In the first part of the paper, we discuss eigenvalue problems of the form
-w"+Pw=Ew with complex potential P and zero boundary conditions at infinity on
two rays in the complex plane. We give sufficient conditions for continuity of
the spectrum when the leading coefficient of P tends to 0. In the second part,
we apply these results to the study of topology and geometry of the real
spectral loci of PT-symmetric families with P of degree 3 and 4, and prove
several related results on the location of zeros of their eigenfunctions.Comment: The main result on singular perturbation is substantially improved,
generalized, and the proof is simplified. 37 pages, 16 figure