Inverse resonance scattering for Dirac operators on the half-line

Abstract

We consider Schrödinger equations with linearly energy-depending potentials which are compactly supported on the half-line. We first provide estimates of the number of eigenvalues and resonances for such complex-valued potentials under suitable regularity assumptions. Then, we consider a specific class of energy-dependent Schrödinger equations without eigenvalues, defined with Miura potentials and boundary conditions at the origin. We solve the inverse resonance problem in this case and describe sets of iso-resonance potentials and boundary condition parameters. Our strategy consists in exploiting a correspondance between Schrödinger and Dirac equations on the half-line. As a byproduct, we describe similar sets for Dirac operators and show that the scattering problem for Schrödinger equation or Dirac operator with an arbitrary boundary condition can be reduced to the scattering problem with the Dirichlet boundary condition. 1.1. The potential class P: Eigenvalues and resonances estimates. Let γ > 0 and define the class of potential perturbations P := {(p, q) ∈ W 1,1 (R + ) × L 1 (R + ) | max supp(|p| + |q|) = γ}.

Inverse resonance scattering for Dirac operators on the half-line

“…There are a lot papers about resonances in the different setting, see articles [8,12,19,30,33] and the book [5] and the references therein. The inverse resonance problem for Schrödinger operators with compactly supported potentials was solved in [21] for the case of the real line and in [19] for the case of the half line.…”

Eigenvalues of Schrödinger operators on finite and infinite intervals

“…see [19] or Lemma 3.4.1 in [25]. Note that if the operator 𝐻 01 has eigenvalue 𝜏 1 ⩾ 0, then from (2.16), (2.4) and (1.6) we deduce that the operator 𝑇 does not have any eigenvalue.…”

“…Proof of Theorem 1.1 i-ii) Dirichlet and Neumann boundary conditions. Recall the identity from [19] (see also Lemma 3.4.1 in [25]):…”

“…We recall the basic result about inverse resonance problem (including the characterization) for compactly supported potentials from [19].…”

“…3. □In order to discuss resonances we define the class of all Jost functions from[19]: Definition J). By  we mean the class of all entire functions 𝑓 having the form𝑓(𝑘) = 1 + F(𝑘) − F(0) 2𝑖𝑘 , 𝑘 ∈ ℂ,(2.24)where F(𝑘) = ∫ 𝐹(𝑥)𝑒 2𝑖𝑥𝑘 𝑑𝑥 is the Fourier transformation of 𝐹 ∈  and a set of zeros 𝐾 = { 𝑘 𝑛 , 𝑛 ∈ ℕ } (counted with multiplicity) of 𝑓 satisfy:1) The set 𝐾 does not have zeros from ℝ ⧵ {0} and has possibly one simple zero at 0.2) The set 𝐾 has a finite number of elements 𝑘 1 , … , 𝑘 𝑚 from ℂ + , which are simple, belong to 𝑖ℝ + and if they are labeled by|𝑘 1 | > |𝑘 2 | > ⋯ > |𝑘 𝑚 | > 0,then the intervals on [0, −𝑖∞) defined by 𝐼 𝑗 = ( − 𝑘 𝑗 , −𝑘 𝑗+1 ) , 𝑗 ∈ ℕ 𝑚−1 , and 𝐼 𝑚 = ( − 𝑘 𝑚 , 0 ] satisfy −𝑘 𝑗 ∉ 𝐾, 𝑗 ∈ ℕ 𝑚 ∶= {1, 2, … , 𝑚},…”