On Eigenvalues of the Schrodinger Operator with an Even Complex-Valued Polynomial Potential

In this paper, we generalize several results in the article "Analytic continuation of eigenvalues of a quartic oscillator" of A. Eremenko and A. Gabrielov [4]. We consider a family of eigenvalue problems for a Schrodinger equation with even polynomial potentials of arbitrary degree d with complex coefficients, and k < (d + 2)/2 boundary conditions. We show that the spectral determinant in this case consists of two components, containing even and odd eigenvalues respectively. In the case with k = (d + 2)/2 boundary conditions, we show that the corresponding parameter space consists of infinitely many connected components.