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

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 r...

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Дата:	2006
Автори:	Soldatov, A.V., Bogolyubov, N.N., Kruchinin, S.P., Jr.
Формат:	Стаття
Мова:	English
Опубліковано:	Інститут фізики конденсованих систем НАН України 2006
Назва видання:	Condensed Matter Physics
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Цитувати:	Method of intermediate problems in the theory of Gaussian quantum dots placed in a magnetic field / A.V. Soldatov, N.N. Bogolyubov, Jr., S.P. Kruchinin // Condensed Matter Physics. — 2006. — Т. 9, № 1(45). — С. 151–159. — Бібліогр.: 18 назв. — англ.

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spelling	irk-123456789-1213092017-06-15T03:05:17Z Method of intermediate problems in the theory of Gaussian quantum dots placed in a magnetic field Soldatov, A.V. Bogolyubov, N.N. Kruchinin, S.P. Jr. 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. 2006 Article Method of intermediate problems in the theory of Gaussian quantum dots placed in a magnetic field / A.V. Soldatov, N.N. Bogolyubov, Jr., S.P. Kruchinin // Condensed Matter Physics. — 2006. — Т. 9, № 1(45). — С. 151–159. — Бібліогр.: 18 назв. — англ. 1607-324X PACS: 73.21.La, 85.35.Be, 75.75.+a, 03.65.Ge, 02.30.Tb DOI:10.5488/CMP.9.1.151 http://dspace.nbuv.gov.ua/handle/123456789/121309 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	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.
format	Article
author	Soldatov, A.V. Bogolyubov, N.N. Kruchinin, S.P. Jr.
spellingShingle	Soldatov, A.V. Bogolyubov, N.N. Kruchinin, S.P. Jr. Method of intermediate problems in the theory of Gaussian quantum dots placed in a magnetic field Condensed Matter Physics
author_facet	Soldatov, A.V. Bogolyubov, N.N. Kruchinin, S.P. Jr.
author_sort	Soldatov, A.V.
title	Method of intermediate problems in the theory of Gaussian quantum dots placed in a magnetic field
title_short	Method of intermediate problems in the theory of Gaussian quantum dots placed in a magnetic field
title_full	Method of intermediate problems in the theory of Gaussian quantum dots placed in a magnetic field
title_fullStr	Method of intermediate problems in the theory of Gaussian quantum dots placed in a magnetic field
title_full_unstemmed	Method of intermediate problems in the theory of Gaussian quantum dots placed in a magnetic field
title_sort	method of intermediate problems in the theory of gaussian quantum dots placed in a magnetic field
publisher	Інститут фізики конденсованих систем НАН України
publishDate	2006
url	http://dspace.nbuv.gov.ua/handle/123456789/121309
citation_txt	Method of intermediate problems in the theory of Gaussian quantum dots placed in a magnetic field / A.V. Soldatov, N.N. Bogolyubov, Jr., S.P. Kruchinin // Condensed Matter Physics. — 2006. — Т. 9, № 1(45). — С. 151–159. — Бібліогр.: 18 назв. — англ.
