Degenerate elliptic operators in one dimension

Let $H$ be the symmetric second-order differential operator on L_2(\Ri) with domain C_c^\infty(\Ri) and action $H\varphi=-(c \varphi')'$ where c\in W^{1,2}_{\rm loc}(\Ri) is a real function which is strictly positive on \Ri\backslash\{0\} but with $c(0)=0$. We give a complete characterization of the self-adjoint extensions and the submarkovian extensions of $H$. In particular if $\nu=\nu_+\vee\nu_-$ where $\nu_\pm(x)=\pm\int^{\pm 1}_{\pm x} c^{-1}$ then $H$ has a unique self-adjoint extension if and only if $\nu\not\in L_2(0,1)$ and a unique submarkovian extension if and only if $\nu\not\in L_\infty(0,1)$. In both cases the corresponding semigroup leaves $L_2(0,\infty)$ and $L_2(-\infty,0)$ invariant. In addition we prove that for a general non-negative c\in W^{1,\infty}_{\rm loc}(\Ri) the corresponding operator $H$ has a unique submarkovian extension.Comment: 28 page

Degenerate elliptic operators: capacity, flux and separation

Let $S=\{S_t\}_{t\geq0}$ be the semigroup generated on L_2(\Ri^d) by a self-adjoint, second-order, divergence-form, elliptic operator $H$ with Lipschitz continuous coefficients. Further let $\Omega$ be an open subset of \Ri^d with Lipschitz continuous boundary $\partial\Omega$. We prove that $S$ leaves $L_2(\Omega)$ invariant if, and only if, the capacity of the boundary with respect to $H$ is zero or if, and only if, the energy flux across the boundary is zero. The global result is based on an analogous local result.Comment: 18 page

Markov uniqueness of degenerate elliptic operators

Let $\Omega$ be an open subset of \Ri^d and $H_\Omega=-\sum^d_{i,j=1}\partial_i c_{ij} \partial_j$ a second-order partial differential operator on $L_2(\Omega)$ with domain $C_c^\infty(\Omega)$ where the coefficients $c_{ij}\in W^{1,\infty}(\Omega)$ are real symmetric and $C=(c_{ij})$ is a strictly positive-definite matrix over $\Omega$. In particular, $H_\Omega$ is locally strongly elliptic. We analyze the submarkovian extensions of $H_\Omega$, i.e. the self-adjoint extensions which generate submarkovian semigroups. Our main result establishes that $H_\Omega$ is Markov unique, i.e. it has a unique submarkovian extension, if and only if \capp_\Omega(\partial\Omega)=0 where \capp_\Omega(\partial\Omega) is the capacity of the boundary of $\Omega$ measured with respect to $H_\Omega$. The second main result establishes that Markov uniqueness of $H_\Omega$ is equivalent to the semigroup generated by the Friedrichs extension of $H_\Omega$ being conservative.Comment: 24 page

Hot Stars With Cool Companions

Young intermediate-mass stars have become high-priority targets for direct-imaging planet searches following the recent discoveries of planets orbiting e.g. HR 8799 and Beta Pictoris. Close stellar companions to these stars can affect the formation and orbital evolution of any planets, and so a census of the multiplicity properties of nearby intermediate mass stars is needed. Additionally, the multiplicity can help constrain the important binary star formation physics. We report initial results from a spectroscopic survey of 400 nearby A- and B-type stars. We search for companions by cross-correlating high resolution and high signal-to-noise ratio echelle spectra of the targets stars against model spectra for F- to M-type stars. We have so far found 18 new candidate companions, and have detected the spectral lines of the secondary in 4 known spectroscopic binary systems. We present the distribution of mass-ratios for close companions, and find that it differs from the distribution for wide ($a < 100$ AU) intermediate-mass binaries, which may indicate a different formation mechanism for the two populations.Comment: Submitted as part of the 18th Cambridge Workshop on Cool Stars, Stellar Systems, and the Sun Proceedings of Lowell Observatory (9-13 June 2014

Correcting For Telluric Absorption: Methods, Case Studies, And Release Of The TelFit Code

Ground-based astronomical spectra are contaminated by the Earth's atmosphere to varying degrees in all spectral regions. We present a Python code that can accurately fit a model to the telluric absorption spectrum present in astronomical data, with residuals of similar to 3%-5% of the continuum for moderately strong lines. We demonstrate the quality of the correction by fitting the telluric spectrum in a nearly featureless A0V star, HIP 20264, as well as to a series of dwarf M star spectra near the 819 nm sodium doublet. We directly compare the results to an empirical telluric correction of HIP 20264 and find that our model-fitting procedure is at least as good and sometimes more accurate. The telluric correction code, which we make freely available to the astronomical community, can be used as a replacement for telluric standard star observations for many purposes.UT Austin Hutchinson fellowshipUniversity of TexasAstronom

The Close Companion Mass-Ratio Distribution of Intermediate-Mass Stars

Binary stars and higher-order multiple systems are an ubiquitous outcome of star formation, especially as the system mass increases. The companion mass-ratio distribution is a unique probe into the conditions of the collapsing cloud core and circumstellar disk(s) of the binary fragments. Inside $a \sim 1000$ AU the disks from the two forming stars can interact, and additionally companions can form directly through disk fragmentation. We should therefore expect the mass-ratio distribution of close companions ($a \lesssim 100$ AU) to differ from that of wide companions. This prediction is difficult to test using traditional methods, especially with intermediate-mass primary stars, for a variety of observational reasons. We present the results of a survey searching for companions to A- and B-type stars using the direct spectral detection method, which is sensitive to late-type companions within $\sim 1"$ of the primary and which has no inner working angle. We estimate the temperatures and surface gravity of most of the 341 sample stars, and derive their masses and ages. We additionally estimate the temperatures and masses of the 64 companions we find, 23 of which are new detections. We find that the mass-ratio distribution for our sample has a maximum near $q \sim 0.3$. Our mass-ratio distribution has a very different form than in previous work, where it is usually well-described by a power law, and indicates that close companions to intermediate-mass stars experience significantly different accretion histories or formation mechanisms than wide companions.Comment: Accepted to the Astronomical Journal. 26 pages, 7 figures, 5 tables. The code to replicate the analysis is available at https://github.com/kgullikson88/BinaryInference All of the spectra used in this work, as well as a significant portion of the intermediate data products and MCMC samples are available at: https://zenodo.org/record/48073 and https://zenodo.org/record/4648