Stability spectrum

Abstract: A new theory of autoionization is presented which makes no reference to decay channels and which treats the process as a bound-state problem. The starting point is a postulative definition, describing autoionizing---or resonance---states as time-stability maxima within the set of bound functions. The outcome of the mathematical formulation of the postulate is a variational expression for the mean-square deviation of energy. The resulting Euler expression is a nonlinear pseudoeigenvalue equation whose solutions include autoionizing as well as stationary bound states. A second equation, similar to the first, is derived, whose solutions correspond to most stable bound functions at any given energy. Using the second equation, it is possible to obtain a plot of lifetime versus energy---the time stability spectrum. The theory is applied to some $^{1}S$ autoionizing states of He below the $n=2, 3, \mathrm{and} 4$ ionization thresholds. The present results are within error bounds of experimental measurements. Agreement with Feshbach-type calculations is also good.

Abstract: We initiate a systematic investigation of the abstract elementary classes that have amalgamation, satisfy tameness (a locality property for orbital types), and are stable (in terms of the number of...

Abstract: We present a study of a phase-field model for diffusion-limited growth. A boundary-layer approximation is used to show that for sharp interface, the first approximation to the phase-field model is the free boundary model, which includes surface tension and a linear kinetic term. The velocity of propagation and the stability spectrum are calculated for a steady-state flat interface. In the case where the phase and the field have similar variation lengths, a stable growth regime is found above a critical value of driving force. We discuss the application of phase-field-like models in the description of the ensemble-average pattern

Abstract: We initiate a systematic investigation of the abstract elementary classes that have amalgamation, satisfy tameness (a locality property for orbital types), and are stable (in terms of the number of orbital types) in some cardinal. Assuming the singular cardinal hypothesis (SCH), we prove a full characterization of the (high-enough) stability cardinals, and connect the stability spectrum with the behavior of saturated models. We deduce (in ZFC) that if a class is stable on a tail of cardinals, then it has no long splitting chains (the converse is known). This indicates that there is a clear notion of superstability in this framework. We also present an application to homogeneous model theory: for $D$ a homogeneous diagram in a first-order theory $T$, if $D$ is both stable in $|T|$ and categorical in $|T|$ then $D$ is stable in all $\lambda \ge |T|$.