series	Condensed Matter Physics
work_keys_str_mv	AT soldatovav methodofintermediateproblemsinthetheoryofgaussianquantumdotsplacedinamagneticfield AT bogolyubovnn methodofintermediateproblemsinthetheoryofgaussianquantumdotsplacedinamagneticfield AT kruchininsp methodofintermediateproblemsinthetheoryofgaussianquantumdotsplacedinamagneticfield AT jr methodofintermediateproblemsinthetheoryofgaussianquantumdotsplacedinamagneticfield
first_indexed	2025-07-08T19:37:41Z
last_indexed	2025-07-08T19:37:41Z
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fulltext	Condensed Matter Physics 2006, Vol. 9, No 1(45), pp. 151–159 Method of intermediate problems in the theory of Gaussian quantum dots placed in a magnetic field A.V.Soldatov1, N.N.Bogolyubov, Jr.,1 S.P.Kruchinin2 1 V.A.Steklov Mathematical Institute, 8 Gubkina Str., 119991 Moscow, Russia 2 Bogolyubov Institute for Theoretical Physics, 14–b Metrologichna Str., 252143, Kiev, Ukraine Received July 18, 2005 Applicability of the method of intermediate problems to the investigation of the energy eigenvalues and eigen- states of a quantum dot (QD) formed by a Gaussian confining potential in the presence of an external mag- netic 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 inter- face 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 spec- trum 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 1. Introduction Various theoretical and experimental aspects of the physics of nano-sized and low-dimensional systems have been under steady uninterrupted development for decades, but only in the last ten- fifteen years of research activities in the field, considerable extra momentum has been gained which can be partly ascribed to the progress in nanofabrication of these systems, partly to the development of new experimental techniques but, most of all, to a general continuous trend and desire to diminish the size of components of integrated electronic circuits down to the so-called mesoscopic scale. Among low-dimensional systems of various kinds, quantum dots are potentially fit for many practical applications and are currently thought to be promising building blocks for novel electronic, spintronic and optoelectronic devices. Reliable estimation of the energy spectrum and eigenstates of a quantum dot is a typical purpose of almost every theoretical study because their properties crucially stipulate the relevant physical characteristics of the quantum dot standing alone as a part of electric circuits or interacting with the environment through its various interfaces. As a matter of fact, nearly all the mathematical methods, developed within the domain of quan- tum mechanics so far, have already been employed in the theory of quantum dots in various specific applications though on a varying scale. The most frequently applicable methods of approximate calculation of eigenvalues and eigenstates of realistic physical models of low-dimensional quantum systems, quantum dots among them, are various numerical methods which permit either direct calculations of the magnitudes in question without proper error estimates or, at best, provide the calculations with nonincreasing or even convergent upper bounds for the eigenvalues. The widely applicable Rayleigh-Ritz method does it but, again, without error estimates. To control the error of the approximations provided by the upper bounds for some quantity it would be enough to derive the corresponding lower bounds which are highly desirable to be c© A.V.Soldatov, N.N.Bogolyubov, Jr., S.P.Kruchinin 151 A.V.Soldatov, N.N.Bogolyubov, Jr., S.P.Kruchinin convergent. Therefore, the development of regular methods to construct such bounds bears a lot of theoretical and practical significance but represents a much more challenging task than deriva- tion of the upper bounds. As to various models of quantum dots, the most suitable ground to fulfill this task would be the method of intermediate problems ascending in its basic idea to the maximum-minimum characterization of eigenvalues of half-bounded Hermitian operators in Hilbert space introduced by H. Weyl [1] and elaborated later by numerous contributors [2,3] with regard to the problems of