Steklov Spectral Geometry for Extrinsic Shape Analysis

We propose using the Dirichlet-to-Neumann operator as an extrinsic alternative to the Laplacian for spectral geometry processing and shape analysis. Intrinsic approaches, usually based on the Laplace-Beltrami operator, cannot capture the spatial embedding of a shape up to rigid motion, and many previous extrinsic methods lack theoretical justification. Instead, we consider the Steklov eigenvalue problem, computing the spectrum of the Dirichlet-to-Neumann operator of a surface bounding a volume. A remarkable property of this operator is that it completely encodes volumetric geometry. We use the boundary element method (BEM) to discretize the operator, accelerated by hierarchical numerical schemes and preconditioning; this pipeline allows us to solve eigenvalue and linear problems on large-scale meshes despite the density of the Dirichlet-to-Neumann discretization. We further demonstrate that our operators naturally fit into existing frameworks for geometry processing, making a shift from intrinsic to extrinsic geometry as simple as substituting the Laplace-Beltrami operator with the Dirichlet-to-Neumann operator.Comment: Additional experiments adde

Multiple Aharonov--Bohm eigenvalues: the case of the first eigenvalue on the disk

It is known that the first eigenvalue for Aharonov--Bohm operators with half-integer circulation in the unit disk is double if the potential's pole is located at the origin. We prove that in fact it is simple as the pole $a\neq 0$

Boundary layers and emitted excitations in nonlinear Schrodinger superflow past a disk

The stability and dynamics of nonlinear Schrodinger superflows past a two-dimensional disk are investigated using a specially adapted pseudo-spectral method based on mapped Chebychev polynomials. This efficient numerical method allows the imposition of both Dirichlet and Neumann boundary conditions at the disk border. Small coherence length boundary-layer approximations to stationary solutions are obtained analytically. Newton branch-following is used to compute the complete bifurcation diagram of stationary solutions. The dependence of the critical Mach number on the coherence length is characterized. Above the critical Mach number, at coherence length larger than fifteen times the diameter of the disk, rarefaction pulses are dynamically nucleated, replacing the vortices that are nucleated at small coherence length

Correlation functions of boundary field theory from bulk Green's functions and phases in the boundary theory

In the context of the bulk-boundary correspondence we study the correlation functions arising on a boundary for different types of boundary conditions. The most general condition is the mixed one interpolating between the Neumann and Dirichlet conditions. We obtain the general expressions for the correlators on a boundary in terms of Green's function in the bulk for the Dirichlet, Neumann and mixed boundary conditions and establish the relations between the correlation functions. As an instructive example we explicitly obtain the boundary correlators corresponding to the mixed condition on a plane boundary $R^d$ of a domain in flat space $R^{d+1}$. The phases of the boundary theory with correlators of the Neumann and Dirichlet types are determined. The boundary correlation functions on sphere $S^d$ are calculated for the Dirichlet and Neumann conditions in two important cases: when sphere is a boundary of a domain in flat space $R^{d+1}$ and when it is a boundary at infinity of Anti-De Sitter space $AdS_{d+1}$. For massless in the bulk theory the Neumann correlator on the boundary of AdS space is shown to have universal logarithmic behavior in all AdS spaces. In the massive case it is found to be finite at the coinciding points. We argue that the Neumann correlator may have a dual two-dimensional description. The structure of the correlators obtained, their conformal nature and some recurrent relations are analyzed. We identify the Dirichlet and Neumann phases living on the boundary of AdS space and discuss their evolution when the location of the boundary changes from infinity to the center of the AdS space.Comment: 32 pages, latex, no figure

D-Branes, Tachyons, and String Field Theory

In these notes we provide a pedagogical introduction to the subject of tachyon condensation in Witten's cubic bosonic open string field theory. We use both the low-energy Yang-Mills description and the language of string field theory to explain the problem of tachyon condensation on unstable D-branes. We give a self-contained introduction to open string field theory using both conformal field theory and overlap integrals. Our main subjects are the Sen conjectures on tachyon condensation in open string field theory and the evidence that supports these conjectures. We conclude with a discussion of vacuum string field theory and projectors of the star-algebra of open string fields. We comment on the possible role of string field theory in the construction of a nonperturbative formulation of string theory that captures all possible string backgrounds.Comment: 103 pages, 11 figures. Lectures presented at TASI 2001. v2: references added. v3: minor errors corrected; reference, discussion added regarding other results on open string spectrum in stable vacuu