arXiv subjects
Publications and source records attributed to Barry Simon.
We prove a Feynman-Kac formula for Schrodinger operators with potentials V(x) that obey (for all ε> 0): V(x) \geq - ε|x|^2 - C_ε. Even though e^{-tH} is an unbounded operator, any ϕ, ψ\in L^2 with compact support lie in D(e^{-tH}) and <ϕ, e^{-tH}ψ> is given by a Feynman-Kac formula.
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.
We study the relative index of two orthogonal infinite dimensional projections which, in the finite dimensional case, is the difference in their dimensions. We relate the relative index to the Fredholm index of appropriate operators, discuss its basic properties, and obtain various formulas for it. We apply the relative index to counting the change in the number of electrons below the Fermi energy of certain quantum systems and interpret it as the charge deficiency. We study the relation of the charge deficiency with the notion of adiabatic charge transport that arises from the consideration of the adiabatic curvature. It is shown that, under a certain covariance, (homogeneity), condition the two are related. The relative index is related to Bellissard's theory of the Integer Hall effect. For Landau Hamiltonians the relative index is computed explicitly for all Landau levels.
In a variety of contexts, we prove that singular continuous spectrum is generic in the sense that for certain natural complete metric spaces of operators, those with singular spectrum are a dense $G_δ$.
We extend the well-known trace formula for Hill's equation to general one-dimensional Schrödinger operators. The new function $ξ$, which we introduce, is used to study absolutely continuous spectrum and inverse problems.