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Search Results: 1 - 10 of 12135 matches for " Barry Simon "
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A Feynman-Kac Formula for Unbounded Semigroups
Barry Simon
Physics , 1999,
Abstract: We prove a Feynman-Kac formula for Schrodinger operators with potentials V(x) that obey (for all \epsilon > 0): V(x) \geq - \epsilon |x|^2 - C_\epsilon. Even though e^{-tH} is an unbounded operator, any \phi, \psi \in L^2 with compact support lie in D(e^{-tH}) and <\phi, e^{-tH}\psi> is given by a Feynman-Kac formula.
The Classical Moment Problem as a Self-Adjoint Finite Difference Operator
Barry Simon
Physics , 1999,
Abstract: This is a comprehensive exposition of the classical moment problem using methods from the theory of finite difference operators. Among the advantages of this approach is that the Nevanlinna functions appear as elements of a transfer matrix and convergence of Pade approximants appears as the strong resolvent convergence of finite matrix approximations to a Jacobi matrix. As a bonus of this, we obtain new results on the convergence of certain Pade approximants for series of Hamburger.
Analogs of the M-Function in the Theory of Orthogonal Polynomials on the Unit Circle
Barry Simon
Mathematics , 2003,
Abstract: We show that the multitude of applications of the Weyl-Titchmarsh m-function leads to a multitude of different functions in the theory of orthogonal polynomials on the unit circle that serve as analogs of the m-function.
Sturm Oscillation and Comparison Theorems
Barry Simon
Mathematics , 2003,
Abstract: This is a celebratory and pedagogical discussion of Sturm oscillation theory. Included is the discussion of the difference equation case via determinants and a renormalized oscillation theorem of Gesztesy, Teschl, and the author.
The sharp form of the strong Szego theorem
Barry Simon
Mathematics , 2004,
Abstract: Let $f$ be a function on the unit circle and $D_n(f)$ be the determinant of the $(n+1)\times (n+1)$ matrix with elements $\{c_{j-i}\}_{0\leq i,j\leq n}$ where $c_m =\hat f_m\equiv \int e^{-im\theta} f(\theta) \f{d\theta}{2\pi}$. The sharp form of the strong Szeg\H{o} theorem says that for any real-valued $L$ on the unit circle with $L,e^L$ in $L^1 (\f{d\theta}{2\pi})$, we have \[ \lim_{n\to\infty} D_n(e^L) e^{-(n+1)\hat L_0} = \exp \biggl(\sum_{k=1}^\infty k\abs{\hat L_k}^2\biggr) \] where the right side may be finite or infinite. We focus on two issues here: a new proof when $e^{i\theta}\to L(\theta)$ is analytic and known simple arguments that go from the analytic case to the general case. We add background material to make this article self-contained.
Meromorphic Szego functions and asymptotic series for Verblunsky coefficients
Barry Simon
Mathematics , 2005,
Abstract: We prove that the Szeg\H{o} function, $D(z)$, of a measure on the unit circle is entire meromorphic if and only if the Verblunsky coefficients have an asymptotic expansion in exponentials. We relate the positions of the poles of $D(z)^{-1}$ to the exponential rates in the asymptotic expansion. Basically, either set is contained in the sets generated from the other by considering products of the form, $z_1 ... z_\ell \bar z_{\ell-1}... \bar z_{2\ell-1}$ with $z_j$ in the set. The proofs use nothing more than iterated Szeg\H{o} recursion at $z$ and $1/\bar z$.
OPUC on One Foot
Barry Simon
Mathematics , 2005,
Abstract: We present an expository introduction to orthogonal polynomials on the unit circle.
Orthogonal polynomials with exponentially decaying recursion coefficients
Barry Simon
Mathematics , 2006,
Abstract: We review recent results on necessary and sufficient conditions for measures on $\mathbb{R}$ and $\partial\mathbb{D}$ to yield exponential decay of the recursion coefficients of the corresponding orthogonal polynomials. We include results on the relation of detailed asymptotics of the recursion coefficients to detailed analyticity of the measures. We present an analog of Carmona's formula for OPRL. A major role is played by the Szego and Jost functions.
CMV matrices: Five years after
Barry Simon
Mathematics , 2006,
Abstract: CMV matrices are the unitary analog of Jacobi matrices; we review their general theory.
Critical Lieb-Thirring bounds for one-dimensional Schrodinger operators and Jacobi matrices with regular ground states
Barry Simon
Mathematics , 2007,
Abstract: This paper has been withdrawn by the author in favor of a stronger result proven by the author with R. Frank and T. Weidl in arXiv:0707.0998
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