A Pedestrian Introduction to Gamow Vectors

The Gamow vector description of resonances is compared with the S-matrix and the Green function descriptions using the example of the square barrier potential. By imposing different boundary conditions on the time independent Schrodinger equation, we obtain either eigenvectors corresponding to real eigenvalues and the physical spectrum or eigenvectors corresponding to complex eigenvalues (Gamow vectors) and the resonance spectrum. We show that the poles of the S matrix are the same as the poles of the Green function and are the complex eigenvalues of the Schrodinger equation subject to a purely outgoing boundary condition. The intrinsic time asymmetry of the purely outgoing boundary condition is discussed. Finally, we show that the probability of detecting the decay within a shell around the origin of the decaying state follows an exponential law if the Gamow vector (resonance) contribution to this probability is the only contribution that is taken into account.Comment: 25 RevTex pages, 3 figure

The Friedrichs Model and its use in resonance phenomena

We present here a relation of different types of Friedrichs models and their use in the description and comprehension of resonance phenomena. We first discuss the basic Friedrichs model and obtain its resonance in the case that this is simple or doubly degenerated. Next, we discuss the model with $N$ levels and show how the probability amplitude has an oscillatory behavior. Two generalizations of the Friedrichs model are suitable to introduce resonance behavior in quantum field theory. We also discuss a discrete version of the Friedrichs model and also a resonant interaction between two systems both with continuous spectrum. In an Appendix, we review the mathematics of rigged Hilbert spaces.Comment: 105 page

Guidelines for the use and interpretation of assays for monitoring autophagy (4th edition)1.

In 2008, we published the first set of guidelines for standardizing research in autophagy. Since then, this topic has received increasing attention, and many scientists have entered the field. Our knowledge base and relevant new technologies have also been expanding. Thus, it is important to formulate on a regular basis updated guidelines for monitoring autophagy in different organisms. Despite numerous reviews, there continues to be confusion regarding acceptable methods to evaluate autophagy, especially in multicellular eukaryotes. Here, we present a set of guidelines for investigators to select and interpret methods to examine autophagy and related processes, and for reviewers to provide realistic and reasonable critiques of reports that are focused on these processes. These guidelines are not meant to be a dogmatic set of rules, because the appropriateness of any assay largely depends on the question being asked and the system being used. Moreover, no individual assay is perfect for every situation, calling for the use of multiple techniques to properly monitor autophagy in each experimental setting. Finally, several core components of the autophagy machinery have been implicated in distinct autophagic processes (canonical and noncanonical autophagy), implying that genetic approaches to block autophagy should rely on targeting two or more autophagy-related genes that ideally participate in distinct steps of the pathway. Along similar lines, because multiple proteins involved in autophagy also regulate other cellular pathways including apoptosis, not all of them can be used as a specific marker for bona fide autophagic responses. Here, we critically discuss current methods of assessing autophagy and the information they can, or cannot, provide. Our ultimate goal is to encourage intellectual and technical innovation in the field

Time Symmetry and Quantum Dephasing

We first stress that the time symmetry in quantum mechanics manifests itself in the analytical properties of the Fourier transform of the evolution operator in the complex E-plane (E being the variable conjugate to time), in such a way that all singularities are distributed only on the real axis in the first Riemannian sheet, and new poles appear, in the N infinite limit (N standing for the number of degrees of freedom of detector or instrument concerned), on the second Riemannian sheet in a symmetric way with respect to the real axis. We then examine the symmetry-breaking phenomenon, such as decay or dissipation, by setting up the initial value problem: The temporal evolution of the transition probability is divided into three parts, the first being Gaussian for very short times, the second exponential for intermediate times and the third of the power type for very long times. We know that the Gaussian decay is directly connected to the so-called quantum Zeno effect, the exponential decay corresponds to a sort of dephasing process, because the time rate of the total transition probability becomes a sum of time rates of partial probabilities, and both the Gaussian-like and power-like decay will disappear, leaving only the exponential one, in the van Hove limit. The dominance of the exponential decay is equivalent to the appearance of a master equation, which tells us that we have no phase-correlation but decoherence or dephasing. All temporal behaviors of quantum-mechanical transition probability and related physics are reflected in the analytical property of the Fourier transform of the evolution operator

Temporal behavior of quantum systems and quantum zeno effect

The temporal behavior of an unstable system is analyzed quantum mechanically and compared to the exponential decay law. The general mathematical features of the quantum evolution, yielding a quadratic region at short times and a power law at long times, are briefly reviewed. The consequences of the short-time quadratic evolution are curious: By performing many measurements in rapid succession on a quantum system, in order to check whether it is still in its initial state, one can hinder its evolution. This phenomenon is known as the quantum Zeno effect and is discussed in detail. In this respect, a specific example involving neutron spin is considered. Finally, we focus our attention on some interesting features of the evolution law