Eigenvalue estimates for non-selfadjoint Dirac operators on the real line

We show that the non-embedded eigenvalues of the Dirac operator on the real line with non-Hermitian potential $V$ lie in the disjoint union of two disks in the right and left half plane, respectively, provided that the $L^1-norm$ of $V$ is bounded from above by the speed of light times the reduced Planck constant. An analogous result for the Schr\"odinger operator, originally proved by Abramov, Aslanyan and Davies, emerges in the nonrelativistic limit. For massless Dirac operators, the condition on $V$ implies the absence of nonreal eigenvalues. Our results are further generalized to potentials with slower decay at infinity. As an application, we determine bounds on resonances and embedded eigenvalues of Dirac operators with Hermitian dilation-analytic potentials

Enhanced hydrogen peroxide generation accompanies the beneficial bioenergetic effects of methylene blue in isolated brain mitochondria

The redox dye methylene blue (MB) is proven to have beneficial effects in various models of neurodegenerative diseases. Here we investigated the effects of MB (100 nM, 300 nM, and 1 μM) on key bioenergetic parameters and on H2O2 production/elimination in isolated guinea pig brain mitochondria under normal as well as respiration-impaired conditions. As measured by high-resolution Oxygraph the rate of resting oxygen consumption was increased, but the ADP-stimulated respiration was unaffected by MB with any of the substrates (glutamate malate, succinate, or α-glycerophosphate) used for supporting mitochondrial respiration. In mitochondria treated with inhibitors of complex I or complex III MB moderately but significantly increased the rate of ATP production, restored ΔΨm, and increased the rate of Ca2+ uptake. The effects of MB are consistent with transferring electrons from upstream components of the electron transport chain to cytochrome c, which is energetically favorable when the flow of electrons in the respiratory chain is compromised. On the other hand, MB significantly increased the production of H2O2 measured by Amplex UltraRed fluorimetry under all conditions, in resting, ATP-synthesizing, and respiration-impaired mitochondria, with each substrate combination supporting respiration. Furthermore, it also decreased the elimination of H2O2. Generation of H2O2 without superoxide formation, observed in the presence of MB, is interpreted as a result of reduction of molecular oxygen to H2O2 by the reduced MB. The elevated generation and impaired elimination of H2O2 should be considered for the overall oxidative state of mitochondria treated with MB

Dissipative Properties of Systems Composed of High-Loss and Lossless Components

We study here dissipative properties of systems composed of two components one of which is highly lossy and the other is lossless. A principal result of our studies is that all the eigenmodes of such a system split into two distinct classes characterized as high-loss and low-loss. Interestingly, this splitting is more pronounced the higher the loss of the lossy component. In addition, the real frequencies of the high-loss eigenmodes can become very small and even can vanish entirely, which is the case of overdamping.Comment: Revision; Improved exposition and typos corrected; 45 pages, 4 figure

Membrane potential and delta pH dependency of reverse electron transport-associated hydrogen peroxide production in brain and heart mitochondria

Succinate-driven reverse electron transport (RET) is one of the main sources of mitochondrial reactive oxygen species (mtROS) in ischemia-reperfusion injury. RET is dependent on mitochondrial membrane potential (Δψm) and transmembrane pH difference (ΔpH), components of the proton motive force (pmf); a decrease in Δψm and/or ΔpH inhibits RET. In this study we aimed to determine which component of the pmf displays the more dominant effect on RET-provoked ROS generation in isolated guinea pig brain and heart mitochondria respiring on succinate or α-glycerophosphate (α-GP). Δψm was detected via safranin fluorescence and a TPP+ electrode, the rate of H2O2 formation was measured by Amplex UltraRed, the intramitochondrial pH (pHin) was assessed via BCECF fluorescence. Ionophores were used to dissect the effects of the two components of pmf. The K+/H+ exchanger, nigericin lowered pHin and ΔpH, followed by a compensatory increase in Δψm that led to an augmented H2O2 production. Valinomycin, a K+ ionophore, at low [K+] increased ΔpH and pHin, decreased Δψm, which resulted in a decline in H2O2 formation. It was concluded that Δψm is dominant over ∆pH in modulating the succinate- and α-GP-evoked RET. The elevation of extramitochondrial pH was accompanied by an enhanced H2O2 release and a decreased ∆pH. This phenomenon reveals that from the pH component not ∆pH, but rather absolute value of pH has higher impact on the rate of mtROS formation. Minor decrease of Δψm might be applied as a therapeutic strategy to attenuate RET-driven ROS generation in ischemia-reperfusion injury

A Survey on the Krein-von Neumann Extension, the corresponding Abstract Buckling Problem, and Weyl-Type Spectral Asymptotics for Perturbed Krein Laplacians in Nonsmooth Domains

In the first (and abstract) part of this survey we prove the unitary equivalence of the inverse of the Krein--von Neumann extension (on the orthogonal complement of its kernel) of a densely defined, closed, strictly positive operator, $S\geq \varepsilon I_{\mathcal{H}}$ for some $\varepsilon >0$ in a Hilbert space $\mathcal{H}$ to an abstract buckling problem operator. This establishes the Krein extension as a natural object in elasticity theory (in analogy to the Friedrichs extension, which found natural applications in quantum mechanics, elasticity, etc.). In the second, and principal part of this survey, we study spectral properties for $H_{K,\Omega}$, the Krein--von Neumann extension of the perturbed Laplacian $-\Delta+V$ (in short, the perturbed Krein Laplacian) defined on $C^\infty_0(\Omega)$, where $V$ is measurable, bounded and nonnegative, in a bounded open set $\Omega\subset\mathbb{R}^n$ belonging to a class of nonsmooth domains which contains all convex domains, along with all domains of class $C^{1,r}$, $r>1/2$.Comment: 68 pages. arXiv admin note: extreme text overlap with arXiv:0907.144