Geodesic
Titchmarsh--Weyl formula for the spectral density of a class of Jacobi matrices in the critical case
Funkcionalʹnyj analiz i ego priloženiâ, Tome 55 (2021) no. 2, pp. 21-43.

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We consider a class of Jacobi matrices with unbounded entries in the so-called critical (double root, Jordan block) case. We prove a formula which relates the spectral density of a matrix to the asymptotics of orthogonal polynomials associated with it. 
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S. N. Naboko; S. A. Simonov. Titchmarsh--Weyl formula for the spectral density of a class of Jacobi matrices in the critical case. Funkcionalʹnyj analiz i ego priloženiâ, Tome 55 (2021) no. 2, pp. 21-43. http://geodesic.mathdoc.fr/item/FAA_2021_55_2_a2/

[1] N. I. Akhiezer, Klassicheskaya problema momentov i nekotorye voprosy analiza, svyazannye s neyu, Fizmatgiz, M., 1961

[2] N. I. Akhiezer, I. M. Glazman, Teoriya lineinykh operatorov v gilbertovom prostranstve, Gostekhizdat, M., 1966 | MR

[3] E. A. Koddington, N. Levinson, Teoriya obyknovennykh differentsialnykh uravnenii, IL, M., 1958

[4] E. Ch. Titchmarsh, Razlozheniya po sobstvennym funktsiyam, svyazannye s differentsialnymi uravneniyami vtorogo poryadka, chast I, IL, M., 1961

[5] W. Van Assche, J. S. Geronimo, “Asymptotics for orthogonal polynomials on and off the essential spectrum”, J. Approx. Theory, 55:2 (1988), 220–231 | DOI | MR | Zbl

[6] A. I. Aptekarev, J. S. Geronimo, “Measures for orthogonal polynomials with unbounded recurrence coefficients”, J. Approx. Theory, 207 (2016), 339–347 | DOI | MR | Zbl

[7] Z. Benzaid, D. A. Lutz, “Asymptotic representation of solutions of perturbed systems of linear difference equations”, Stud. Appl. Math., 77:3 (1987), 195–221 | DOI | MR | Zbl

[8] S. Bodine, D. A. Lutz, Asymptotic Integration of Differential and Difference Equations, Springer, Cham, 2015 | MR | Zbl

[9] D. Damanik, S. Naboko, “Unbounded Jacobi matrices at critical coupling”, J. Approx. Theory, 145:2 (2007), 221–236 | DOI | MR | Zbl

[10] D. J. Gilbert, D. B. Pearson, “On subordinacy and analysis of the spectrum of one-dimensional Schrödinger operators”, J. Math. Anal. Appl., 128:1 (1987), 30–56 | DOI | MR | Zbl

[11] W. A. Harris, Jr., D. A. Lutz, “Asymptotic integration of adiabatic oscillators”, J. Math. Anal. Appl., 51 (1975), 76–93 | DOI | MR | Zbl

[12] E. Ianovich, “On one condition of absolutely continuous spectrum for self-adjoint operators and its applications”, Opuscula Math., 38:5 (2018), 699–718 | DOI | MR | Zbl

[13] J. Janas, “The asymptotic analysis of generalized eigenvectors of some Jacobi operators. Jordan box case”, J. Difference Equ. Appl., 12:6 (2006), 597–618 | DOI | MR | Zbl

[14] J. Janas, M. Moszyński, “Spectral properties of Jacobi matrices by asymptotic analysis”, J. Approx. Theory, 120:2 (2003), 309–336 | DOI | MR | Zbl

[15] J. Janas, S. Naboko, “Multithreshold spectral phase transitions for a class of Jacobi matrices”, Recent Advances in Operator Theory, Groningen, 1998, Oper. Theory Adv. Appl., 124, Birkäuser, Basel, 2001, 267–285 | MR | Zbl

[16] J. Janas, S. Naboko, “Spectral analysis of selfadjoint Jacobi matrices with periodically modulated entries”, J. Funct. Anal., 191:2 (2002), 318–342 | DOI | MR | Zbl

[17] J. Janas, S. Naboko, E. Sheronova, “Asymptotic behavior of generalized eigenvectors of Jacobi matrices in the critical (“double root”) case”, Z. Anal. Anwend., 28:4 (2009), 411–430 | DOI | MR | Zbl

