Mathematical Colloquium - Lyapunov operators and irreversible representations of quantum dynamics with application to scattering resonances . Speaker: Dr. Yossi Strauss

00:00 09-06-2010
<p class="MsoNormal" style="MARGIN: 0cm 0cm 0pt; DIRECTION: ltr; unicode-bidi: embed; TEXT-ALIGN: left"><span style="FONT-SIZE: 14px; COLOR: black; FONT-FAMILY: 'Arial','sans-serif'"><font face="Arial, Helvetica, sans-serif"><font size="2"><span style="FONT-FAMILY: 'Arial','sans-serif'"></span></font></font></span> </p> <span style="COLOR: black"> <p align="left"><font face="Arial, Helvetica, sans-serif" size="2"><strong>Abstract:</strong> In non-relativistic quantum mechanics time enters as a parameter </font><font face="Arial, Helvetica, sans-serif" size="2">in the Schr¨odinger equation. However, there are various situations </font><font face="Arial, Helvetica, sans-serif" size="2">where the need arises to view time as a dynamical variable. In this talk we </font><font face="Arial, Helvetica, sans-serif" size="2">consider the dynamical role of time through the construction of a Lyapunov </font><font face="Arial, Helvetica, sans-serif" size="2">variable - i.e., a self-adjoint quantum observable whose expectation value </font><font face="Arial, Helvetica, sans-serif" size="2">varies monotonically as time increases. It can be shown that a certain class </font><font face="Arial, Helvetica, sans-serif" size="2">of models admits the existence of a lyapunov variable and that the existence </font><font face="Arial, Helvetica, sans-serif" size="2">of such an operator leads to the construction of an irreversible representation </font><font face="Arial, Helvetica, sans-serif" size="2">of the quantum dynamics in which evolution is given in terms of a dynamical </font><font face="Arial, Helvetica, sans-serif" size="2">semigroup. As an application of the formalism the irreversible representation </font><font face="Arial, Helvetica, sans-serif" size="2">is combined with a time dependent framework for the description </font><font face="Arial, Helvetica, sans-serif" size="2">of scattering resonances known as the semigroup decomposition formalism. </font><font face="Arial, Helvetica, sans-serif" size="2">This combination results in a definition of physically meaningful exact resonance </font><font face="Arial, Helvetica, sans-serif" size="2">states of a scattering system. In addition, a novel representation of </font><font face="Arial, Helvetica, sans-serif" size="2">a scattering process is obtained, in which the Heisenberg evolution A(t) of </font><font face="Arial, Helvetica, sans-serif" size="2">each quantum observable A is decomposed into a sum A(t) = A−(t)+A+(t) </font><font face="Arial, Helvetica, sans-serif" size="2">where A−(t) vanishes in the far future and A+(t) vanishes in the far past.<br /></font><font face="Arial, Helvetica, sans-serif" size="2">Thus, a scattering process is described in terms of a transition from A−(t) </font><font face="Arial, Helvetica, sans-serif" size="2">to A+(t).</font></p> </span><SPAN STYLE="display: none"> <P>Acetylenide minutes, pointedly? 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