arXiv subjects

Rowan Killip

Publications and source records attributed to Rowan Killip.

80 records · Page 5Linked to original sources

arXiv2004

Bounds on the spectral shift function and the density of states

We study spectra of Schrödinger operators on $\RR^d$. First we consider a pair of operators which differ by a compactly supported potential, as well as the corresponding semigroups. We prove almost exponential decay of the singular values $μ_n$ of the difference of the semigroups as $n\to \infty$ and deduce bounds on the spectral shift function of the pair of operators. Thereafter we consider alloy type random Schrödinger operators. The single site potential $u$ is assumed to be non-negative and of compact support. The distributions of the random coupling constants are assumed to be Hölder continuous. Based on the estimates for the spectral shift function, we prove a Wegner estimate which implies Hölder continuity of the integrated density of states.

math-ph↗

arXiv2004

Sum Rules for Jacobi Matrices and Their Applications to Spectral Theory

We discuss the proof of and systematic application of Case's sum rules for Jacobi matrices. Of special interest is a linear combination of two of his sum rules which has strictly positive terms. Among our results are a complete classification of the spectral measures of all Jacobi matrices J for which J-J_0 is Hilbert--Schmidt, and a proof of Nevai's conjecture that the Szego condition holds if J-J_0 is trace class.

math-ph↗

arXiv2004

Ergodic Potentials With a Discontinuous Sampling Function Are Non-Deterministic

We prove absence of absolutely continuous spectrum for discrete one-dimensional Schrödinger operators on the whole line with certain ergodic potentials, $V_ω(n) = f(T^n(ω))$, where $T$ is an ergodic transformation acting on a space $Ω$ and $f: Ω\to \R$. The key hypothesis, however, is that $f$ is discontinuous. In particular, we are able to settle a conjecture of Aubry and Jitomirskaya--Mandel'shtam regarding potentials generated by irrational rotations on the torus. The proof relies on a theorem of Kotani, which shows that non-deterministic potentials give rise to operators that have no absolutely continuous spectrum.

math-ph↗

arXiv2003

Necessary and Sufficient Conditions in the Spectral Theory of Jacobi Matrices and Schrödinger Operators

We announce three results in the theory of Jacobi matrices and Schrödinger operators. First, we give necessary and sufficient conditions for a measure to be the spectral measure of a Schrödinger operator $-\f{d^2}{dx^2} +V(x)$ on $L^2 (0,\infty)$ with $V\in L^2 (0,\infty)$ and $u(0)=0$ boundary condition. Second, we give necessary and sufficient conditions on the Jacobi parameters for the associated orthogonal polynomials to have Szegő asymptotics. Finally, we provide necessary and sufficient conditions on a measure to be the spectral measure of a Jacobi matrix with exponential decay at a given rate.

math.SP↗

arXiv2003

Half-line Schrodinger Operators With No Bound States

We consider Schödinger operators on the half-line, both discrete and continuous, and show that the absence of bound states implies the absence of embedded singular spectrum. More precisely, in the discrete case we prove that if $Δ+ V$ has no spectrum outside of the interval $[-2,2]$, then it has purely absolutely continuous spectrum. In the continuum case we show that if both $-Δ+ V$ and $-Δ- V$ have no spectrum outside $[0,\infty)$, then both operators are purely absolutely continuous. These results extend to operators with finitely many bound states.

math-ph↗

arXiv2002

Variational Estimates for Discrete Schrödinger Operators with Potentials of Indefinite Sign

Let $H$ be a one-dimensional discrete Schrödinger operator. We prove that if $σ_{\ess} (H)\subset [-2,2]$, then $H-H_0$ is compact and $σ_{\ess}(H)=[-2,2]$. We also prove that if $H_0 + \frac14 V^2$ has at least one bound state, then the same is true for $H_0 +V$. Further, if $H_0 + \frac14 V^2$ has infinitely many bound states, then so does $H_0 +V$. Consequences include the fact that for decaying potential $V$ with $\liminf_{|n|\to\infty} |nV(n)| > 1$, $H_0 +V$ has infinitely many bound states; the signs of $V$ are irrelevant. Higher-dimensional analogues are also discussed.

math-ph↗

arXiv2000

Reflection symmetries of almost periodic functions

We study global reflection symmetries of almost periodic functions. In the non-limit periodic case, we establish an upper bound on the Haar measure of the set of those elements in the hull which are almost symmetric about the origin. As an application of this result we prove that in the non-limit periodic case, the criterion of Jitomirskaya and Simon ensuring absence of eigenvalues for almost periodic Schrödinger operators is only applicable on a set of zero Haar measure. We complement this by giving examples of limit periodic functions where the Jitomirskaya-Simon criterion can be applied to every element of the hull.

math-ph↗

arXiv1999

Uniform spectral properties of one-dimensional quasicrystals, III. $α$-continuity

We study the spectral properties of discrete one-dimensional Schrödinger operators with Sturmian potentials. It is shown that the point spectrum is always empty. Moreover, for rotation numbers with bounded density, we establish purely $α$-continuous spectrum, uniformly for all phases. The proofs rely on the unique decomposition property of Sturmian potentials, a mass-reproduction technique based upon a Gordon-type argument, and on the Jitomirskaya-Last extension of the Gilbert-Pearson theory of subordinacy.

math-ph↗