Some Remarks on Upper Bounds for Weierstrass Primary Factors and Their Application in Spectral Theory

Abstract

Abstract. We study upper bounds on Weierstrass primary factors and discuss their application in spectral theory. One of the main aims of this note is to draw attention to works of Blumenthal and Denjoy from 1910, but we also provide some new results and some numerical computations of our own.

Eigenvalues of compactly perturbed operators via entropy numbers

“…(1) p (as follows from results in [15]) and K p,∞ ≤ K p for K ∈ S (e) p (X), we see that even for perturbations in the smaller class S (11) is stronger than (12). Moreover, for perturbations K from the still smaller class S…”

“…Remark 3. The idea to apply Carl's inequality (15) with an m depending on n has also been used in [7] in a non-perturbative setting.…”

Eigenvalue bounds for Stark operators with complex potentials

“…Moreover, there exists a constant C n > 0 (see [9]) depending only on n such that | det n (I + X)| e Cn X n Sn , ∀X ∈ S n .…”

$L_p$-spectrum and Lieb-Thirring inequalities for Schrödinger operators on the hyperbolic plane

“…I , where µ r denotes the eigenvalue constant of I and Γ r is a universal rdependent constant, see [28], (p3) d(u) = 0 iff u ∈ σ(A + K), (p4) if u ∈ (A) ∩ σ d (A + K), then its algebraic multiplicity (as an eigenvalue) coincides with its order as a zero of d.…”