Bounds for Schrödinger Operators on the Half-Line Perturbed by Dissipative Barriers

Abstract

We consider Schrödinger operators of the form $$H_R = - \,\text {{d}}^2/\,\text {{d}}x^2 + q + i \gamma \chi _{[0,R]}$$ H R = - d 2 / d x 2 + q + i γ χ [ 0 , R ] for large $$R>0$$ R > 0 , where $$q \in L^1(0,\infty )$$ q ∈ L 1 ( 0 , ∞ ) and $$\gamma > 0$$ γ > 0 . Bounds for the maximum magnitude of an eigenvalue and for the number of eigenvalues are proved. These bounds complement existing general bounds applied to this system, for sufficiently large R.

Schrödinger operators with complex sparse potentials

“…Hence, (10) shows that, at the scale considered here, a substantial fraction of resonances are actual eigenvalues. In the special case of a step potential Stepanenko [76] proved that the total number of eigenvalues is bounded by N 2 / log N . The lower bound (10) shows that this is sharp up to constants; this was observed independently by Stepanenko [75].…”

“…Since we are free to choose κ, set κ = ±1 + iǫσ with σ = 0, which is will yield an eigenvalue with Re E = 1 + O(ǫ) and |Im E| = O(ǫ). Going back to (76) we see that we must have…”

“…It is quickly checked that this is consistent with the bound of Abramov et al [1] since V L 1 log 1 ǫ and |E| ≍ 1. In view of (76) we may write e 2iκR = ǫu, where u = Ce 2iRe κR . Using the Taylor approximation sec 2 (κR) = −4ǫu(1 + 2ǫu + O(ǫ 2 )) (78) we obtain from (74) that…”