Method of intermediate problems in the theory of Gaussian quantum dots placed in a magnetic field

Abstract

Applicability of the method of intermediate problems to the investigation of the energy eigenvalues and eigenstates of a quantum dot (QD) formed by a Gaussian confining potential in the presence of an external magnetic field is discussed. Being smooth at the QD boundaries and of finite depth and range, this potential can only confine a finite number of excess electrons thus forming a realistic model of a QD with smooth interface between the QD and its embedding environment. It is argued that the method of intermediate problems, which provides convergent improvable lower bound estimates for eigenvalues of linear half-bound Hermitian operators in Hilbert space, can be fused with the classical Rayleigh-Ritz variational method and stochastic variational method thus resulting in an efficient tool for analytical and numerical studies of the energy spectrum and eigenstates of the Gaussian quantum dots, confining small-to-medium number of excess electrons, with controllable or prescribed precision.

A Rigorous Approach for Calculating the Chemical Potential of an Ultrarelativistic Spherical Degenerate Fermi Gas Interacting with Nuclear Matter

“…In the context of the above behaviour, finding the chemical potential of the aforementioned gases is really useful from both the theoretical and practical points of view. It is well-known that elucidating the physics of Fermi gases is crucial in fields as, for instance, nanophysics [8] and, particularly, nanoscale superconductivity. [9,10] The optical potential, a concept arising from the many-body theory, appears as a very important piece of the chemical potential (see relations (1) and (8)).…”

“…At this point, it is interesting to remark that the Fermi gas in question is dilute if |b(r)| k F (r) ≪ 1 and very dilute if |b(r)| k F (r) ≪ 1. Consequently, if the gas is dilute, then U(r; T ) may be neglected (see formulae (1) and (9)) so that the chemical potential is approximately equal to the Fermi level plus the optical potential whereas, if the gas is very dilute, then both U(r; T ) and V (r; T ) may be neglected (see formulae (1), (8) and (9)), so now the chemical potential equals approximately the Fermi level.…”

“…[1]- [8] Relativistic Fermi gases occupy a significant place in Relativistic Quantum Statistical Mechanics. In particular, finding the chemical potential of an ultrarelativistic degenerate Fermi gas is a useful task given the relevance of the concept of chemical potential in several branches of Physics.…”

New Results on the Density of Electronic States, Superexchange Interaction, and Electrical Conduction in Transition-Metal Salts

“…[14]) so, from relation (15) it follows that γ dp ≈ t. Of course, relation (15) holds in the paramagnetic case. After formula (13), we have written E ≈ −t 2 /W so, in practice, we have that E ≈ −γ 2 dp /W .…”

Lipmann-Schwinger Equations for a System of Multiband Energy-Eigenvalue Spectrum

“…Really, the physics of condensed matter exhibits a number of examples of quantum systems with multiband eigenvalue spectra. Consider, for instance, a significant two-band eigenvalue spectrum consisting of spin-up and spin-down electronic bands (as, for example, in metamagnetic systems [4,5] and nanophysics [6][7][8][9][10]) and a typical system with three-band spectrum as, for instance, a semiconductor with the conduction band, the valence band, and the forbidden band (band gap). Certainly, systems with two or three eigenvalue bands are relatively frequent in Physics.…”