If spectrum of a Schr\"{o}dinger oparator with a non-Hermitian potential contains a spectral singularity (SS), the latter requires exact matching of the parameters characterizing the potential. We provide a necessary and sufficient condition for a potential to have a SS at a given wavelength. It is shown that potentials with SS at prescribed wavelengths can be obtained by a simple and effective procedure. In particular, the developed approach allows one to obtain potentials with several SSs or with SSs of the second order and potentials obeying a given symmetry, say, PT-symmetric potentials. Also, the problem can be solved when it is required to obtain a potential obeying a given symmetry, say, $\mathcal{PT}-$symmetric potential. We illustrate all opportunities with examples. We also describe splitting of the SSs of the second order, under change of the potential parameters, and discuss possibilities of experimental observation of SSs of different orders. 查看全文>>