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.

The Bogolyubov model of liquid helium is considered. The validity of substituting a c-number for the k = 0 mode operatorâ0 is established rigorously. The domain of stability of the Bogolyubov's Hamiltonian is found. We derive sufficient conditions which ensure the appearance of the Bose condensate in the model. For some temperatures and some positive values of the chemical potential, there is a gapless Bogolyubov spectrum of elementary excitations, leading to a proper microscopic interpretation of superfluidity. Jp, 03.75.Fi, The modelLet us consider a system of N spinless identical nonrelativistic bosons of mass m enclosed in a centered cubic box Λ ⊂ R 3 of volume V = |Λ| = L 3 with periodic boundary conditions for wave functions. The Hamiltonian of the system can be written in the second quantized form aŝHereâ # k = {â † k orâ k } are the usual boson (creation or annihilation) operators for the one-particle state ψ k (x) = V −1/2 exp(ikx), k ∈ Λ * , x ∈ Λ, acting on the Fock spacesymm is the symmetrized n-particle Hilbert space appropriate for bosons, andThe sums in (1) run over the dual set2m) is the one-particle energy spectrum of free bosons in the modes k ∈ Λ * (we propose = 1),N Λ = k∈Λ * â † kâ k is the total particle-number operator, µ is the chemical potential, ν(k) is the Fourier transform of the interaction pair potential Φ(x). We suppose that Φ(x) = Φ(|x|) ∈ L 1 (R 3 ) and ν(k) is a real function with a compact support such that 0 ν(k) = ν(−k) ν(0) for all k ∈ R 3 . Under these conditions the Hamiltonian (1) is superstable [1]. So long as the rigorous analysis of the Hamiltonian (1) is very knotty problem, Bogolyubov introduced the model Hamiltonian of superfluidity theory [2,3]. He proposed to disregard the terms of the third and fourth order in operatorsâ # k , k = 0 in the Hamiltonian (1),c N.N. Bogolyubov, Jr., D.P. Sankovich

Рассмотрены точно решаемые полностью асимметричные модели маломерной неравновесной статистической физики на периодической цепочке, а именно полностью асимметричный простой процесс с запретом и полностью асимметричный простой процесс с нулевым радиусом. Описан метод вычисления корреляционных функций моделей на периодической решетке. Скалярные произведения векторов состояний моделей представлены в виде определителей. Ключевые слова: асимметричные процессы, интегрируемые системы, корреляционные функции.