classical and quantum mechanics. Being supplemented with regular Rayleigh-Ritz or newly developed stochastic variational method [4], the method of intermediate problems may serve as a powerful tool for investigation of the energy spectrum of few-body quantum systems with prescribed or controllable precision and would permit to verify the results obtained by other numerical and analytical methods [5]. A stochastic variational method can also be developed on its basis [6] permitting, in principle, the construction of improvable lower bounds for energy eigen- values of realistic models of quantum dots confining a relatively large number of excess electrons. Numerical studies of the latter systems proved to be especially difficult and time-consuming if un- dertaken by means of direct numerical solution of the underlying Schrödinger equation leading to highly multidimensional grids, or by regular variational approaches resulting in prohibitively large sets of basis states. It is worth noting, that whenever a particular algorithm based on the method of intermediate problems is applied in order to get lower bound estimates for energy eigenvalues, the corresponding eigenstates would result immediately as a concomitant outcome of the calcula- tions. Error estimates for the so derived eigenstates can also be obtained but, which appears to be more effective, such error estimates can also be derived for the approximate eigenstates calculated within the framework of the Rayleigh-Ritz regular or stochastic variational method [4] provided that the lower bound estimates for the corresponding energy eigenvalues are known. Equally, under the same premises, expectation values (matrix elements) of physically relevant operator variables (dipole or quadruple moments of a quantum dot, for example) calculated with respect to these Rayleigh-Ritz approximate eigenstates can be given their respective error bound estimates too. 2. Quantum dot formed by a Gaussian confining potential In what follows, a model of a quantum dot confined by a Gaussian potential in an external uniform magnetic field B will be considered H = N ∑ i=1 1 2m∗ e ( p̂i + e c Â(ri) )2 + N ∑ i=1 V̂ (ri) + N ∑ i<j e2 εr̂ij + N ∑ i=1 g∗µB ~ B · Ŝi . (2.1) Here m∗ e is the electron conduction-band mass, e is the absolute value of the electron charge, g∗ is the effective g factor and indices i, j label the electrons. The Coulomb interaction between the excess electrons is assumed to be proportional to the inverse effective dielectric constant of the bulk material ε and an the inverse interelectron distance rij = \|ri − rj \|. The vector potential Â(r), (B = rotA) is introduced in the symmetric gauge, Â(r) = B/2(−y, x, 0), and V̂ (r) is the attractive Gaussian potential V̂ (r) = −V0 exp(−r2/2R2) (2.2) with the depth of the potential well V0 > 0 and the range R, and r = \|r\|, where r = (x, y) for 2D quantum dots and r = (x, y, z) for 3D quantum dots. For the sake of simplicity and clarity, we limited our present research to the case of two- dimensional quantum dots, though, as will be seen later, generalization of the outlined approach for the case of three-dimensional spherical quantum dots embedded into a supporting matrix is straightforward. The usage of a Gaussian potential is advantageous in comparison with other commonly em- ployed types of potentials. This potential possesses the finite depth and range so that it can confine only a finite number of the excess electrons, which fact mimics the actual experimental situation properly. In the vicinity of the dot center this potential can be approximated by the parabolic po- tential routinely applicable in countless dot model studies so far. Therefore, the whole body of the 152 Method of intermediate problems theoretical results obtained for parabolic potentials presently would hold at least approximately for Gaussian potentials for some range of model parameters. 