Anharmonic Oscillators with Infinitely Many Real Eigenvalues and PT-Symmetry

We study the eigenvalue problem $-u"+V(z)u=\lambda u$ in the complex plane with the boundary condition that $u(z)$ decays to zero as $z$ tends to infinity along the two rays $\arg z=-\frac{\pi}{2} \pm \frac{2\pi}{m+2}$, where $V(z)=-(iz)^m-P(iz)$ for complex-valued polynomials $P$ of degree at most $m-1\geq 2$. We provide an asymptotic formula for eigenvalues and a necessary and sufficient condition for the anharmonic oscillator to have infinitely many real eigenvalues

Eigenvalues of PT-symmetric oscillators with polynomial potentials

We study the eigenvalue problem $-u^{\prime\prime}(z)-[(iz)^m+P_{m-1}(iz)]u(z)=\lambda u(z)$ with the boundary conditions that $u(z)$ decays to zero as $z$ tends to infinity along the rays $\arg z=-\frac{\pi}{2}\pm \frac{2\pi}{m+2}$, where $P_{m-1}(z)=a_1 z^{m-1}+a_2 z^{m-2}+...+a_{m-1} z$ is a polynomial and integers $m\geq 3$. We provide an asymptotic expansion of the eigenvalues $\lambda_n$ as $n\to+\infty$, and prove that for each {\it real} polynomial $P_{m-1}$, the eigenvalues are all real and positive, with only finitely many exceptions.Comment: 23 pages, 1 figure. v2: equation (14) as well as a few subsequent equations has been changed. v3: typos correcte

The potential (iz)^m generates real eigenvalues only, under symmetric rapid decay conditions

We consider the eigenvalue problems -u"(z) +/- (iz)^m u(z) = lambda u(z), m >= 3, under every rapid decay boundary condition that is symmetric with respect to the imaginary axis in the complex z-plane. We prove that the eigenvalues lambda are all positive real.Comment: 23 pages and 1 figur

A quartic system and a quintic system with fine focus of order 18

AbstractBy using an effective complex algorithm to calculate the Lyapunov constants of polynomial systems En: z˙=iz+Rn(z,z¯), where Rn is a homogeneous polynomial of degree n, in this note we construct two concrete examples, E4 and E5, such that in both cases, the corresponding orders of fine focus can be as high as 18. The systems are given, respectively, by the following ordinary differential equations:E4:z˙=iz+2iz4+izz¯3+5227820723eiθz¯4, where θ∉{kπ±π6,kπ+π2,k∈Z}, andE5:z˙=iz+3z5+20(c+3)9c2−15z4z¯+zz¯4+20(c+3)c29c2−15z¯5, where c is the root between (−3,−5/3) of the equation4155c6−10716c5−63285c4−18070c3+168075c2+205450c+60375=0