Exotics of the Schroedinger problem on the line

Unusual studies on the Schroedinger equation on the line were done in recent years. A bouquet of these exotic flowers is tersely surveyed in the present lecture. We recall impedance scattering problems with discontinuous or step-like coefficients, and we review scattering and spectral problems with asymptotically or locally singular potentials, inverse problems with unusual entries, and finally, relations between the spectral problem and trapped water waves. (orig.)