3. The one-electron eigenvalue problem 3.1. The one-electron Gaussian quantum dot model To our knowledge, no analytical solutions have been obtained so far for the one-electron eigen- value problem set by the Hamiltonian Ĥ(i) = 1 2m∗ e ( p̂i + e c Â(ri) )2 + V̂ (ri) + g∗µB ~ B · Ŝi (3.1) with the attractive Gaussian potential (2.2) even without an external magnetic field though several attempts have been undertaken to solve this problem for the latter case approximately [7–11] due to the role the potentials of this type play in nuclear physics describing the scattering of complex nuclei [12]. Contrariwise, normalized eigenstates of the Hamiltonian Ĥ0(i) = Ĥi0 + Ĥsi0 = 1 2m∗ e ( p̂i + e c Â(ri) )2 + g∗µB ~ B · Ŝi (3.2) are well known [13] and can be presented as Ψnmσ(r, φ, s) = ψnm(r, φ)χσ(s) = 1√ 2π eimφRnm ( r2 2a2 0 ) χσ(s), (3.3) where χσ(s) is a spin function and the radial part is given by Rnm(x) = 1 a0 √ n! (n+ \|m\|)! exp ( −x 2 ) x \|m\| 2 L\|m\| n (x), (3.4) where L \|m\| n (x) is a generalized Laguerre polynomial. The corresponding eigenvalues are Enmσ = Enm + Eσ = ~ωc ( n+ \|m\| +m 2 + 1 2 ) + g∗µB ~ Bσ, (3.5) where σ = ±1/2, n = 0, 1, . . . , m = 0,±1, . . . and µB is the Bohr magneton. It is seen that the magnitudes ~ωc = (~eB)/(m∗ ec) and a0 = √ (~c)/(eB) provide natural characteristic energy and length scales. The energy eigenvalues (3.5) are infinitely degenerate and form the so-called Landau levels. Due to this feature, the method of the intermediate problems cannot be applied directly to this type of models. At the same time, the total angular momentum component ∑ i L̂ i z is preserved for the models (2.1) and (3.1) as well as the total spin component ∑ i Ŝ i z . Therefore, instead of the model (3.2) with infinitely degenerate energy spectrum (3.5) as well as instead of the model (3.1) one can consider a sequence of models reduced to subspaces labeled with the angular momentum quantum number m and the spin quantum number σ. The energy contribution of the spin part of the Hamiltonian (3.2) is trivially accountable and thus can be omitted. Therefore, further studies can be limited to the treatment of the reduced Hamiltonian Ĥi(m) = ∞ ∑ n,n′=0 \|nm〉〈nm\| ( Ĥi0 + ˆ̃V (ri) − V0 ) \|n′m〉〈n′m\| (3.6) using the method of intermediate problems. Here an identity transformation of the Gaussian po- tential was introduced V̂ (r) ≡ V0 ( 1 − exp(−r2/2R2) ) − V0 ≡ ˆ̃V (r) − V0 , (3.7) where the addition energy V0 can be currently dropped and then restored only upon the completion of all the calculations. As will be seen in the next section, the necessity of this transformation stems from the positivity condition (3.13). 153 A.V.Soldatov, N.N.Bogolyubov, Jr., S.P.Kruchinin 3.2. Basics of the method of intermediate problems To make this proceeding self-contained, some technicalities of the method of intermediate prob- lems are outlined here in brief. The starting point of the method is the standard time-independent Schrödinger equation Hψ = Eψ, (3.8) where H is some Hermitian operator with respect to the inner product (φ, ψ) = ∫ φ∗ψdτ in Hilbert space. It is assumed that all continuous energy levels of H are higher than the lowest discrete energy levels of one’s interest. Let us also assume that these discrete eigenvalues of H can be ordered in a nondecreasing sequence, E1 6 E2 6 · · · . (3.9) Eigenstates ψi, corresponding to the eigenvalues Ei, satisfy the equation Hψi = Eiψi , (3.10) and are assumed to be orthonormalized, so that (ψi, ψj) = δij , (3.11) where δij is Kronecker’s delta. It is further assumed that the Hamiltonian H can be decomposed as H = H0 +H ′, (3.12) where H0 has the known eigenvalues and eigenstates and H ′ is an arbitrary Hermitian operator which is to be positively definite in the sense that (ψ,H ′ψ) = ∫ ψ∗H ′ψdτ > 0, (ψ 6= 0) (3.13) for every ψ in the domain of H . Hereafter, it is assumed that the lowest part of the discrete spectrum of H0 is below its continuous spectrum and that the corresponding discrete eigenvalues can be ordered in the same manner (3.9) as the ones belonging to the total Hamiltonian H E0 1 6 E0 2 6 · · · . (3.14) The corresponding orthonormalized eigenstates ψ0 i satisfy the equation H0ψ0 i = E0 i ψ 0 i , (ψ0 i , ψ 0 j ) = δij . (3.15) Since H0 6 H in the sense of inequality (ψ,H0ψ) 6 (ψ,Hψ) (3.16) for every ψ in the domain of H , it follows from the Weyl comparison theorem [1] that E0 i 6 Ei, (i = 1, 2, . . . ). (3.17) Therefore,the