2019/12/08 by V. L. Geynts, Geynts, V. L., А. А. Шкаликов +1
Mathematics · #Spectral Theory in Mathematical Physics #Numerical methods in inverse problems #Differential Equations and Boundary Problems
paper · pdf · doi:10.48550/arxiv.1912.03678
We work with the Schr" odinger equation Hq y = -y'' + q(x)y = z2y, x∈ [0,∞), where q∈ L1((0,∞), xdx), and asssume that the corresponding operator Hq is defined by the Dirihlet condition y(0) = 0 The function ψ(z) = y(0,z) where y(x,z) is the Jost solution of the above equation is analytic in the whole complex plane, provided that the support of the potential q is finite. The zeros of ψ are called the resonances. It is known that q is uniquely determined by the sequence of resonances. Using only finitely many resonances lying in the disk |z|≤ r we can recover the potential q with accuracy ε(r)→ 0 as r → ∞. The main result of the paper is the estimate ε(r) ≤ Cr-α with some constants C and α>0 which are defined by a priori information about the potential q.