Spectral asymptotics for canonical systems

Jonathan Eckhardt; Aleksey Kostenko; Gerald Teschl

doi:10.1515/crelle-2015-0034

Article

Spectral asymptotics for canonical systems

Jonathan Eckhardt , Aleksey Kostenko and Gerald Teschl

Published/Copyright: July 31, 2015

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From the journal Journal für die reine und angewandte Mathematik (Crelles Journal) Volume 2018 Issue 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.

References

[1] S. Albeverio, R. Hryniv and Y. Mykytyuk, Inverse spectral problems for Bessel operators, J. Differential Equations 241 (2007), 130–159. 10.1016/j.jde.2007.06.017Search in Google Scholar

[2] S. Albeverio, R. Hryniv and Y. Mykytyuk, Reconstruction of radial Dirac operators, J. Math. Phys. 48 (2007), Article ID 043501. 10.1063/1.2709847Search in Google Scholar

[3] F. V. Atkinson and A. B. Mingarelli, Asymptotics of the number of zeros and eigenvalues of general weighted Sturm–Liouville problems, J. reine angew. Math. 375/376 (1987), 380–393. 10.1515/crll.1987.375-376.380Search in Google Scholar

[4] A. Beigl, J. Eckhardt, A. Kostenko and G. Teschl, On spectral deformations and singular Weyl functions for one-dimensional Dirac operators, J. Math. Phys. 56 (2015), no. 1, Article ID 012102. 10.1063/1.4905166Search in Google Scholar

[5] C. Bennewitz, The HELP inequality in the regular case, Internat. Schriftenreihe Numer. Math. 80 (1987), 337–346. 10.1007/978-3-0348-7192-1_27Search in Google Scholar

[6] C. Bennewitz, Spectral asymptotics for Sturm–Liouville equations, Proc. Lond. Math. Soc. (3) 59 (1989), no. 2, 294–338. 10.1112/plms/s3-59.2.294Search in Google Scholar

[7] N. H. Bingham, C. M. Goldie and J. L. Teugels, Regular variation, 2nd ed., Encyclopedia Math. Appl. 27, Cambridge University Press, Cambridge 2001. Search in Google Scholar

[8] R. Brunnhuber, J. Eckhardt, A. Kostenko and G. Teschl, Singular Weyl–Titchmarsh–Kodaira theory for one-dimensional Dirac operators, Monatsh. Math. 174 (2014), 515–547. 10.1007/s00605-013-0563-5Search in Google Scholar

[9] S. Clark and F. Gesztesy, Weyl–Titchmarsh M-function asymptotics, local uniqueness results, trace formulas, and Borg-type theorems for Dirac operators, Trans. Amer. Math. Soc. 358 (2002), 3475–3534. 10.1090/S0002-9947-02-03025-8Search in Google Scholar

[10] L. de Branges, Some Hilbert spaces of entire functions. II, Trans. Amer. Math. Soc. 99 (1961), 118–152. 10.1090/S0002-9947-1961-0133456-2Search in Google Scholar

[11] D. Djurčić, A. Togašev and J. Ješić, The strong asymptotic equivalence and the generalized inverse, Sib. Math. J. 49 (2008), 628–636. 10.1007/s11202-008-0059-zSearch in Google Scholar

[12] J. Eckhardt, F. Gesztesy, R. Nichols and G. Teschl, Weyl–Titchmarsh theory for Sturm–Liouville operators with distributional potential coefficients, Opuscula Math. 33 (2013), 467–563. 10.7494/OpMath.2013.33.3.467Search in Google Scholar

[13] J. Eckhardt, F. Gesztesy, R. Nichols and G. Teschl, Supersymmetry and Schrödinger-type operators with distributional matrix-valued potentials, J. Spectr. Theory 4 (2014), no. 4, 715–768. 10.4171/JST/84Search in Google Scholar

[14] J. Eckhardt and A. Kostenko, An isospectral problem for global conservative multi-peakon solutions of the Camassa–Holm equation, Comm. Math. Phys. 239 (2014), 893–918. 10.1007/s00220-014-1905-4Search in Google Scholar

[15] J. Eckhardt and A. Kostenko, A quadratic operator pencil associated with the conservative Camassa–Holm flow, preprint (2014), http://arxiv.org/abs/1406.3703. Search in Google Scholar

[16] J. Eckhardt and A. Kostenko, The inverse spectral problem for indefinite strings, preprint (2014), http://arxiv.org/abs/1409.0139. 10.1007/s00222-015-0629-1Search in Google Scholar

[17] J. Eckhardt, A. Kostenko and G. Teschl, Inverse uniqueness results for one-dimensional weighted Dirac operators, Spectral theory and differential equations: V. A. Marchenko 90th anniversary collection, Adv. Math. Sci. 233, American Mathematical Society, Providence (2014), 117–133. 10.1090/trans2/233/07Search in Google Scholar

