Condensed Matter Physics, 2006, vol. 9, No. 1(45), p. 151159, English
DOI:10.5488/CMP.9.1.151
Title:
Method of intermediate problems in the theory of
Gaussian quantum dots placed in a magnetic field
Author(s):
 A.V.Soldatov
(V.A.Steklov Mathematical Institute, 8 Gubkina
Str., 119991 Moscow, Russia)
,

 N.N.Bogolyubov, Jr.
(V.A.Steklov Mathematical Institute, 8 Gubkina
Str., 119991 Moscow, Russia)
,

 S.P.Kruchinin
(Bogolyubov Institute for Theoretical Physics,
14b Metrologichna Str., 252143, Kiev, Ukraine)

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 halfbound Hermitian operators in
Hilbert space, can be fused with the classical RayleighRitz 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 smalltomedium
number of excess electrons, with controllable or prescribed precision.
Key words: quantum dots, eigenvalues, eigenstates, upper and lower bounds
PACS: 73.21.La, 85.35.Be, 75.75.+a, 03.65.Ge, 02.30.Tb