eigenvalues of H0 already provide a rough lower bound to the eigenvalues of H . The Hamiltonian H0 is called the base Hamiltonian as usual. It is worth noting that the decomposition (3.12) is not unique and can be tailored to meet the requirements of a particular problem in question. The basic idea of the method of intermediate problems is to approximate the original Hamiltoni- anH from below by a non-decreasing sequence of the so-called truncated intermediate Hamiltonians H l,k. These Hamiltonians are to be constructed to satisfy the inequalities H l,k 6 H l+1,k 6 Hk 6 H, (l, k = 1, 2, . . . ), (3.18) H l,k 6 H l,k+1 6 H, (l, k = 1, 2, . . . ). (3.19) 154 Method of intermediate problems Therefore, the Hamiltonians H l,k increase whatever index k or l is increased and thus should give improvable lower bounds for the lowest eigenvalues of the original Hamiltonian H . It was shown [15] that the truncated Hamiltonians H l,k can be represented in a general form H l,k = H l,0 +H ′P k, (l, k = 1, 2, . . . ). (3.20) Here the Hamiltonian H l,0 is a truncation of the base Hamiltonian H0 of the order l defined as H l,0ψ = l ∑ i=1 ( ψ0 i , ψ ) E0 i ψ 0 i + E0 l+1 [ ψ − l ∑ i=1 ( ψ0 i , ψ ) ψ0 i ] , (l = 1, 2, . . . ), (3.21) or, alternatively, in Dirac’s more transparent bra and ket notations H l,0 = l ∑ i=1 E0 i \|E0 i 〉〈E0 i \| + E0 l+1 [ Î − l ∑ i=1 ∣ ∣E0 i 〉〈E0 i ∣ ∣ ] , (l = 1, 2, . . . ) , (3.22) where Î stands for the identity operator. Truncations of H0 satisfy the inequalities H l,0 6 H l+1,0 6 H0, (l = 1, 2, . . . ), (3.23) which were proved in general case in [17]. The operator P k defines a projection of an arbitrary vector φ in the domain of H onto the subspace formed by a sequence of vectors p1, p2, . . . , pk : P kφ = k ∑ i=1 αipi , (3.24) where constants αi should satisfy the equations [ pj , P kφ ] = [pj , φ] = k ∑ i=1 αi [pj , pi] , (j = 1, 2, . . . , k). (3.25) Here an auxiliary inner product with respect to the metric operator H ′ was introduced as [ψ, φ] = (ψ,H ′φ) = ∫ ψ∗H ′φdτ (3.26) for every pair of vectors ψ, φ for which H ′ψ and H ′φ are defined. Vectors p1, p2, . . . , pk are to be chosen linearly independent in the vector space with an inner product (3.26). These vectors are to be normalizable but their explicit normalization is not required. Projections P k become larger with the increase of the number k of the elements pi involved. As a consequence, the following inequality holds 0 6 [φ, P kφ] 6 [φ, P k+1φ] 6 [φ, φ], (k = 1, 2, . . . ), (3.27) which in the original vector space reads as 0 6 (φ,H ′P kφ) 6 (φ,H ′P k+1φ) 6 (φ,H ′φ), (k = 1, 2, . . . ). (3.28) From equations (3.24, 3.25) it follows that H ′P kφ = k ∑ i,j=1 (H ′pi, φ)bijH ′pj , (3.29) where bij are the elements of the matrix inverse to the matrix with terms [pj, pi]. As a consequence of equation (3.28) H ′P k 6 H ′P k+1 6 H ′, (k = 1, 2, . . . ), (3.30) 155 A.V.Soldatov, N.N.Bogolyubov, Jr., S.P.Kruchinin and the intermediate truncated Hamiltonians Hk defined as Hk = H0 +H ′P k, (k = 1, 2, . . . ) (3.31) satisfy inequalities Hk 6 Hk+1 6 H (3.32) by construction if inequalities (3.30) are taken into account. According to equations (3.23) and (3.30) H l,k 6 H l,k+1 6 H l,0 +H ′ 6 H, (l, k = 1, 2, . . . ). (3.33) Therefore, the lowest ordered eigenvalues El,k i of H l,k should satisfy the parallel inequalities El,k i 6 El+1,k i 6 Ek i 6 Ei, (i, l, k = 1, 2, . . . ), (3.34) and El,k i 6 El,k+1 i 6 Ei, (i, l, k = 1, 2, . . . ), (3.35) thus providing improvable lower bounds for the original eigenvalues Ei of the Hamiltonian H . As was proved in [17], the so constructed Hamiltonian H l,k can have no continuous spectrum and must have E0 l+1 as an eigenvalue of infinite multiplicity. Therefore, only those eigenvalues of H l,k that are smaller or equal to E0 l+1 can be considered as lower bound estimates for the eigenvalues of the initial Hamiltonian H . The truncation procedure (3.21) can be significantly improved from the point of view of practical calculations if the original Hamiltonian H is formally decomposed as H = H l,0 + (H0 −H l,0) +H ′ = H l,0 +H ′ +H ′′ = H l,0 + H̃ ′, (l = 1, 2, . . . ), (3.36) where the difference H̃ ′ = H −H l,0 is obviously positive and can play the role played before by the metric operator H ′. In this case the positive contributions from the operator H ′′ = H0 −H l,0, (l = 1, 2, . . . ) (3.37) to lower bound estimates are not simply neglected at will but rather carefully taken into consid- eration on common grounds with the contributions stemming from H ′, thus making these bounds higher than they might have been otherwise under the original truncation procedure (3.21). 