Calculating resonance positions and widths using the Siegert approximation method

Here we present complex resonance states (or Siegert states), that describe the tunneling decay of a trapped quantum particle, from an intuitive point of view which naturally leads to the easily applicable Siegert approximation method that can be used for analytical and numerical calculations of complex resonances of both the linear and nonlinear Schr\"odinger equation. Our approach thus complements other treatments of the subject that mostly focus on methods based on continuation in the complex plane or on semiclassical approximations.Comment: 15 pages, 1 figure, contains MATLAB source code; new version with additional illustration

Barrier transmission for the one-dimensional nonlinear Schr\"odinger equation: resonances and transmission profiles

The stationary nonlinear Schr\"odinger equation (or Gross-Pitaevskii equation) for one-dimensional potential scattering is studied. The nonlinear transmission function shows a distorted profile, which differs from the Lorentzian one found in the linear case. This nonlinear profile function is analyzed and related to Siegert type complex resonances. It is shown, that the characteristic nonlinear profile function can be conveniently described in terms of skeleton functions depending on a few instructive parameters. These skeleton functions also determine the decay behavior of the underlying resonance state. Furthermore we extend the Siegert method for calculating resonances, which provides a convenient recipe for calculating nonlinear resonances. Applications to a double Gaussian barrier and a square well potential illustrate our analysis.Comment: 9 pages, 6 figures, 1 tabl

Heuristic approach to BEC self-trapping in double wells beyond mean-field

We present a technically simple treatment of self-trapping of Bose-Einstein condensates in double well traps based on intuitive semi-classical approximations. Our analysis finally leads to a convenient closed form approximation for the time-averaged population imbalance valid in both the mean-field case and in the case of finite particle numbers for short times.Comment: 11 pages, 4 figure

Barrier transmission for the Nonlinear Schr\"odinger Equation: Surprises of nonlinear transport

In this communication we report on a peculiar property of barrier transmission that systems governed by the nonlinear Schroedinger equation share with the linear one: For unit transmission the potential can be divided at an arbitrary point into two sub-potentials, a left and a right one, which have exactly the same transmission. This is a rare case of an exact property of a nonlinear wave function which will be of interest, e.g., for studies of coherent transport of Bose-Einstein condensates through mesoscopic waveguideComment: 7 pages, 2 figure

Nonlinear resonant tunneling of Bose-Einstein condensates in tilted optical lattices

We study the tunneling decay of a Bose-Einstein condensate out of tilted optical lattices within the mean-field approximation. We introduce a novel method to calculate also excited resonance eigenstates of the Gross-Pitaevskii equation, based on a grid relaxation procedure with complex absorbing potentials. This algorithm works efficiently in a wide range of parameters where established methods fail. It allows us to study the effects of the nonlinearity in detail in the regime of resonant tunneling, where the decay rate is enhanced by resonant coupling to excited unstable states.Comment: Revised and enlarged version, including 1 additional figur

Elektrophysiologische Untersuchungen zur Regulation rekombinanter Kaliumkanäle durch Metabolite des Fettsäure- und Phosphoinositidstoffwechsels

Die Familie der einwärtsgleichrichtenden (Kir) K+ Kanäle ist eine wichtige Klasse von K+ Kanälen, die an der Aufrechterhaltung des Ruhemembranpotentials, der Festlegung des Schwellenwertes für Aktionspotentiale und der K+ Homöostase maßgeblich beteiligt ist. In dieser Arbeit wurde die Regulation von Kir-Kanälen durch langkettige Fettsäure-Coenzym A Ester (LC-CoA) untersucht, deren intrazelluläre Konzentration in bestimmten Stoffwechselsituationen erhöht sein kann (z. B. bei Adipositas) und so möglicherweise die Entstehung eines Diabetes mellitus vom Typ II begünstigt. Unter Anwendung der “patch-clamp“-Technik konnte gezeigt werden, dass LC-CoA potente Inhibitoren aller getesteten Kir-Kanäle (Kir1.1, Kir2.1, Kir3.4, Kir7.1) darstellen und dass die Präsenz einer bestimmten Phosphatgruppe im Coenzym A Molekül entscheidend für die Wirkung an Kir-Kanälen ist. Des Weiteren sind die Länge der Fettsäurekette und die Potenz der Inhibition direkt korreliert. Am Kanalprotein selbst führt das Binden von LC-CoA zu strukturellen Veränderungen des Kanals, es konnten sowohl intra- als auch extrazelluläre Abschnitte als Teil einer Gating-Maschinerie identifiziert werden, die das Öffnen und Schließen des Kanals vermittelt. Die Untersuchungen implizieren eine direkte Verbindung aller Kir-Kanäle mit dem zellulären Fettsäuremetabolismus und dienen dem besseren Verständnis des komplexen Schaltverhaltens von Kir-Kanälen

State-Dependent Network Connectivity Determines Gating in a K+ Channel

YesX-ray crystallography has provided tremendous insight into the different structural states of membrane proteins and, in particular, of ion channels. However, the molecular forces that determine the thermodynamic stability of a particular state are poorly understood. Here we analyze the different X-ray structures of an inwardly rectifying potassium channel (Kir1.1) in relation to functional data we obtained for over 190 mutants in Kir1.1. This mutagenic perturbation analysis uncovered an extensive, state-dependent network of physically interacting residues that stabilizes the pre-open and open states of the channel, but fragments upon channel closure. We demonstrate that this gating network is an important structural determinant of the thermodynamic stability of these different gating states and determines the impact of individual mutations on channel function. These results have important implications for our understanding of not only K+ channel gating but also the more general nature of conformational transitions that occur in other allosteric proteins.Wellcome Trus

Nonlinear Schr\"odinger equation for a PT symmetric delta-functions double well

The time-independent nonlinear Schr\"odinger equation is solved for two attractive delta-function shaped potential wells where an imaginary loss term is added in one well, and a gain term of the same size but with opposite sign in the other. We show that for vanishing nonlinearity the model captures all the features known from studies of PT symmetric optical wave guides, e.g., the coalescence of modes in an exceptional point at a critical value of the loss/gain parameter, and the breaking of PT symmetry beyond. With the nonlinearity present, the equation is a model for a Bose-Einstein condensate with loss and gain in a double well potential. We find that the nonlinear Hamiltonian picks as stationary eigenstates exactly such solutions which render the nonlinear Hamiltonian itself PT symmetric, but observe coalescence and bifurcation scenarios different from those known from linear PT symmetric Hamiltonians.Comment: 16 pages, 9 figures, to be published in Journal of Physics

Resonance solutions of the nonlinear Schr\"odinger equation in an open double-well potential

The resonance states and the decay dynamics of the nonlinear Schr\"odinger (or Gross-Pitaevskii) equation are studied for a simple, however flexible model system, the double delta-shell potential. This model allows analytical solutions and provides insight into the influence of the nonlinearity on the decay dynamics. The bifurcation scenario of the resonance states is discussed, as well as their dynamical stability properties. A discrete approximation using a biorthogonal basis is suggested which allows an accurate description even for only two basis states in terms of a nonlinear, nonhermitian matrix problem.Comment: 21 pages, 14 figure