment of the widespread belief among some undergraduate students that the amplitude of any harmonic oscillator in the presence of any type of friction, decays exponentially in time. To dispel that notion, we compare the amplitude decay for a harmonic oscillator in the presence of (i) viscous friction and (ii) dry friction. It is shown that, in the first case, the amplitude decays exponentially with time while in the second case, it decays linearly with time.Comment: 3 pages, 1 figure, accepted in Phys. Teac