Widths of certain classes of entire functions

Spectral Asymptotics of Pauli Operators and Orthogonal Polynomials in Complex Domains

“…Asymptotics of the n-widths of the embedding F 2 ⊂ C(Ω), where Ω is a compact set in C, was studied in [13]. Below we repeat the arguments of [13] (with trivial modifications) to obtain the required asymptotics.…”

“…We shall assume B 0 = 2; the general case can be reduced to this one by a linear change of coordinates. Asymptotics of the n-widths of the embedding F 2 ⊂ C(Ω), where Ω is a compact set in C, was studied in [13]. Below we repeat the arguments of [13] (with trivial modifications) to obtain the required asymptotics.…”

“…Next, we observe that the eigenvalues of P 0 vP 0 coincide with the singular numbers of a certain embedding operator (see (4.1)). Using the approach of [13], we relate the singular numbers of this embedding operator to some sequence of orthogonal polynomials in the complex domain (see below). Finally, application of the results of [22] concerning the asymptotics of these orthogonal polynomials leads to (1.9).…”

Eigenvalue clusters of the Landau Hamiltonian in the exterior of a compact domain

“…Take a smooth cut-off function ω ∈ C ∞ 0 (R 2 ) such that ω(x) = 1 in the neighborhood of D. Then we can replace u, v by u 1 = ωu, v 1 = ωv in the r.h.s. of (14). By the local elliptic regularity we have…”

“…In [3], the spectral asymptotics of P q W P q was related to the asymptotics of the singular numbers of the embedding of the Segal-Bargmann space F 2 (see section 4.2 below) into an L 2 space with the weight related to W . Following the technique of [14], in [3] the analysis of this asymptotics is then reduced to the analysis of the sequence of polynomials of a complex variable, orthogonal with respect to the relevant weight. After this, the results of [18] ensure that the asymptotics of these polynomials is determined by the logarithmic capacity of the support of the weight.…”