The Distribution Of Resonances And Inverse Resonance Problem For One-dimensional Schr(?)dinger Operators With Compactly Supported Potential

In quantum physics, resonances correspond to quasi-stationary states that only exist for a finite time, and their existing time is proportional to the inverse of the imaginary part of the resonance and energy is proportional to the real part of the resonance. Eigenvalues, however, represent the states in which the particles can be permanently localized. Resonances and eigenvalues are directly observable in spectrometers. The inverse resonance problems aims to recover the potential by the resonances and eigenvalues. In this paper, we first discuss the inverse resonance problem with separated boundary condition. With the method of trans-forming the resonance problem to a eigenvalue problem with eigen-parameter in the bound-ary condition, we obtain the uniqueness theorem of the inverse resonance problem, which means that the eigenvalues and resonances can uniquely determine the potential function, generalize the results of the resonance problem with Dirichlet boundary condition developed by W. Rundell and P. Sacks in [31]. We also discuss the distribution of resonances with sep-arated boundary condition, and obtain the asymptotic formula of resonances, and finally we give a new proof of an important property about the distribution of antibound states, and in the proof we get the alternativity of antibound states and the eigenvalues of some eigenvalue problems.