Convergence Radii for Eigenvalues of Tri-Diagonal Matrices

Abstract

Abstract. Consider a family of infinite tri-diagonal matrices of the form L + zB, where the matrix L is diagonal with entries L kk = k 2 , and the matrix B is off-diagonal, with nonzero entries B k,k+1 = B k+1,k = k α , 0 ≤ α < 2. The spectrum of L + zB is discrete. For small |z| the n-th eigenvalue En(z), En(0) = n 2 , is a well-defined analytic function. Let Rn be the convergence radius of its Taylor's series about z = 0. It is proved that Rn ≤ C(α)n 2−α if 0 ≤ α < 11/6.

Spectral gaps of Schrödinger operators with periodic singular potentials

“…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].…”

Local form-subordination condition and Riesz basisness of root systems

“…Proof. Formulas for λ (2) n and φ (1) n follow from (3.15) and (3.18), respectively, by the calculation of residues. To derive the formula for λ (1) n , we can use (3.25); like e.g.…”

“…Formulas for λ (2) n and φ (1) n follow from (3.15) and (3.18), respectively, by the calculation of residues. To derive the formula for λ (1) n , we can use (3.25); like e.g. in [11,Lemmas 8,9], it is important to integrate before taking the trace (or norm in the proof of Theorem 3.2) since all but one term in (3.24) are zero after the integration.…”

“…Since |b V (ψ m , ψ n )| ≤ V L 1 (−l,l) /l, see (5.13), the condition (1.10) is satisfied with α = 0. Notice that the classical formula is recovered by using Riemann-Lebesgue lemma, namely, the first correction λ (1) n to eigenvalues of A, see Theorem 3.2, reads…”

“…Some of the conclusions in Sections 5.2, 5.3 are certainly not new, see e.g. [1] for eigenvalue analysis of tridiagonal matrices, [12, Chap.XIX.3] or [24] for Riesz basis property of perturbations of −d 2 /dx 2 in boundary conditions or [8,9,10] for perturbations by singular potentials, nevertheless, the goal of these sections is to demonstrate the flexibility of our approach. For instance, −d 2 /dx 2 with an infinite number of complex δ-interactions can be treated with the same amount of effort as the perturbation by a bounded potential.…”