Resonance free regions and non-Hermitian spectral optimization for Schrödinger point interactions

Abstract

Abstract. Resonances of Schrödinger Hamiltonians with point interactions are considered. The main object under the study is the resonance free region under the assumption that the centers, where the point interactions are located, are known and the associated "strength" parameters are unknown and allowed to bear additional dissipative effects. To this end we consider the boundary of the resonance free region as a Pareto optimal frontier and study the corresponding optimization problem for resonances. It is shown that upper logarithmic bound on resonances can be made uniform with respect to the strength parameters. The necessary conditions on optimality are obtained in terms of first principal minors of the characteristic determinant. We demonstrate the applicability of these optimality conditions on the case of 4 equidistant centers by computing explicitly the resonances of minimal decay for all frequencies. This example shows that a resonance of minimal decay is not necessarily simple, and in some cases it is generated by an infinite family of feasible resonators.

On the multilevel internal structure of the asymptotic distribution of resonances

“…The first goal of the present paper is to prove that Σ(H) essentially decomposes into a finite number of sequences with logarithmic asymptotics for an arbitrary point interaction Hamiltonian H = H a,Y corresponding to the formal differential expression − ∆u(x) + " N j=1 µ(a j )δ(x − y j )u(x)", x ∈ R 3 , N ∈ N, (1.1) with a finite number N ≥ 2 of interaction centers y j ∈ R 3 and the tuple a = (a j ) N j=1 ∈ C N of 'strength' parameters (see [4,2,7,5] and Section 2 for the definition of H a,Y ). The second goal is to connect the leading parameters of these asymptotic sequences to the geometry of the set Y = {y j } N j=1 (Section 4).…”

“…The Krein-type formula (2.1) for the difference of the perturbed and unperturbed resolvents of operators H a,Y and −∆ can be used as a definition of H a,Y (see [2]). For other equivalent definitions of H and for the meaning of µ(a j ) and a j in (1.1), we refer to [1,2,7] in the case a j ∈ R, and to [3,5] in the case a j ∈ R. Note that, in the case a ∈ R N , the operator H a,Y is self-adjoint in L 2 C (R 3 ); and in the case a ∈ (C − ∪ R) N , H a,Y is closed and maximal dissipative (in the sense of [20], or in the sense that iH a,Y is maximal accretive).…”

On real resonances for three-dimensional Schrödinger operators with point interactions

“…A zero resonance may occur, and one of us [38] qualified this possibility in terms of a convenient low-energy resolvent expansion which is at the basis of our definition 2.5 below. Complex resonances (Imz < 0) have been investigated by Albeverio and Karabash [7,8,9] and Lipovský and Lotoreichik [34], using techniques on the localisation of zeroes of exponential polynomials, and turn out to lie mostly within certain logarithmic strips in the complex z-plane. Real positive resonances (thus, z ∈ R \ {0}) have been recently excluded by Galtbayar and Yajima [19], and implicitly also by Goloshchapova, Malamud, and Zastavnyi [22,23].…”

Asymptotics of Resonances Induced by Point Interactions

“…Resonances of H α,X were discussed in the monograph [1] and in several more recent publications, e.g. [2][3][4], see also the review [5] and the references therein.…”

“…To this aim, we define the size of X by 2) where Π N is the family of all the permutations of the set {1, 2, . .…”