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

Dobrushin's ergodicity coefficient for Markov operators on cones

We give a characterization of the contraction ratio of bounded linear maps in Banach space with respect to Hopf's oscillation seminorm, which is the infinitesimal distance associated to Hilbert's projective metric, in terms of the extreme points of a certain abstract "simplex". The formula is then applied to abstract Markov operators defined on arbitrary cones, which extend the row stochastic matrices acting on the standard positive cone and the completely positive unital maps acting on the cone of positive semidefinite matrices. When applying our characterization to a stochastic matrix, we recover the formula of Dobrushin's ergodicity coefficient. When applying our result to a completely positive unital map, we therefore obtain a noncommutative version of Dobrushin's ergodicity coefficient, which gives the contraction ratio of the map (representing a quantum channel or a "noncommutative Markov chain") with respect to the diameter of the spectrum. The contraction ratio of the dual operator (Kraus map) with respect to the total variation distance will be shown to be given by the same coefficient. We derive from the noncommutative Dobrushin's ergodicity coefficient an algebraic characterization of the convergence of a noncommutative consensus system or equivalently the ergodicity of a noncommutative Markov chain.Comment: An announcement of some of the present results has appeared in the Proceedings of the ECC 2013 conference (Zurich). Further results can be found in the companion arXiv:1302.522

The operational meaning of min- and max-entropy

We show that the conditional min-entropy Hmin(A|B) of a bipartite state rho_AB is directly related to the maximum achievable overlap with a maximally entangled state if only local actions on the B-part of rho_AB are allowed. In the special case where A is classical, this overlap corresponds to the probability of guessing A given B. In a similar vein, we connect the conditional max-entropy Hmax(A|B) to the maximum fidelity of rho_AB with a product state that is completely mixed on A. In the case where A is classical, this corresponds to the security of A when used as a secret key in the presence of an adversary holding B. Because min- and max-entropies are known to characterize information-processing tasks such as randomness extraction and state merging, our results establish a direct connection between these tasks and basic operational problems. For example, they imply that the (logarithm of the) probability of guessing A given B is a lower bound on the number of uniform secret bits that can be extracted from A relative to an adversary holding B.Comment: 12 pages, v2: no change in content, some typos corrected (including the definition of fidelity in footnote 8), now closer to the published versio