On Local Borg-Marchenko Uniqueness Results

We provide a new short proof of the following fact, first proved by one of us in 1998: If two Weyl-Titchmarsh m-functions, $m_j(z)$, of two Schr\"odinger operators H_j = -\f{d^2}{dx^2} + q_j, j=1,2 in $L^2 ((0,R))$, $0<R\leq \infty$, are exponentially close, that is, |m_1(z)- m_2(z)| \underset{|z|\to\infty}{=} O(e^{-2\Ima (z^{1/2})a}), 0<a<R, then $q_1 = q_2$ a.e.~on $[0,a]$. The result applies to any boundary conditions at x=0 and x=R and should be considered a local version of the celebrated Borg-Marchenko uniqueness result (which is quickly recovered as a corollary to our proof). Moreover, we extend the local uniqueness result to matrix-valued Schr\"odinger operators.Comment: LaTeX, 18 page

Uniqueness theorems in inverse spectral theory for one-dimensional Schrödinger operators

New unique characterization results for the potential V(x) in connection with Schrödinger operators on R and on the half-line [0,∞)are proven in terms of appropriate Krein spectral shift functions. Particular results obtained include a generalization of a well-known uniqueness theorem of Borg and Marchenko for Schrödinger operators on the half-line with purely discrete spectra to arbitrary spectral types and a new uniqueness result for Schrödinger operators with confining potentials on the entire real line

Electron spin dynamics and electron spin resonance in graphene

A theory of spin relaxation in graphene including intrinsic, Bychkov-Rashba, and ripple spin-orbit coupling is presented. We find from spin relaxation data by Tombros et al. [Nature 448, 571 (2007).] that intrinsic spin-orbit coupling dominates over other contributions with a coupling constant of 3.7 meV. Although it is 1-3 orders of magnitude larger than those obtained from first principles, we show that comparable values are found for other honeycomb systems, MgB2 and LiC6; the latter is studied herein by electron spin resonance (ESR). We predict that spin coherence is longer preserved for spins perpendicular to the graphene plane, which is beneficial for spintronics. We identify experimental conditions when bulk ESR is realizable on graphene

Core drill's bit is replaceable without withdrawal of drill stem - A concept

Drill bit is divided into several sectors. When collapsed, the outside diameter is forced down the drill stem, when it reaches bottom the sectors are forced outward and form a cutting bit. A dulled bit is retracted by reversal of this procedure

A Possible Nanometer-scale Computing Device Based on an Adding Cellular Automaton

We present a simple one-dimensional Cellular Automaton (CA) which has the property that an initial state composed of two binary numbers evolves quickly into a final state which is their sum. We call this CA the Adding Cellular Automaton (ACA). The ACA requires only 2N two-state cells in order to add any two N-1 bit binary numbers. The ACA could be directly realized as a wireless nanometer-scale computing device - a possible implementation using coupled quantum dots is outlined.Comment: 8 pages, RevTex, 3 Postscript figures. This version to appear in App. Phys. Let

Singular Continuous Spectrum for the Laplacian on Certain Sparse Trees

We present examples of rooted tree graphs for which the Laplacian has singular continuous spectral measures. For some of these examples we further establish fractional Hausdorff dimensions. The singular continuous components, in these models, have an interesting multiplicity structure. The results are obtained via a decomposition of the Laplacian into a direct sum of Jacobi matrices