3.3. Reduced linear algebraic problems for the one-electro n model The eigenvalues and eigenstates of the intermediate Hamiltonians H l,k of any order (i.e. for arbitrary magnitudes of the indices l and/or k) can be expressed analytically or calculated nu- merically in terms of the known eigenvalues and eigenstates of H0 and an arbitrarily chosen set of linearly independent vectors pi, (i = 1, . . . , k). For numerical calculations it is especially conveni- ent to follow the observation made by W. Börsch-Supan [15] that any intermediate Hamiltonian H l,k can be reduced to the finite-dimensional space M generated by the known eigenstates ψ0 i , (i = 1, . . . , l) of the base Hamiltonian H0 and the vectors H ′pi, (i = 1, . . . , k). This reduction makes it possible to determine the eigenstates of H l,k lying in M along with the corresponding eigenvalues from a linear algebraic problem of the order l+ k for one symmetric indefinite matrix relative to another. This problem can be formulated in terms of the system of equations with respect to γν k+l ∑ ν=1 γν ( Cµν − ( E − E0 l+1 ) Dµν ) = 0, (µ = 1, 2, . . . , k + l) , (3.38) where (Cµν) = ( δµν (ψ0 µ, H ′pν−l) (H ′pµ−l, ψ0 ν) (H ′pµ−l, H ′pν−l ) , (3.39) 156 Method of intermediate problems and (Dµν) = ( δµν/(E 0 ν − E0 l+1 ) 0 0 (H ′pµ−l, pν−l) ) . (3.40) For the eigenvalues E which are not equal to E0 l+1 the corresponding eigenstates of H l,k are given by ψ (E) = ( l ∑ ν=1 γνψ 0 ν + l+k ∑ ν=l+1 γνH ′pν−l ) ( E − E0 l+1 )−1 , (3.41) and E0 l+1 is an eigenvalue of H l,k of infinite multiplicity. Sometimes, another special choice for vectors pi is possible pi = (H ′)−1ψ0 i , i = 1, 2, . . . , k. (3.42) As was shown in [14], k eigenvalues of the intermediate Hamiltonian Hk can be found by the solution of the matrix eigenvalue problem k ∑ j=1 { [E0 t − E]δjt + bjt } γj = 0, 1 6 t 6 k, (3.43) where bji is the matrix inverse to that with elements (ψ0 i , pj) = (ψ0 i , (H ′)−1ψ0 j ). (3.44) For the thus found eigenvalues E the corresponding eigenstates of Hk are given by ψ(E) = k ∑ t=1 γt(E)ψ0 t . (3.45) Moreover, if an eigenfunction ψ0 t of H0 with an eigenvalue E0 t is not employed in forming any element pi, then ψ0 t is also an eigenfunction of the intermediate Hamiltonian Hk with the same eigenvalue E0 t . Unlike the Hamiltonian (3.20), the Hamiltonian Hk for the special choice of pi vec- tors (3.42) may also possess a continuous spectrum which can be shown to be identical to that ofH0. 0 5 10 15 20 25 −20 −15 −10 −5 0 5 10 15 scaled potential range R/a 0 sc al ed e ne rg y ei ge nv al ue s Figure 1. Scaled energy eigenvalues E/~wc via scaled potential range R/a0; circles o: m=0; crosses + : m=1; crosses x: m=-1; V0/~wc = 20. For the particular single electron model (3.6), the HamiltonianH ′ in (3.12) can be iden- tified with ˆ̃V (ri) thus satisfying the positivity condition (3.13). For any fixed angular quan- tum number m and any given order l of the in- termediate Hamiltonian (3.20) the vectors ψ0 i must be identified with the first l + 1 func- tions ψnm, n = 0, 1, . . . , l. Correspondingly, the eigenvalues E0 i , i = 1, . . . , l + 1 are Enm, n = 0, 1, . . . , l defined in equation (3.5). In our present numerical calculations the vectors pi, i = 1, . . . , k coincide with the first k eigenstates ψnm, n = 1, . . . , k. Actually, this choice ensures that the contribution to all the requisite matrix elements from the operator of the kind (3.37) is zero. What seems to be much more impor- tant is the fact that this choice made possible an analytical calculation of all matrix elements appearing in (3.38), thus increasing the precision of the results. For convenience, the condition k = l was fulfilled throughout all the