Singular perturbation of polynomial potentials with applications to $PT$-symmetric families

We discuss eigenvalue problems of the form $-w''+Pw=\lambda w$ with complex polynomial potential $P(z)=tz^d+\ldots$, where $t$ is a parameter, with zero boundary conditions at infinity on two rays in the complex plane. In the first part of the paper we give sufficient conditions for continuity of the spectrum at $t=0$. 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.