Variations on a Theme of Jost and Pais

We explore the extent to which a variant of a celebrated formula due to Jost and Pais, which reduces the Fredholm perturbation determinant associated with the Schr\"odinger operator on a half-line to a simple Wronski determinant of appropriate distributional solutions of the underlying Schr\"odinger equation, generalizes to higher dimensions. In this multi-dimensional extension the half-line is replaced by an open set \Omega\subset\bbR^n, n\in\bbN, $n\geq 2$, where $\Omega$ has a compact, nonempty boundary $\partial\Omega$ satisfying certain regularity conditions. Our variant involves ratios of perturbation determinants corresponding to Dirichlet and Neumann boundary conditions on $\partial\Omega$ and invokes the corresponding Dirichlet-to-Neumann map. As a result, we succeed in reducing a certain ratio of modified Fredholm perturbation determinants associated with operators in $L^2(\Omega; d^n x)$, n\in\bbN, to modified Fredholm determinants associated with operators in $L^2(\partial\Omega; d^{n-1}\sigma)$, $n\geq 2$. Applications involving the Birman-Schwinger principle and eigenvalue counting functions are discussed.Comment: 40 pages. To appear in J. Funct. Ana

Playing with Duality: An Overview of Recent Primal-Dual Approaches for Solving Large-Scale Optimization Problems

Optimization methods are at the core of many problems in signal/image processing, computer vision, and machine learning. For a long time, it has been recognized that looking at the dual of an optimization problem may drastically simplify its solution. Deriving efficient strategies which jointly brings into play the primal and the dual problems is however a more recent idea which has generated many important new contributions in the last years. These novel developments are grounded on recent advances in convex analysis, discrete optimization, parallel processing, and non-smooth optimization with emphasis on sparsity issues. In this paper, we aim at presenting the principles of primal-dual approaches, while giving an overview of numerical methods which have been proposed in different contexts. We show the benefits which can be drawn from primal-dual algorithms both for solving large-scale convex optimization problems and discrete ones, and we provide various application examples to illustrate their usefulness