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

Neuronal glucose transporter isoform 3 deficient mice demonstrate features of autism spectrum disorders.

Neuronal glucose transporter (GLUT) isoform 3 deficiency in null heterozygous mice led to abnormal spatial learning and working memory but normal acquisition and retrieval during contextual conditioning, abnormal cognitive flexibility with intact gross motor ability, electroencephalographic seizures, perturbed social behavior with reduced vocalization and stereotypies at low frequency. This phenotypic expression is unique as it combines the neurobehavioral with the epileptiform characteristics of autism spectrum disorders. This clinical presentation occurred despite metabolic adaptations consisting of an increase in microvascular/glial GLUT1, neuronal GLUT8 and monocarboxylate transporter isoform 2 concentrations, with minimal to no change in brain glucose uptake but an increase in lactate uptake. Neuron-specific glucose deficiency has a negative impact on neurodevelopment interfering with functional competence. This is the first description of GLUT3 deficiency that forms a possible novel genetic mechanism for pervasive developmental disorders, such as the neuropsychiatric autism spectrum disorders, requiring further investigation in humans