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

Trace Formulas for Non-Self-Adjoint Periodic Schr\"odinger Operators and some Applications

Recently, a trace formula for non-self-adjoint periodic Schr\"odinger operators in $L^2(\mathbb{R})$ associated with Dirichlet eigenvalues was proved in [9]. Here we prove a corresponding trace formula associated with Neumann eigenvalues. In addition we investigate Dirichlet and Neumann eigenvalues of such operators. In particular, using the Dirichlet and Neumann trace formulas we provide detailed information on location of the Dirichlet and Neumann eigenvalues for the model operator with the potential $Ke^{2ix}$, where $K\in\mathbb{C}$.Comment: 26 pages, no figure

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

Neutrino Flavor Ratio on Earth and at Astrophysical Sources

We present the reconstruction of neutrino flavor ratios at astrophysical sources. For distinguishing the pion source and the muon-damped source to the 3$\sigma$ level, the neutrino flux ratios, $R\equiv\phi(\nu_\mu)/(\phi(\nu_e)+\phi(\nu_\tau))$ and $S\equiv\phi(\nu_e)/\phi(\nu_\tau)$, need to be measured in accuracies better than 10%.Comment: 3 pages, 8 figures. Talk presented by T.C. Liu in ERICE 2009, Sicily