[18] J. Eckhardt and G. Teschl, Sturm–Liouville operators with measure-valued coefficients, J. Anal. Math. 120 (2013), no. 1, 151–224. 10.1007/s11854-013-0018-xSearch in Google Scholar

[19] W. N. Everitt, On a property of the m-coefficient of a second order linear differential equation, J. Lond. Math. Soc. (2) 4 (1972), 443–457. 10.1112/jlms/s2-4.3.443Search in Google Scholar

[20] W. N. Everitt, On an extension to an integro–differential inequality of Hardy, Littlewood and Polya, Proc. Roy. Soc. Edinburgh A 69 (1972), 295–333. 10.1017/S0080454100008797Search in Google Scholar

[21] W. N. Everitt, D. B. Hinton and J. K. Shaw, The asymptotic form of the Titchmarsh–Weyl coefficient for Dirac systems, J. Lond. Math. Soc. (2) 27 (1983), 465–476. 10.1112/jlms/s2-27.3.465Search in Google Scholar

[22] W. N. Everitt and A. Zettl, On a class of integral inequalities, J. Lond. Math. Soc. (2) 17 (1978), 291–303. 10.1112/jlms/s2-17.2.291Search in Google Scholar

[23] A. Fleige, The critical point infinity associated with indefinite Sturm–Liouville problems, Operator theory, Springer, Berlin (2014), 10.1007/978-3-0348-0692-3_44-1. 10.1007/978-3-0348-0692-3_44-1Search in Google Scholar

[24] I. Gohberg and M. Krein, Theory and applications of Volterra operators in Hilbert spaces, Transl. Math. Monographs 24, American Mathematical Society, Providence 1970. Search in Google Scholar

[25] I. S. Kac, On the behavior of spectral functions of second-order differential systems (Russian), Dokl. Akad. Nauk SSSR 106 (1956), 183–186.Search in Google Scholar

[26] I. S. Kac, Integral characterization of the growth of the spectral functions of second order generalized boundary value problems with boundary conditions at the regular end (Russian), Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 154–184. Search in Google Scholar

[27] I. S. Kac, Power-asymptotic estimates for spectral functions of generalized boundary value problems of second order, Sov. Math. Dokl. 13 (1972), 453–457. Search in Google Scholar

[28] I. S. Kac, A generalization of the asymptotic formula of V. A. Marchenko for the spectral function of a second order boundary value problem, Izv. Akad. Nauk SSSR Ser. Mat. 37 (1973), 422–436. Search in Google Scholar

[29] I. S. Kac, Linear relations generated by canonical differential equations, Funct. Anal. Appl. 17 (1983), 315–317. 10.1007/BF01076728Search in Google Scholar

[30] I. S. Kac and M. G. Krein, On the spectral functions of the string, Amer. Math. Soc. Transl. (2) 103 (1974), 19–102. 10.1090/trans2/103/02Search in Google Scholar

[31] M. Kaltenbäck, H. Winkler and H. Woracek, Strings, dual strings, and related canonical systems, Math. Nachr. 280 (2007), no. 13–14, 1518–1536. 10.1002/mana.200410562Search in Google Scholar

[32] I. M. Karabash, A. S. Kostenko and M. M. Malamud, The similarity problem for J-nonnegative Sturm–Liouville operators, J. Differential Equations 246 (2009), 964–997. 10.1016/j.jde.2008.04.021Search in Google Scholar

[33] Y. Kasahara, Spectral theory of generalized second order differential operators and its applications to Markov processes, Japan. J. Math. (N.S.) 1 (1975/76), 67–84. 10.4099/math1924.1.67Search in Google Scholar

[34] Y. Kasahara, Spectral function of Krein’s and Kotani’s string in the class Γ, Proc. Japan Acad. Ser. A Math. Sci. 88 (2012), no. 10, 173–177. 10.3792/pjaa.88.173Search in Google Scholar

[35] Y. Kasahara and S. Watanabe, Asymptotic behavior of spectral measures of Krein’s and Kotani’s strings, Kyoto J. Math. 50 (2010), 623–644. 10.1215/0023608X-2010-007Search in Google Scholar

[36] K. Kodaira, The eigenvalue problem for ordinary differential equations of the second order and Heisenberg’s theory of S-matrices, Amer. J. Math. 71 (1949), 921–945. 10.2307/2372377Search in Google Scholar

[37] J. Korevaar, Tauberian theory: A century of developments, Grundlehren Math. Wiss. 329, Springer, Berlin 2004. 10.1007/978-3-662-10225-1Search in Google Scholar

[38] A. Kostenko, The similarity problem for indefinite Sturm–Liouville operators and the HELP inequality, Adv. Math. 246 (2013), 368–413. 10.1016/j.aim.2013.05.025Search in Google Scholar

