arXiv subjects
Publications and source records attributed to Marvin R. Schulz.
We prove that the best constant in the one-dimensional Lieb-Thirring inequality with exponent one is $4/(3\sqrt{3}π)$, confirming the Lieb-Thirring conjecture in this case. Moreover, we extend the inequality to operator-valued potentials and obtain the bound $L_{1,d}\leq(2/\sqrt3)L_{1,d}^{\mathrm{cl}}$ in every dimension.
For Gamow's liquid drop model, we improve the nonexistence bound of Frank-Killip-Nam from 8 to 7.5. The key additional input is a geometric perimeter inequality arising from the capillarity problem. This yields a quantitative gain in the averaged slicing argument and shows that the variational problem admits no minimizer for $A\geq 7.5$.
We prove that the Schrödinger operator describing four particles in two dimensions, interacting solely through short-range three-body forces, can possess infinitely many bound states. This holds under the assumption that each three-body subsystem has a virtual level at zero energy. Our result establishes an analog of the Efimov effect for such four-particle systems in two dimensions.
We study a system of three bosons interacting with short-range potentials which can move along three different lines. Two of these lines are parallel to each other within one plane. The third line is constrained to a plane perpendicular to the first one. Recently it was predicted in physics literature that such a system exhibits the so-called confinement induced Efimov effect. We prove that this prediction is not correct by showing that this system has at most finitely many bound-states.
In this paper the determination of material properties such as Sieverts' constant (solubility) and diffusivity (transport rate) via so-called gas release experiments is discussed. In order to simulate the time-dependent hydrogen fluxes and concentration profiles efficiently, we make use of an analytical method, namely we provide an analytical solution for the corresponding diffusion equations on a cylindrical specimen and a cylindrical container for three boundary conditions. These conditions occur in three phases -- loading phase, evacuation phase and gas release phase. In the loading phase the specimen is charged with hydrogen assuring a constant partial pressure of hydrogen. Then the gas will be quickly removed by a vacuum pump in the second phase, and finally in the third time interval, the hydrogen is released from the specimen to the gaseous phase, where the pressure increase will be measured by an equipment which is attached to the cylindrical container. The investigated diffusion equation in each phase is a simple homogeneous equation, but due to the complex time-dependent boundary conditions which include the Sieverts' constant and the pressure, we transform the homogeneous equations to the non-homogeneous ones with a zero Dirichlet boundary condition. Compared with the time consuming numerical methods our analytical approach has an advantage that the flux of desorbed hydrogen can be explicitly given and therefore can be evaluated efficiently. Our analytical solution also assures that the time-dependent boundary conditions are exactly satisfied and furthermore that the interaction between specimen and container is correctly taken into account.