Condensed Matter Physics, 2006, vol. 9, No. 1(45), p. 151-159, English

Title: Method of intermediate problems in the theory of Gaussian quantum dots placed in a magnetic field
  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, 14--b 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 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.

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

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