Narrow Results

We consider a Schrödinger operator with a Hermitian $2\times 2$ matrix-valued potential which is lattice-periodic and can be diagonalized smoothly on the whole ${ℝ}^n.$ In the case of the potential taking its minimum only on the lattice, we prove that the well-known semiclassical asymptotics of the first band spectrum for a scalar potential remains valid for our model.

We investigate Bose-Einstein condensation of a gas of non-interacting Bose particles moving in the background of a periodic lattice of delta functions. In the one-dimensional case, where one has no condensation in the free case, we showed that this property persist also in the presence of the lattice. In addition we formulated some conditions on the spectral functions which would allow for condensation.

In the optimization of the quadrupole mass spectrometer (QP-MS), the understanding of ion motion in terms of the phase space (the combined representation of the trajectory coordinate and momentum) is useful. The phase space representation gives an “ensemble” behavior of ions inside the filter. Even though each ion trajectory does not have the RF periodicity of the applied field, the phase space evolution does. It is only when appropriate ensemble ions are considered together that a proper QP filter characterization is possible. We here report a new framework for the phase space calculation of the QP-MS. The Mathieu–Hill equation is first solved for “complex eigen-trajectory” that has pseudo RF periodicity (the Bloch function). It is then shown that the acceptance phase space can be derived from the Bloch function without a need to calculate each ion trajectory. The ensemble behavior of ions can be estimated from a single Bloch function by expressing the trajectory phase space point by the complex amplitude (coefficient) of the Bloch function. The application of the Bloch function method to the auxiliary (pre-rod) field reveals that the ion injection efficiency may significantly be improved by optimizing the number of RF periods the ions spend in the pre-rod section.

We study the dynamics of overdamped systems in a symmetric periodic potential against an external load. Using numerical and analytical methods, we show that a symmetric driving signal cannot rectify these systems while an asymmetric one can rectify them, causing its unidirectional rotation. We consider the case where the driving signal is multiharmonic. We find that the mean velocity of the system increases to get the highest value, by appropriate tuning of parameters.

The main purpose of this paper is to establish the existence of solutions for the semilinear Schrödinger equation