Trace formula and Spectral Riemann Surfaces for a class of tri-diagonal matrices

Abstract

For tri-diagonal matrices arising in the simplified Jaynes-Cummings model, we give an asymptotics of the eigenvalues, prove a trace formula and show that the Spectral Riemann Surface is irreducible.

Spectral gaps of Schrödinger operators with periodic singular potentials

“…The next theorem generalizes a series of results about asymptotic behavior of γ n (see and compare Theorem 41 in [13] in the case of L 2 -potentials v).…”

“…They gave upper bounds of the radii of these discs (see Satz 8, Section 1.5 in [56] and p. 87, the last paragraph, in [57]). This is an interesting topic of its own (see [82,85] and [13]). We'll extend this analysis to families of tri-diagonal matrix operators in the paper [1].…”

Convergence Radii for Eigenvalues of Tri-Diagonal Matrices

“…where R 0 λ = (λ − L) −1 , e j is the j th unit vector, and Π is the square centered at n 2 of width 2n. This formula appears in [5] only in the case of α ∈ [0, 1), but its proof therein holds for α < 2 as well. It follows from (1.1) that for each j ∈ N ,…”

“…On the other hand, the set ∆ Rn \ U is at most countable and has no finite accumulation points (see Section 5.1 in [5]).…”

On the spectral surface of a model two-parameter Sturm–Liouville problem

“…We also refer to Djakov and Mityagin [5] for the discussion of a spectral surface generated by infinite tri-diagonal matrices, and Bender and Turbiner [6] for a problem on analytic continuation of eigenvalues.…”