Asymptotics of Resonances Induced by Point Interactions

Abstract

We consider the resonances of the self-adjoint three-dimensional Schrödinger operator with point interactions of constant strength supported on the set X = {xn} N n=1 . The size of X is defined by VX = maxπ∈Π N N n=1 |xn−x π(n) |, where ΠN is the family of all the permutations of the set {1, 2, . . . , N }. We prove that the number of resonances counted with multiplicities and lying inside the disc of radius R behaves asymptotically linearwhere the constant WX ∈ [0, VX ] can be seen as the effective size of X. Moreover, we show that there exist a configuration of any number of points such that WX = VX . Finally, we construct an example for N = 4 with WX < VX , which can be viewed as an analogue of a quantum graph with non-Weyl asymptotics of resonances.

On the multilevel internal structure of the asymptotic distribution of resonances

“…The logic behind this is that the equality s 1 := (s 2 + s 0 )/2 simplifies the formulation of Theorem 4.4. The N-size s N was called in [40] the size of Y and used to define Weyl and non-Weyl types of asymptototics for N H a,Y (·) (see the beginning of Section 4).…”

“…However this theorem give a hint how one can define the structural parameters even if the decomposition of Σ(H) into asymptotic sequences is not available. With this aim we use counting functions in 'shaped strips' similar to that of [47,48] The above definition of Ad log (+∞) is natural because, for H a,Y and for quantum graphs, this limit exists and has a finite value that was studied in [14,13,40,6] in connection with Weyl-type asymptotics of N H (·). For H a,Y and for quantum graphs, the function Ad log : (−∞, +∞] → [0, +∞) is bounded, nondecreasing and satisfies Ad log (µ) = 0 for µ < 0 and Ad log (µ) = Ad log (+∞) for large enough µ (this follows from the definition, results of [13,14,40], and Theorem 3.4).…”

“…Recall that the counting function N H (·) for resonances is defined by N H (R) := #{k ∈ Σ(H) : |k| ≤ R}, (1.2) where #E is the number of elements of a multiset E. The study of the asymptotics of N(R) as R → ∞ for scattering poles associated with compactly supported potentials in R m with odd m ≥ 3 was initiated in [41] (for the relation between the notions of scattering poles and resonances, see [17,56]). This study was continued [53,23,45,11,50] and extended to a geometric scattering (see reviews in [55,17,56]), to quantum graphs [13,14,39,22], and to point interaction Hamiltonians H a,Y [40,6]. For any unbounded set S ⊂ C, one can define a restricted version of the counting function N S H (R) := #{k ∈ S ∪ Σ(H) : |k| ≤ R}.…”

“…In the process of summation of exp-monomials of (2.3) some of the terms may cancel so that, for a certain permutation σ ∈ S N , α(σ) is not a frequency of D. If this is the case, we say that there is frequency cancellation for the pair {a, Y } (for two different examples of frequency cancellation see [40] and Section 4.3).…”

“…For the following considerations let us recall that the m-sizes s m of Y were defined in Section 1 and that the value s N was introduced in [40] and called the size of Y . Let N H a,Y (·) be the resonance counting function defined by (1.2) with H = H a,Y .…”