[18] J. Janas, S. Simonov, “Weyl–Titchmarsh type formula for discrete Schrödinger operator with Wigner–von Neumann potential”, Studia Math., 201:2 (2010), 167–189 | DOI | MR | Zbl

[19] S. Khan, D. B. Pearson, “Subordinacy and spectral theory for infinite matrices”, Helv. Phys. Acta, 65:4 (1992), 505–527 | MR

[20] R.-J. Kooman, “An asymptotic formula for solutions of linear second-order difference equations with regularly behaving coefficients”, J. Difference Equ. Appl., 13:11 (2007), 1037–1049 | DOI | MR | Zbl

[21] P. Kurasov, S. Simonov, “Weyl–Titchmarsh-type formula for periodic Schrödinger operator with Wigner–von Neumann potential”, Proc. Roy. Soc. Edinburgh, Sect. A, 143 (2013), 401–425 | DOI | MR | Zbl

[22] A. Máté, P. Nevai, V. Totik, “Asymptotics for orthogonal polynomials defined by a recurrence relation”, Constr. Approx., 1:3 (1985), 231–248 | DOI | MR | Zbl

[23] S. Naboko, S. Simonov, “Spectral analysis of a class of Hermitian Jacobi matrices in a critical (double root) hyperbolic case”, Proc. Edinb. Math. Soc. (2), 53:1 (2010), 239–254 | DOI | MR | Zbl

[24] S. Naboko, S. Simonov, “Zeroes of the spectral density of the periodic Schrödinger operator with Wigner–von Neumann potential”, Math. Proc. Cambridge Philos. Soc., 153:1 (2012), 33–58 | DOI | MR | Zbl

[25] S. Naboko, M. Solomyak, “On the absolutely continuous spectrum in a model of an irreversible quantum graph”, Proc. London Math. Soc. (3), 92:1 (2006), 251–272 | DOI | MR | Zbl

[26] W. Rudin, Real and Complex Analysis, McGraw-Hill, New York, 1987 | MR | Zbl

[27] L. O. Silva, “Uniform and smooth Benzaid–Lutz type theorems and applications to Jacobi matrices”, Operator Theory, Analysis and Mathematical Physics, Oper. Theory Adv. Appl., 174, Birkhäuser, Basel, 2007, 173–186 | MR | Zbl

[28] S. Simonov, “An example of spectral phase transition phenomenon in a class of Jacobi matrices with periodically modulated weights”, Operator Theory, Analysis and Mathematical Physics, Oper. Theory Adv. Appl., 174, Birkhäuser, Basel, 2007, 187–203 | MR | Zbl

[29] S. Simonov, “Weyl–Titchmarsh type formula for Hermite operator with small perturbation”, Opuscula Math., 29:2 (2009), 187–207 | DOI | MR | Zbl

[30] S. Simonov, “Zeroes of the spectral density of discrete Schrödinger operator with Wigner–von Neumann potential”, Integral Equations Operator Theory, 73:3 (2012), 351–364 | DOI | MR | Zbl

[31] S. Simonov, “Zeroes of the spectral density of the Schrödinger operator with the slowly decaying Wigner–von Neumann potential”, Math. Z., 284:1–2 (2016), 335–411 | DOI | MR | Zbl

[32] G. Świderski, “Spectral properties of unbounded Jacobi matrices with almost monotonic weights”, Constr. Approx., 44:1 (2016), 141–157 | DOI | MR | Zbl

[33] G. Świderski, B. Trojan, “Periodic perturbations of unbounded Jacobi matrices I: Asymptotics of generalized eigenvectors”, J. Approx. Theory, 216 (2017), 38–66 | DOI | MR | Zbl

[34] G. Świderski, “Periodic perturbations of unbounded Jacobi matrices II: Formulas for density”, J. Approx. Theory, 216 (2017), 67–85 | DOI | MR | Zbl

[35] G. Świderski, “Periodic perturbations of unbounded Jacobi matrices III: The soft edge regime”, J. Approx. Theory, 233 (2018), 1–36 | DOI | MR | Zbl

[36] R. Szwarc, “Absolute continuity of spectral measure for certain unbounded Jacobi matrices”, Advanced Problems in Constructive Approximation, Internat. Ser. Numer. Math., 142, Birkhäuser, Basel, 2002, 255–262