On the multilevel internal structure of the asymptotic distribution of resonances

Abstract

We prove that the asymptotic distribution of resonances has a multilevel internal structure for the following classes of Hamiltonians H: Schrödinger operators with point interactions in R 3 , quantum graphs, and 1-D photonic crystals. In the case of N ≥ 2 point interactions, the set of resonances Σ(H) essentially consists of a finite number of sequences with logarithmic asymptotics. We show how the leading parameters µ of these sequences are connected with the geometry of the set Y = {y j } N j=1 of interaction centers y j ∈ R 3 . The minimal parameter µ min corresponds to the sequences with 'more narrow' and so more observable resonances. The asymptotic density of such narrow resonances can be expressed via the multiplicity of µ min , which occurs to be connected with the symmetries of Y and naturally introduces a finite number of classes of configurations of Y . In the case of quantum graphs and 1-D photonic crystals, the decomposition of Σ(H) into a finite number of asymptotic sequences is proved under additional commensurability conditions. To address the case of a general quantum graph, we introduce families of special asymptotic density functions for two classes of strips in C. The obtained results and effects are compared with those of obstacle scattering.

Asymptotics of random resonances generated by a point process of delta-interactions

“…Then the multiset of resonances Σ(H Y ) has the global structure of a finite number of sequences going to ∞ with prescribed asymptotics [7]. Namely, there exists a sequence (In [7] a more precise asymptotic formula is given, but we do not need it in the present paper.)…”

“…2 and is a countable sequence with an accumulation point at ∞. A more detailed description of the set of zeros of this transcedental function of z in terms of α and ℓ(ω) can be found in [5,3,7] (see also Section 4).…”

“…Finally, assume that 2 ≤ #Υ(ω) < ∞ and Υ(ω) is simple. Then it is easy to see that, in the Leibniz expansion of D Υ(ω) , the exp-monomials (2.5) with positive frequencies cannot completely cancel each other [6,7]. So, additionally to the zero frequency (2.7), the exponential polynomial D Υ(ω) (•) has at least one positive frequency.…”

“…We use in the present paper the terminology of [7] and say that Ad(H Y ) := C/π is the total asymptotic density of resonances of H Y . This is motivated by the equality (see (3.1))…”

“…In Section 4, we consider the fine structure of asymptotical behaviour of the resonance point process Σ(H Υ ) near k = ∞. This structure can be seen with the use of the logarithmic (asymptotic) density function Ad log (h) [7], i.e., the normalized density corresponding to semi-logarithmic strips {−h ln(| Re z| + 1) ≤ Im z}, which is defined by…”