Spectral asymptotics for canonical systems

Jonathan Eckhardt; Aleksey Kostenko; Gerald Teschl

doi:10.1515/crelle-2015-0034

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Spectral asymptotics for canonical systems

Jonathan Eckhardt , Aleksey Kostenko und Gerald Teschl

Veröffentlicht/Copyright: 31. Juli 2015

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Aus der Zeitschrift Journal für die reine und angewandte Mathematik (Crelles Journal) Band 2018 Heft 736

Abstract

Based on continuity properties of the de Branges correspondence, we develop a new approach to study the high-energy behavior of Weyl–Titchmarsh and spectral functions of 2×2 first order canonical systems. Our results improve several classical results and solve open problems posed by previous authors. Furthermore, they are applied to radial Dirac and radial Schrödinger operators as well as to Krein strings and generalized indefinite strings.

Funding source: Austrian Science Fund

Award Identifier / Grant number: J3455

Award Identifier / Grant number: P26060

Award Identifier / Grant number: Y330

Funding statement: Research supported by the Austrian Science Fund (FWF) under Grants No. J3455, P26060, and Y330.

A Regularly varying functions

Let us recall the concept of regularly varying functions (see, e.g., [7, 37]).

Definition A.1.

Let f:(a,∞)→[0,∞) be measurable and eventually positive. The function f is called slowly varying at ∞ if

limx→∞⁡f⁢(x⁢t)f⁢(x)=1,t>0.

The function f is called regularly varying at ∞ with index α∈ℝ (we shall write f∈RVα⁡(∞)) if

(A.1)limx→∞⁡f⁢(x⁢t)f⁢(x)=tα,t>0.

If the limit in (A.1) equals ∞ for all t>1, then f is called rapidly varying at ∞ (we shall write f∈RV∞⁡(∞)).

The function f:(0,b)→[0,∞) is called regularly (rapidly) varying at 0 if the function x↦1/f⁢(1/x) is regularly (rapidly) varying at ∞. The corresponding notation in this case is f∈RVα⁡(0) (f∈RV∞⁡(0)).

Clearly, the class of slowly varying functions coincides with the class of regularly varying functions with index 0. Note also that a regularly varying function with index α admits the representation f⁢(x)=xα⁢f~⁢(x), where f~ is a slowly varying function. Moreover, by the Karamata representation theorem [37, Theorem IV.2.2], f is slowly varying at infinity precisely if there is x0≥a such that

(A.2)f⁢(x)=exp⁡{η⁢(x)+∫x0xε⁢(t)t⁢𝑑t},x≥x0>0,

where η is a bounded measurable function on (x0,∞) such that limx→∞⁡η⁢(x)=η0, and ε is a continuous function satisfying limx→∞⁡ε⁢(x)=0. Regularly varying functions can be characterized by their behavior under integration against powers and this is the content of the Karamata characterization theorem (see [37, Section IV.5]).

Concerning rapidly varying functions, it immediately follows from Definition A.1 that the limit in (A.1) exists and equals 0 whenever t∈(0,1). Moreover, the convergence is uniform on every interval (λ,∞) and (0,λ-1), λ>1. It also follows from [7, Theorems 2.4.7 (ii) and 1.5.6] that for every γ>0 there is a constant C>0 such that f⁢(x)≥C⁢xγ as x→∞ (f⁢(x)≤C⁢xγ as x→0 if f∈RV∞⁡(0)).

We also need the following fact on the asymptotic behavior of functions and their inverses (see [7, Chapter 1.5.7] and [11]).

Lemma A.2.

Let F0 and F be two strictly increasing and positive functions on (a,∞). Let also f0 and f denote their inverses, respectively. Suppose that F0 and F are regularly varying at infinity with index α∈(0,∞] and limx→0⁡F⁢(x)/F0⁢(x)=1. Then f0,f∈RV1/α⁡(∞) and

limy→∞⁡f⁢(y)f0⁢(y)=1.

Acknowledgements

We thank the referee for the careful reading of our manuscript and several critical remarks.

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Received: 2014-12-12

Revised: 2015-3-31

Published Online: 2015-7-31

Published in Print: 2018-3-1

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https://doi.org/10.1515/crelle-2015-0034