In this work we discuss 3D selfconsistent solution of Poisson and Schrödinger equations for electrostatically formed quantum dot. 3D simulations give detailed insight into the energy spectrum of the device and allow us to find values of respective voltages ensuring given number of electrons in the dot. We performed calculations for fully 3D potential and apart from that calculations for the same potential separated into two independent parts, i.e. regarding to the plane of 2DEG and to the direction perpendicular to the meant plane. We found that calculations done for the two independent parts of the potential give good information about quantum dot properties and they are much faster compared to fully 3D simulations.

In this paper we have studied the driftless control system on a Lie group which arises due to the invariance of Black-Scholes equation by conformal transformations. These type of studies are possible as Black-Scholes equation can be mapped to one dimensional free Schrödinger equation. In particular we have studied the controllability, optimal control of the resulting dynamics as well as stability aspects of this system.We have also found out the trajectories of the states of the system through two unconventional integrators along with conventional Runge-Kutta integrator.