Inverse Spectral Theory for Sturm-Liouville Operators with Distributional Potentials

We discuss inverse spectral theory for singular differential operators on arbitrary intervals $(a,b) \subseteq \mathbb{R}$ associated with rather general differential expressions of the type $$\tau f = \frac{1}{r} \left(- \big(p[f' + s f]\big)' + s p[f' + s f] + qf\right),$$ where the coefficients $p$, $q$, $r$, $s$ are Lebesgue measurable on $(a,b)$ with $p^{-1}$, $q$, $r$, $s \in L^1_{\text{loc}}((a,b); dx)$ and real-valued with $p\not=0$ and $r>0$ a.e.\ on $(a,b)$. In particular, we explicitly permit certain distributional potential coefficients. The inverse spectral theory results derived in this paper include those implied by the spectral measure, by two-spectra and three-spectra, as well as local Borg-Marchenko-type inverse spectral results. The special cases of Schr\"odinger operators with distributional potentials and Sturm--Liouville operators in impedance form are isolated, in particular.Comment: 29 page

Supersymmetry and Schr\"odinger-type operators with distributional matrix-valued potentials

Building on work on Miura's transformation by Kappeler, Perry, Shubin, and Topalov, we develop a detailed spectral theoretic treatment of Schr\"odinger operators with matrix-valued potentials, with special emphasis on distributional potential coefficients. Our principal method relies on a supersymmetric (factorization) formalism underlying Miura's transformation, which intimately connects the triple of operators $(D, H_1, H_2)$ of the form [D= (0 & A^*, A & 0) \text{in} L^2(\mathbb{R})^{2m} \text{and} H_1 = A^* A, H_2 = A A^* \text{in} L^2(\mathbb{R})^m.] Here $A= I_m (d/dx) + \phi$ in $L^2(\mathbb{R})^m$, with a matrix-valued coefficient $\phi = \phi^* \in L^1_{\text{loc}}(\mathbb{R})^{m \times m}$, $m \in \mathbb{N}$, thus explicitly permitting distributional potential coefficients $V_j$ in $H_j$, $j=1,2$, where [H_j = - I_m \frac{d^2}{dx^2} + V_j(x), \quad V_j(x) = \phi(x)^2 + (-1)^{j} \phi'(x), j=1,2.] Upon developing Weyl--Titchmarsh theory for these generalized Schr\"odinger operators $H_j$, with (possibly, distributional) matrix-valued potentials $V_j$, we provide some spectral theoretic applications, including a derivation of the corresponding spectral representations for $H_j$, $j=1,2$. Finally, we derive a local Borg--Marchenko uniqueness theorem for $H_j$, $j=1,2$, by employing the underlying supersymmetric structure and reducing it to the known local Borg--Marchenko uniqueness theorem for $D$.Comment: 36 page

Ariadne: An interface to support collaborative database browsing:Technical Report CSEG/3/1995

This paper outlines issues in the learning of information searching skills. We report on our observations of the learning of browsing skills and the subsequent iterative development and testing of the Ariadne system – intended to investigate and support the collaborative learning of search skills. A key part of this support is a mechanism for recording an interaction history and providing students with a visualisation of that history that they can reflect and comment upon

Inverse Spectral Problems for Schr\"odinger-Type Operators with Distributional Matrix-Valued Potentials

The principal purpose of this note is to provide a reconstruction procedure for distributional matrix-valued potential coefficients of Schr\"odinger-type operators on a half-line from the underlying Weyl-Titchmarsh function.Comment: 14 pages. arXiv admin note: text overlap with arXiv:1206.496

Normativity and epistemic intuitions

Journal ArticleIn this paper we propose to argue for two claims. The first is that a sizable group of epistemological projects -- a group which includes much of what has been done in epistemology in the analytic tradition -- would be seriously undermined if one or more of a cluster of empirical hypotheses about epistemic intuitions turns out to be true. The basis for this claim will be set out in section 2. The second claim is that while the jury is still out, there is now a substantial body of evidence suggesting that some of those empirical hypotheses are true. Much of this evidence derives from an ongoing series of experimental studies of epistemic intuitions that we have been conducting. A preliminary report on these studies will be presented in section 3. In light of these studies, we think it is incumbent on those who pursue the epistemological projects in question to either explain why the truth of the hypotheses does not undermine their projects, or to say why, in light of the evidence we will present, they nonetheless assume that the hypotheses are false. In section 4, which is devoted to Objections and Replies, we'll consider some of the ways in which defenders of the projects we are criticizing might reply to our challenge. Our goal is not to offer a conclusive argument demonstrating that the epistemological projects we will be criticizing are untenable. Rather, our aim is to shift the burden of argument. For far too long, epistemologists who rely heavily on epistemic intuitions have proceeded as though they could simply ignore the empirical hypotheses we will set out. We will be well satisfied if we succeed in making a plausible case for the claim that this approach is no longer acceptable

Aca-Media Podcast Episode 70: Jordan Sjol on Medium Specificity

If you’re feeling sluggish from the holiday season, press play on this rich conversation between Jonathan Nichols-Pethick and Jordan Sjol to get your brain sparked and ready for a new year of smart conversations about media. The two DePauw colleagues talk about Sjol’s JCMS article, “A Diachronic, Scale-Flexible, Relational, Perspectival Operation: In Defense of (Always-Reforming) Medium Specificity” (don’t worry, they break it down word-by-word), as well as the recent feature film that Sjol co-wrote, How to Blow Up a Pipeline. Then Chris and Michael chat about how to name a department and how not to title a podcast

Weyl-Titchmarsh Theory for Sturm-Liouville Operators with Distributional Potentials

We systematically develop Weyl-Titchmarsh theory for singular differential operators on arbitrary intervals $(a,b) \subseteq \mathbb{R}$ associated with rather general differential expressions of the type \[ \tau f = \frac{1}{r} (- \big(p[f' + s f]\big)' + s p[f' + s f] + qf),] where the coefficients $p$, $q$, $r$, $s$ are real-valued and Lebesgue measurable on $(a,b)$, with $p\neq 0$, $r>0$ a.e.\ on $(a,b)$, and $p^{-1}$, $q$, $r$, $s \in L^1_{\text{loc}}((a,b); dx)$, and $f$ is supposed to satisfy [f \in AC_{\text{loc}}((a,b)), \; p[f' + s f] \in AC_{\text{loc}}((a,b)).] In particular, this setup implies that $\tau$ permits a distributional potential coefficient, including potentials in $H^{-1}_{\text{loc}}((a,b))$. We study maximal and minimal Sturm-Liouville operators, all self-adjoint restrictions of the maximal operator $T_{\text{max}}$, or equivalently, all self-adjoint extensions of the minimal operator $T_{\text{min}}$, all self-adjoint boundary conditions (separated and coupled ones), and describe the resolvent of any self-adjoint extension of $T_{\text{min}}$. In addition, we characterize the principal object of this paper, the singular Weyl-Titchmarsh-Kodaira $m$-function corresponding to any self-adjoint extension with separated boundary conditions and derive the corresponding spectral transformation, including a characterization of spectral multiplicities and minimal supports of standard subsets of the spectrum. We also deal with principal solutions and characterize the Friedrichs extension of $T_{\text{min}}$. Finally, in the special case where $\tau$ is regular, we characterize the Krein-von Neumann extension of $T_{\text{min}}$ and also characterize all boundary conditions that lead to positivity preserving, equivalently, improving, resolvents (and hence semigroups).Comment: 80 pages. arXiv admin note: text overlap with arXiv:1105.375