Localization of quantum wave packets

We study the semiclassical propagation of squeezed Gau{\ss}ian states. We do so by considering the propagation theorem introduced by Combescure and Robert \cite{CR97} approximating the evolution generated by the Weyl-quantization of symbols $H$. We examine the particular case when the Hessian $H^{\prime\prime}(X_{t})$ evaluated at the corresponding solution $X_{t}$ of Hamilton's equations of motion is periodic in time. Under this assumption, we show that the width of the wave packet can remain small up to the Ehrenfest time. We also determine conditions for ``classical revivals'' in that case. More generally, we may define recurrences of the initial width. Some of these results include the case of unbounded classical motion. In the classically unstable case we recover an exponential spreading of the wave packet as in \cite{CR97}

Semiclassics, adiabatic decoupling and perturbed periodic structures

This thesis deals with the consequences of periodic structures in quantum mechanics in different semiclassical regimes. The first chapter introduces the Floquet theory. It is applied on the Hill equation. As a special case, the solutions to the Mathieu equation are studied exemplifying the properties of the spectrum of periodic Schrödinger operators. Special attention is brought to the properties of the spectrum and the corresponding Floquet exponent of solutions. We present new results concerning the relation between the eigenvalues and the Floquet exponent. We introduce the Weyl quantization in the second chapter. In particular, we introduce semiclassical propagation results which, up to some known errors, describes the propagation of an initial Gaußian under quantum evolution (Combescure, Robert 1997). This approximation breaks down at what is commonly known as the Ehrenfest time. We use these results to investigate the localization properties of an initial Gaußian. We can show localization of the state up to the Ehrenfest time under the assumption that the flow differential of the corresponding classical motion admits Floquet solutions with purely imaginary Floquet exponents. In this case, we also show the existence of what we call classical revivals. The last part is a study of the spectral properties of perturbed periodic Schrödinger operators. This is done in the limit of weak perturbation epsilon . As the structure of the band spectrum of the unperturbed operator remains unchanged under small enough perturbations, the effective dynamics can be described adiabatically under certain prerequisites. We consider the construction of WKB ansätze (Littlejohn, Flynn, 1991, Emmrich, Weinstein 1996). Our results include Bohr-Sommerfeld quantization conditions modulo O(epsilon²). This is made possible by including geometric terms. The Wannier-Stark system is studied as an example

On the characteristic exponents of Floquet solutions to the Mathieu equation

We study the Floquet solutions of the Mathieu equation. In order to find an explicit relation between the characteristic exponents and their corresponding eigenvalues of the Mathieu operator, we consider the Whittaker-Hill formula. This gives an explicit relation between the eigenvalue and its characteristic exponent. The equation is explicit up to a determinant of an infinite dimensional matrix. We find a third-order linear recursion for which this determinant is exactly the limit. An explicit solution for third-order linear recursions is obtained which enables us to write the determinant explicitly.Sträng Jan-Eric. On the characteristic exponents of Floquet solutions to the Mathieu equation. In: Bulletin de la Classe des sciences, tome 16, n°7-12, 2005. pp. 269-287

Parametric fixed points.

<p>An example of parametric dependency of fixed points is shown. <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0131832#pone.0131832.t001" target="_blank">Table 1</a> shows the BNE.</p

Phenotypes predicted by the BNE.

<p>The phenotype profile used for the mapping is based on the 11 genes present in both the Boolean model and the <i>Xenopus</i> analysis. The figure in the left panel shows the distance curves for the nearest phenotypes and fixed points. The x-axis denotes the values of the parameter <i>exogen</i>_<i>canWnt</i>_<i>I</i> and the phenotypes to which the fixed points were mapped. The y-axis shows the actual distance. The phenotypes are ordered by increasing <i>exogen</i>_<i>canWnt</i>_<i>I</i> expression propensity (right panel). Activated genes are shown in green and deactivated genes are shown in red. The framed box shows the gene expression propensity pattern for the four genes Isl1, Nkx2.5, Tbx1, and Tbx5 that corresponds to the the RT–PCR phenotypes of the FHF and SHF from the <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0131832#pone.0131832.g003" target="_blank">Fig 3</a>. The <i>FHF</i>_<i>BOOL</i> and <i>SHF</i>_<i>BOOL</i> phenotypes correspond to the phenotypes found in the Boolean model [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0131832#pone.0131832.ref016" target="_blank">16</a>]. The SHF1 phenotype is split in three sub-phenotypes <i>PH-1076</i>, <i>PH-564</i>, and <i>PH-692</i> that differ by the gene expression propensity of canWnt, Bmp2, and Fgf8. The expression of the Fgf8 gene was not reported in <i>Xenopus</i>. Its activation pattern is a prediction of the BNE.</p

Genes, proteins and model variables to BN model of cardiac development.

<p>The first column shows the variables used in the BN model. The second column, function, describes the type and location of the expressed protein or the purpose of the variable in the BN.</p

Predicting Variabilities in Cardiac Gene Expression with a Boolean Network Incorporating Uncertainty

<div><p>Gene interactions in cells can be represented by gene regulatory networks. A Boolean network models gene interactions according to rules where gene expression is represented by binary values (<i>on / off</i> or {1, 0}). In reality, however, the gene’s state can have multiple values due to biological properties. Furthermore, the noisy nature of the experimental design results in uncertainty about a state of the gene. Here we present a new Boolean network paradigm to allow intermediate values on the interval [0, 1]. As in the Boolean network, fixed points or attractors of such a model correspond to biological phenotypes or states. We use our new extension of the Boolean network paradigm to model gene expression in first and second heart field lineages which are cardiac progenitor cell populations involved in early vertebrate heart development. By this we are able to predict additional biological phenotypes that the Boolean model alone is not able to identify without utilizing additional biological knowledge. The additional phenotypes predicted by the model were confirmed by published biological experiments. Furthermore, the new method predicts gene expression propensities for modelled but yet to be analyzed genes.</p></div

Schematic drawing of cardiac tissue in Xenopus laevis at stage 24—Expression of genes in heart fields (left) and RT–PCR analysis of selected genes (right).

<p>Panels adapted from Gessert and Kühl [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0131832#pone.0131832.ref019" target="_blank">19</a>]. The left panel shows the genes expressed in different domains of the first heart field (FHF) and second heart field (SHF). The SHF is shown at the top and the FHF is shown at the bottom. Common genes expressed in all regions of the FHF and SHF, respectively, are shown on the left. Genes expressed in particular domains are shown on the right. Colors indicate different domains and corresponding expressed genes. The right figure shows the results of single cell RT–PCR analysis of gene expression for the four genes Nkx2.5, Isl1, Tbx1, and Tbx5. Values (0 and 1) and colors red/green represent inactive or active genes. The panel shows the gene expression of different single cell samples (numbered and named at the bottom). FHF and SHF are distinguished by the expression of the Isl1.</p