TY - GEN
T1 - Eigenvalues of Tridiagonal Hermitian Toeplitz Matrices with Perturbations in the Off-diagonal Corners
AU - Grudsky, Sergei M.
AU - Maximenko, Egor A.
AU - Soto-González, Alejandro
N1 - Publisher Copyright:
© 2021, The Author(s), under exclusive license to Springer Nature Switzerland AG.
PY - 2021
Y1 - 2021
N2 - In this paper we study the asymptotic behavior of the eigenvalues of Hermitian Toeplitz matrices with the entries 2, −1, 0, …, 0, −α in the first column. Notice that the generating symbol depends on the order n of the matrix. This matrix family is a particular case of periodic Jacobi matrices. If |α|≤ 1, then the eigenvalues belong to [0, 4] and are asymptotically distributed as the function g(x) = 4 sin 2(x∕ 2 ) on [0, π]. The situation changes drastically when |α| > 1 and n tends to infinity. For this case, we prove that the two extreme eigenvalues (the minimal and the maximal one) lay out of [0, 4] and converge rapidly to certain limits determined by the value of α, whilst all others belong to [0, 4] and are asymptotically distributed as g. In all cases, we derive asymptotic formulas for the eigenvalues and transform the characteristic equation to a form convenient to solve by numerical methods.
AB - In this paper we study the asymptotic behavior of the eigenvalues of Hermitian Toeplitz matrices with the entries 2, −1, 0, …, 0, −α in the first column. Notice that the generating symbol depends on the order n of the matrix. This matrix family is a particular case of periodic Jacobi matrices. If |α|≤ 1, then the eigenvalues belong to [0, 4] and are asymptotically distributed as the function g(x) = 4 sin 2(x∕ 2 ) on [0, π]. The situation changes drastically when |α| > 1 and n tends to infinity. For this case, we prove that the two extreme eigenvalues (the minimal and the maximal one) lay out of [0, 4] and converge rapidly to certain limits determined by the value of α, whilst all others belong to [0, 4] and are asymptotically distributed as g. In all cases, we derive asymptotic formulas for the eigenvalues and transform the characteristic equation to a form convenient to solve by numerical methods.
KW - Asymptotic expansion
KW - Eigenvalue
KW - Periodic Jacobi matrix
KW - Perturbation
KW - Toeplitz matrix
KW - Tridiagonal matrix
UR - http://www.scopus.com/inward/record.url?scp=85116795943&partnerID=8YFLogxK
U2 - 10.1007/978-3-030-77493-6_11
DO - 10.1007/978-3-030-77493-6_11
M3 - Contribución a la conferencia
AN - SCOPUS:85116795943
SN - 9783030774929
T3 - Springer Proceedings in Mathematics and Statistics
SP - 179
EP - 202
BT - Operator Theory and Harmonic Analysis, OTHA 2020
A2 - Karapetyants, Alexey N.
A2 - Kravchenko, Vladislav V.
A2 - Liflyand, Elijah
A2 - Malonek, Helmuth R.
PB - Springer
T2 - International Scientific Conference on Modern Methods, Problems and Applications of Operator Theory and Harmonic Analysis, OTHA 2020
Y2 - 26 April 2020 through 30 April 2020
ER -