Inverse resonance problems for the Schrödinger operator on the real line with mixed given data

Xiao-Chuan Xu, Chuan-Fu Yang

Published 2017-03-06Version 1

In this work, we study inverse resonance problems for the Schr\"odinger operator on the real line with the potential supported in $[0,1]$. In general, all eigenvalues and resonances can not uniquely determine the potential (Zworski, SIAM J. Math. Anal. 32, 2001). (i) It is shown that if the potential is known a priori on $[0,1/2]$, then the unique recovery of the potential on the whole interval from all eigenvalues and resonances is valid. (ii) If the potential is known a priori on $[0,a]$, then for the case $a>1/2$, infinitely many eigenvalues and resonances can be missing for the unique determination of the potential, and for the case $a<1/2$, all eigenvalues and resonances plus a part of so-called sign-set can uniquely determine the potential. (iii) It is also shown that all eigenvalues and resonances, together with a set of logarithmic derivative values of eigenfunctions and wave-functions at $1/2$, can uniquely determine the potential.