Two extensions of Lubinsky’s universality theorem

“…This alone is su¢ cient for universality at x. Subsequently, Totik [26], his student Findley [3], and Simon [22] presented far reaching extensions of this result. For example, Totik showed that the same result holds for regular measures on a general compact subset of the real line, instead of [ 1; 1], and moreover, we may relax the requirement of continuity of w. We only need log w to be integrable in a neighborhood of the points where universality is desired, together with a Lebesgue point type condition on a certain local Szeg½ o function.…”

Section: Introduction and Resultsmentioning

confidence: 97%

Universality limits in the bulk for arbitrary measures on compact sets

Lubinsky

2008

J Anal Math

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Abstract. We present a new method for establishing universality limits in the bulk, based on the theory of entire functions of exponential type. Let be a measure on a compact subset of the real line. Assume that is absolutely continuous in a neighborhood of some point x in the support, and that 0 is bounded above and below near x, which is assumed to be a Lebesgue point of 0 . Then universality holds at x i¤ it holds "along the diagonal", that is lim n!1 Kn x + a n ; x + a n Kn (x; x) = 1;for all real a. The method does not require regularity of the measure as did earlier methods. Moreover, the assumption on the diagonal is certainly satis…ed in the case of regular measures, so that we obtain another proof of some recent results of Simon and Totik.

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“…We will show the analogue of Lemma 3.1 in Simon [Sim08b]. Assume regularity bounds (1.4.1) on the measure dµ.…”

Section: Bounds On the Diagonal Kernelmentioning

confidence: 95%

“…It relates a fundamental object to the sine kernel and implies that the left hand side of (2.5.1) only depends on the continuity and positivity of the measure dη at x 0 and its essential support. Simon [Sim08b] and Totik [Tot] extend this argument to measures with supp ess (dη) = ∪I j a finite union of intervals. In this thesis I adapt all the steps to Schrödinger operators.…”

Section: Christoffel-darboux Formulamentioning

confidence: 99%

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Universality Limits of a Reproducing Kernel for a Half-Line Schrödinger Operator and Clock Behavior of Eigenvalues

Maltsev

2010

Commun. Math. Phys.

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In particular, we define a reproducing kernel S L for Schrödinger operators and we use it to study the fine spacing of eigenvalues in a box of the half-line Schrödinger operator with perturbed periodic potential. We show that if solutions u(ξ, x) are bounded in x by e x uniformly for ξ near the spectrum in an average sense and the spectral measure is positive and absolutely continuous in a bounded interval I in the interior of the spectrum with ξ 0 ∈ I, then uniformly in Iwhere ρ(ξ)dξ is the density of states. We deduce that the eigenvalues near ξ 0 in a large box of size L are spaced asymptotically as 1 Lρ . We adapt the methods used to show similar results for orthogonal polynomials.vi

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Two extensions of Lubinsky’s universality theorem

Cited by 64 publications

References 39 publications

Universality Limits at the Hard Edge of the Spectrum for Measures with Compact Support

Universality Limits at the Hard Edge of the Spectrum for Measures with Compact Support

Universality limits in the bulk for arbitrary measures on compact sets

Universality Limits of a Reproducing Kernel for a Half-Line Schrödinger Operator and Clock Behavior of Eigenvalues

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