A stochastic controlled Schroedinger equation: convergence and robust stability for numerical solutions

In this paper we study the stochastic stability of numerical solutions of a stochastic controlled Schr¨odinger equation. We investigate the boundedness in second moment, the convergence and the stability of the zero solution for this equation, using two new definitions of almost sure exponential robust stability and asymptotic stability, for the Euler-Maruyama numerical scheme. Considering that the diffusion term is controlled, by using the method of Lyapunov functions and the corresponding diffusion operator associated, we apply techniques of X. Mao and A. Tsoi for achieve our task. Finally, we illustrate this method with a problem in Nuclear Magnetic Resonance (NMR).

On quantum analogue of instable oscillating states of an inverted oscillator in external poly-harmonic field

Nonstationary Schroedinger equation (NSE) is solved analytically and numerically to study a phenomenon of dynamical stabilization of the inverted oscillator driven by polyharmonic in time and spatially uniform force with specially chosen phase shifts. It is shown that for Gaussian wave packet asymptotically fitting the initial condition (IC) it occurs temporary delay of the packet center about top of the parabolic potential for about 2 fundamental time periods followed by the center bifurcation.

SCHROEDINGER EQUATION USE FOR INVESTIGATIONS OF PHENOMENA TAKING PLACE IN PLASMA GENERATOR OF GLOW DISCHARGE

The work purpose is development of the theory ensuring creation of an efficient system of fast process control in the plasma generator of a glow discharge and contributing to the development of new techniques and equipment for their realization under conditions of controllable automated tool production. There are used regulations of quantum mechanics lying in that any system can be described by setting in a general case a complex wave function of the kind of . A possibility to find out a charged particle at the time t in some point of the near-cathode area of the closed volume of the plasma generator with the radius-vector was defined by probability density which is presented by a module square of the wave function of . In the course of the investigation carried out there are obtained the following results. The formation of charged particle flows in the plasma generator of a glow discharge has a probabilistic character. The Schroedinger equation use to obtain analytical dependences describing the processes of charged particle flows formation in the plasma generator of a glow charge is the most corresponding as it allows defining a value of their energy depending on the gas technological environment used. The rate change of gas technological environment pumping allows forming a corresponding volume of ions having specified energy and frequency, in the flow taking into account their mass and energy according to the adopted exponential distribution of ion mass in the flow. In automated technological environment having changed a kind of gas technological environment and a rate of its pumping it is possible to obtain predictable results of the impact of glow discharge plasma upon a surface of products worked explaining the effect of defect generation. As a result of high-energy ion bombardment of the surface of products under processing in the plasma generator of the glow discharge there is discovered the presence of a dissipative process with the elements of self-organization. The low-energy ion presence in the flow ensures an ion current transfer which results in the change of chemical and phase structure in the surface volume of material, its modification and reduction of a crystalline structure and also in amorphism on the surface.

SCHROEDINGER EQUATION USE FOR INVESTIGATIONS OF PHENOMENA TAKING PLACE IN PLASMA GENERATOR OF GLOW DISCHARGE

The work purpose is development of the theory ensuring creation of an efficient system of fast process control in the plasma generator of a glow discharge and contributing to the development of new techniques and equipment for their realization under conditions of controllable automated tool production. There are used regulations of quantum mechanics lying in that any system can be described by setting in a general case a complex wave function of the kind of . A possibility to find out a charged particle at the time t in some point of the near-cathode area of the closed volume of the plasma generator with the radius-vector was defined by probability density which is presented by a module square of the wave function of . In the course of the investigation carried out there are obtained the following results. The formation of charged particle flows in the plasma generator of a glow discharge has a probabilistic character. The Schroedinger equation use to obtain analytical dependences describing the processes of charged particle flows formation in the plasma generator of a glow charge is the most corresponding as it allows defining a value of their energy depending on the gas technological environment used. The rate change of gas technological environment pumping allows forming a corresponding volume of ions having specified energy and frequency, in the flow taking into account their mass and energy according to the adopted exponential distribution of ion mass in the flow. In automated technological environment having changed a kind of gas technological environment and a rate of its pumping it is possible to obtain predictable results of the impact of glow discharge plasma upon a surface of products worked explaining the effect of defect generation. As a result of high-energy ion bombardment of the surface of products under processing in the plasma generator of the glow discharge there is discovered the presence of a dissipative process with the elements of self-organization. The low-energy ion presence in the flow ensures an ion current transfer which results in the change of chemical and phase structure in the surface volume of material, its modification and reduction of a crystalline structure and also in amorphism on the surface.

A Comparison of Methods for Determining the Time Step When Propagating with the Lanczos Algorithm

To use the short iterative Lanczos algorithm to solve the time-dependent Schroedinger equation, one must choose, for a given Lanczos space size, a time step. We compare the derivation of the well-known Lubich and Hochbruck time step from SIAM J. Numer. Anal. 34 (1997) 1911 with the a priori time step we proposed in Mohankumar and Carrington (MC) Comput. Phys. Commun., 181 (2010) 1859 and demonstrate that the MC time step is somewhat larger, i.e., that the MC error bound is tighter. In addition, we use the MC approach to derive an error bound and time step for imaginary time propagation. The error bound we derive is much tighter than the error bound of Stewart and Leyk.

Cooperative Motion of Electrons on the Graphene Surface

Introduction. Today, the development of the graphene theory to control its physical and mechanical properties is a relevant objective. The paper deals with the conducting properties of graphene. In particular, the paper investigates the linear law of electron dispersion and traces its corollaries. Materials and Methods. The development of the theory is based on the verified experimental data and on the foundamental principles of the solid body theory and quantum mechanics. The study follows the universal synergetic principle according to which, there have been developed two split-level mathematical models of the quasi-particle motion in graphene on exposure to the electric field. On the macroscopic level, we suggest that graphene should be analyzed as a crystal consisting of three parallel planes. Two of them are electron gas. The remaining one is the main body of the crystal. On the microscopic level, the quasi-particle motion of the electron wave is described through the Schroedinger equation. Results. The study has developed the alternative method for the explanation of the linear dispersion law in graphene on the macroscopic level. Basing on the analysis of the model, the paper provides a hypothesis of the cooperative motion of the electron pairs, which make up a boson particle. The given hypothesis is different from the traditional one. In accordance with the latter, quasi-particles in graphene are Dirac fermions. To prove the hypothesis consilience, the study examines Hall’s effect in grapheme. The linear dispersion law for a pair of electrons is also deduced from the Schroedinger equation. Both the macroscopic and microscopic models are in a reasonable agreement with the experimental data. Discussion and Conclusion. The main result of the research is the development of the multi-level mathematical model which properly features the conducting properties of graphene (linear dispersion law, anomalous Hall effect). The practical relevance consists in revealing the possibility to control the conducting properties of graphene through impacts on electron pairs.