Asymptotics of the discrete spectrum for complex Jacobi matrices

• Autorzy: Malejki M.
• Języki publikacji: EN

Abstrakty

EN

The spectral properties and the asymptotic behaviour of the discrete spectrum for a special class of infinite tridiagonal matrices are given. We derive the asymptotic formulae for eigenvalues of unbounded complex Jacobi matrices acting in l2(N).

Słowa kluczowe

EN

tridiagonal matrix; complex Jacobi matrix; discrete spectrum; eigenvalue; asymptotic formula; unbounded operator; Riesz projection

• Wydawca: AGH University of Science and Technology Press
• Czasopismo: Opuscula Mathematica
• Rocznik: 2014
• Tom: Vol. 34, no. 1
• Strony: 139--160
• Opis fizyczny: Bibliogr. 25 poz.

Twórcy

autor

Malejki M.

• AGH University of Science and Technology Faculty of Applied Mathematics al. Mickiewicza 30, 30-059 Krakow, Poland

Bibliografia

• [1] B. Beckermann, Complex Jacobi matrices, J. Comput. Appl. Math. 127 (2001), 17–65.
• [2] B. Beckermann, V. Kaliaguine, The diagonal of the Padé table and the approximation of the Weyl function of second-order difference operators, Constr. Approx. 13 (1997), 481–510.
• [3] Yu.M. Berezansky, L.Ya. Ivasiuk, O.A. Mokhonko, Recursion relation for orthogonal polynomials on the complex plane, Methods Funct. Anal. Topol. 14 (2008) 2, 108–116.
• [4] A. Boutet de Monvel, S. Naboko, L. Silva, The asympthotic behaviour of eigenvalues of modified Jaynes-Cummings model, Asymptot. Anal. 47 (2006) 3–4, 291–315.
• [5] A. Boutet de Monvel, L. Zielinski, Explicit error estimates for eigenvalues of some unbounded Jacobi matrices, Operator Theory: Adv. Appl. 221 (2012), 189–217.
• [6] P. Cojuhari, J. Janas, Discreteness of the spectrum for some unbounded Jacobi matrices, Acta Sci. Math. (Szeged) 73 (2007), 649–667.
• [7] P. Djakov, B. Mityagin, Simple and double eigenvalues of the Hill operator with a two-term potential, J. Approx. Theory 135 (2005), 70–104.
• [8] P. Djakov, B. Mityagin, Trace formula and spectral Riemann surface for a class of tri-diagonal matrices, J. Approx. Theory 139 (2006), 293–326.
• [9] J. Edward, Spectra of Jacobi matrices, differential equations on the circle, and the su(1, 1) Lie algebra, SIAM J. Math. Anal. 24 (1993) 3, 824–831.
• [10] I. Egorova, L. Golinskii, Discrete spectrum for complex perturbations of periodic Jacobi matrices, J. Difference Equ. Appl. 11 (2005) 14, 1185–1203.
• [11] I. Egorova, L. Golinskii, On the location of the discrete spectrum for complex Jacobi matrices, Proc. Am. Math. Soc. 133 (2005) 12, 3635–3641.
• [12] L. Golinskii, M. Kudryavtsev, On the discrete spectrum of complex banded matrices, Zh. Mat. Fiz. Anal. Geom. 2 (2006) 4, 396–423.
• [13] Y. Ikebe, N. Asai, Y. Miyazaki, D. Cai, The eigenvalue problem for infinite complex symmetric tridiagonal matrices with application, Linear Algebra Appl. 241–243 (1996),
• 599–618. [14] J. Janas, M. Malejki, Y. Mykytyuk, Similarity and the point spectrum of some non-selfadjoint Jacobi matrices, Proc. Edinb. Math. Soc., II. Ser. 46 (2003) 3, 575–595.
• [15] J. Janas, M. Malejki, Alternative approaches to asymptotic behaviour of eigenvalues of some unbounded Jacobi matrices, J. Comput. Appl. Math. 200 (2007), 342–356.
• [16] J. Janas, S. Naboko, Infinite Jacobi matrices with unbounded entries: asymptotics of eigenvalues and the transformation operator approach, SIAM J. Math. Anal. 36 (2004) 2, 643–658.
• [17] J. Janas, S. Naboko, Multithreshold spectral phase transitions for a class of Jacobi matrices, Operator Theory: Adv. Appl. 124 (2001), 267–285.
• [18] M. Malejki, Asymptotics of large eigenvalues for some discrete unbounded Jacobi matrices, Linear Algebra Appl. 431 (2009), 1952–1970.
• [19] M. Malejki, Approximation and asymptotics of eigenvalues of unbounded self-adjoint Jacobi matrices acting in l2 by the use of finite submatrices, Cent. Eur. J. Math. 8 (2010) 1, 114–128.
• [20] G.V. Rozenbljum, Near-similarity of operators and the spectral asymptotic behaviour of pseudodifferential operators on the circle, [in Russian] Trudy Maskov. Mat. Obshch. 36 (1978), 59–84.
• [21] P.N. Shivakumar, Chuanxiang Li, Upper and lower bounds for inverse elements of infinite tridiagonal matrices, Linear Algebra Appl. 247 (1996), 297-316.
• [22] P.N. Shivakumar, J.J. Wiliams, N. Rudraiah, Eigenvalues for infinite matrices, Linear Algebra Appl. 96 (1987), 35–63.
• [23] H. Volkmer, Approximation of eigenvalues of some differential equations by zeros of orthogonal polynomials, J. Comput. Appl. Math. 213 (2008), 488–500.
• [24] L. Zielinski, Eigenvalue asymptotics for a class of Jacobi matrices, Hot topics in operator theory, Theta Ser. Adv. Math., 9, Theta, Bucharest, 2008, 217–229.
• [25] C.H. Ziener, M. Rückl, T. Kampf, W.R. Bauer, H.P. Schlemmer, Mathieu functions for purely imaginary parameters, J. Comput. Appl. Math. 236 (2012), 4513–4524.