OPTIMALITY OF INCREASING STABILITY FOR AN INVERSE BOUNDARY VALUE PROBLEM

In this work we study the optimality of increasing stability of the inverse boundary value problem (IBVP) for the Schr\" odinger equation. The rigorous justification of increasing stability for the IBVP for the Schr\" odinger equation were established by Isakov [Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), pp. 631-640] and by Isakov et al. [Inverse Problems and Applications, Contemp. Math. 615, American Math Society, Providence, RI, 2014, pp. 131-141]. In [Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), pp. 631-640] and [Inverse Problems and Applications, Contemp. Math. 615, American Math Society, Providence, RI, 2014, pp. 131-141], the authors showed that the stability of this IBVP increases as the frequency increases in the sense that the stability estimate changes from a logarithmic type to a H\"older type. In this work, we prove that the instability changes from an exponential type to a H\" older type when the frequency increases. This result verifies that results in [Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), pp. 631-640] and [Inverse Problems and Applications, Contemp. Math. 615, American Math Society, Providence, RI, 2014, pp. 131-141] are optimal.