[39] A. Kostenko, A. Sakhnovich and G. Teschl, Commutation methods for Schrödinger operators with strongly singular potentials, Math. Nachr. 285 (2012), 392–410. 10.1002/mana.201000108Search in Google Scholar

[40] A. Kostenko, A. Sakhnovich and G. Teschl, Weyl–Titchmarsh theory for Schrödinger operators with strongly singular potentials, Int. Math. Res. Not. IMRN 2012 (2012), 1699–1747. 10.1093/imrn/rnr065Search in Google Scholar

[41] A. Kostenko and G. Teschl, On the singular Weyl–Titchmarsh function of perturbed spherical Schrödinger operators, J. Differential Equations 250 (2011), 3701–3739. 10.1016/j.jde.2010.10.026Search in Google Scholar

[42] A. Kostenko and G. Teschl, Spectral asymptotics for perturbed spherical Schrödinger operators and applications to quantum scattering, Comm. Math. Phys. 322 (2013), 255–275. 10.1007/s00220-013-1698-xSearch in Google Scholar

[43] M. Langer and H. Woracek, Indefinite Hamiltonian systems whose Titchmarsh–Weyl coefficients have no finite generalized poles of non-negative type, Oper. Matrices 7 (2013), 477–555. 10.7153/oam-07-29Search in Google Scholar

[44] M. Langer and H. Woracek, Direct and inverse spectral theorems for a class of Hamiltonian systems with two singular endpoints, in preparation. Search in Google Scholar

[45] M. Lesch and M. M. Malamud, On the deficiency indices and self-adjointness of symmetric Hamiltonian systems, J. Differential Equations 189 (2003), 556–615. 10.1016/S0022-0396(02)00099-2Search in Google Scholar

[46] B. M. Levitan, A remark on one theorem of V. A. Marchenko (Russian), Trudy Moskov. Mat. Obsch. 1 (1952), 421–422. Search in Google Scholar

[47] B. M. Levitan, Inverse Sturm–Liouville problems, VNU Science Press, Utrecht 1987. 10.1515/9783110941937Search in Google Scholar

[48] B. M. Levitan and I. S. Sargsjan, Sturm–Liouville and Dirac operators (Russian), Nauka, Moscow 1988. Search in Google Scholar

[49] V. A. Marchenko, Some questions in the theory of one-dimensional second-order linear differential operators. I (Russian), Trudy Moskov. Mat. Obsch. 1 (1952), 327–420; translation in Amer. Math. Soc. Transl. (2) 101 (1973), 1–104. Search in Google Scholar

[50] V. A. Marchenko, Sturm–Liouville operators and applications, rev. ed., American Mathematical Society, Providence 2011. 10.1090/chel/373Search in Google Scholar

[51] F. W. J. Olver, D. W. Lozier, R. F. Boisvert and C. W. Clark, NIST handbook of mathematical functions, Cambridge University Press, Cambridge 2010. Search in Google Scholar

[52] R. Romanov, Canonical systems and de Branges spaces, preprint (2014), http://arxiv.org/abs/1408.6022. 10.1007/978-3-0348-0692-3_9-1Search in Google Scholar

[53] A. M. Savchuk and A. A. Shkalikov, Sturm–Liouville operators with singular potentials, Math. Notes 66 (1999), no. 5–6, 741–753. 10.1007/BF02674332Search in Google Scholar

[54] G. Teschl, Mathematical methods in quantum mechanics. With applications to Schrödinger operators, 2nd ed., American Mathematical Society, Providence 2014. Search in Google Scholar

[55] B. Thaller, The Dirac equation, Springer, Berlin 1991. 10.1007/978-3-662-02753-0Search in Google Scholar

[56] E. C. Titchmarsh, Eigenfunction expansions associated with second order differential equations. Part I, 2nd ed., Clarendon Press, Oxford 1962. 10.1063/1.3058324Search in Google Scholar

[57] H. Weyl, Über gewöhnliche Differentialgleichungen mit Singularitäten und die zugehörigen Entwicklungen willkürlicher Funktionen, Math. Ann. 68 (1910), 220–269. 10.1007/BF01474161Search in Google Scholar

[58] H. Winkler, The inverse spectral problem for canonical systems, Integral Equations Operator Theory 22 (1995), 360–374. 10.1007/BF01378784Search in Google Scholar

[59] H. Winkler, Spectral estimations for canonical systems, Math. Nachr. 220 (2000), 115–141. 10.1002/1522-2616(200012)220:1<115::AID-MANA115>3.0.CO;2-ISearch in Google Scholar

[60] H. Winkler and H. Woracek, Reparametrizations of non trace-normed Hamiltonians, Spectral theory, mathematical system theory, evolution equations, differential and difference equations (Berlin 2010), Oper. Theory Adv. Appl. 221, Birkhäuser, Basel (2012), 667–690. 10.1007/978-3-0348-0297-0_40Search in Google Scholar

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