We extend to finite temperature the time-dependent effective theory for the Goldstone field ͑the phase of the pair field͒ which is appropriate for a superfluid containing one species of fermions with s-wave interactions, described by the BCS Lagrangian. We show that, when Landau damping is neglected, the effective theory can be written as a local time-dependent nonlinear Schrödinger Lagrangian which preserves the Galilean invariance of the zero-temperature effective theory and is identified with the superfluid component. We then calculate the relevant Landau terms that are nonlocal and that destroy the Galilean invariance. We show that the retarded propagator ͑in momentum space͒ can be well represented by two poles in the lower-half frequency plane, describing damping with a predicted temperature, frequency, and momentum dependence. It is argued that the real parts of the Landau terms can be approximately interpreted as contributing to the normal fluid component.

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We apply perturbative renormalization group theory to the symmetric phase of a dilute interacting Bose gas which is trapped in a three-dimensional harmonic potential. Using Wilsonian energy-shell renormalization and the ε-expansion, we derive the flow equations for the system. We relate these equations to the flow for the homogeneous Bose gas. In the thermodynamic limit, we apply our results to study the transition temperature as a function of the scattering length. Our results compare well to previous studies of the problem.

We summarize unusual bound or localized states in quantum mechanics. Our guide through these intriguing phenomena is the classical physics of the upside-down pendulum, taking advantage of the analogy between the corresponding Newton's equation of motion and the time independent Schrödinger equation. We discuss the zero-energy ground state in a three-dimensional, spatially oscillating, potential. Moreover, we focus on the effect of the attractive quantum anti-centrifugal potential that only occurs in a two-dimensional situation.Keywords: Quantum Mechanics; Bound States; Parametric Oscillator; Periodic Potential. 1 THE PENDULUM: A GUIDE TO QUANTUM PHYSICSDetermining the energy levels of a quantum system was a desperate enterprise before the advent of the BohrSommerfeld quantization conditions. Max Planck had emphasized that the energy of the harmonic oscillator is quantized in units of a fundamental energy given by the product of what we now call Dirac's constant and the frequency of the oscillator. However, this rule failed miserably when used in other quantum systems. Paul Ehrenfest's adiabatic principle applied to the pendulum [1] whose length slowly changes as a function of time, made clear that it is not the energy, but the action that is quantized, but why? When we change the length of the pendulum adiabatically, the amplitude of oscillation does not stay constant, neither does the frequency nor the energy. What stays constant is the action, that is, the area in phase space. For the founding fathers of quantum mechanics it must have been a miracle, an amazing fact symbolizing in post-Schrödinger language that the number of nodes in an energy wave function stays constant under adiabatic changes.The classical dynamics of the pendulum serves as an excellent guide for many quantum phenomena. For example, the upside-down pendulum yields insight into the energy wave function of a periodic potential [2]. Here we do not focus on the familiar Bloch states which appear when the potential enjoys a strict periodicity over the whole space. Our states occur when the modulation of the potential extends only over a finite domain of space. Due to the shape of this potential we refer to it as the accordion potential. Extensions of these one-dimensional considerations to two and three dimensions lead us to the effect of the quantum anti-centrifugal force [3].In the present paper we take seriously the joke 'physics takes mathematics and makes it understandable', therefore we shall suppress all the mathematics and highlight the essential ideas. This approach is justified by the fact that we are not inventing or applying new mathematics, but attempt to draw together phenomena of different fields exposing a common thread. We shall focus on a general point of view but emphasize that the modern tools of cold atoms in a standing electromagnetic wave can demonstrate these unusual bound or localized states predicted in this paper.We follow one 'Leitmotif', that is, a theme common to all phenomena discussed in this article: The classica...