calculations. The upper bound estimates on the energy eigenvalues of the Hamiltonian (3.6) were obtained by the 157 A.V.Soldatov, N.N.Bogolyubov, Jr., S.P.Kruchinin standard Rayleigh-Ritz method using a set of p eigenfunctions ψnm as the base set. The numbers l, k and p of the functions involved in all these calculations were chosen to ensure the coincidence of the corresponding upper and lower energy spectrum bound estimates up to the sixth digit. Ex- emplary results of these calculations are shown in figure 1 where ten lowest eigenvalues for each of the three different values of angular momentum m = −1,m = 0 and m = 1 are plotted via the Gaussian potential range R. It is seen that the originally degenerate at R = 0 eigenvalues become nondegenerate as the potential range R increases to the magnitudes comparable with the characteristic magnetic length a0 and return to degeneracy again upon a further increase of the potential range, thus restoring the structure of Landau levels. 4. Two- and many-electron Gaussian quantum dots In general, Gaussian quantum dots in a magnetic field containing two or more excess electrons can be treated using the method of intermediate problems in the same way as the simple one- electron quantum dot was treated above with the only difference that now the electron-electron interaction is to be taken into account. This interaction is repulsive and the additional interacti- on term ∑N i<j V̂ (rij) in H ′ satisfies the positivity condition (3.42). Therefore, all the prospected difficulties in applying the method would not be of conceptual but rather of technical nature due to cumbersome and laborious calculations. Of course, the Fermi statistics should be accounted for when constructing many-electron functions ψ0 i and pi. Thus, the functions ψ0 i should be anti- symmetric with respect to pairwise electron permutations eigenfunctions of the total angular and spin momentum built of the one-electron eigenstates (3.3). The amount of the required calculations drastically depends on the proper choice of the functions pi. It is quite reasonable to anticipate that the stochastic variational method for the calculation of the lower bound estimates to eigenvalues, proposed in [6], will be useful in significantly cutting down the size of the {pi} sets. For the two-electron quantum dot the special choice (3.42) of pi functions looks advantageous. In this case (H ′)−1 = ( ˆ̃V (r1) + ˆ̃V (r2) + V̂ (r12)) −1 (4.1) and all matrix elements (3.44) can be reduced to calculating the three-dimensional integrals. For- mally, further reduction to two-dimensional integrals is also possible but at the expense of elliptic functions showing up in the integrands. It was found that either three- or two-dimensional integrals can be calculated numerically by means of the Gauss-Kronrod adaptive method. The same choice (3.42) can be employed in general case of a quantum dot containing N electrons. The resulting matrix elements should be calculated by the methods of numerical multidimensional integration. An alternative approach (3.38) can also be employed in the case of the two-electron Gaussian quantum dot. In this case it is convenient to choose the pi functions in the form pi = r12ψ 0 i , but such a choice may be not optimal and can lead to prohibitively large sets of pi functions. An expediency of using the correlated Gaussians [4,18], as a much more effective choice for pi, is currently being investigated. This choice looks promising for Gaussian quantum dots containing small-to-medium number of electrons. 5. Conclusion It was shown that the energy eigenvalues of practically important and realistic models of quan- tum dots formed by a Gaussian confining potential and placed in an external magnetic field can be estimated with arbitrarily prescribed precision by means of the standard Rayleigh-Ritz method, the method of intermediate problems being complementary to each other. Two-side improvable error bound estimates can be found for the eigenvalues while the corresponding eigenstates come out as a by-product of the applied methods together with their error bound estimates as well. 158 Method of intermediate problems Acknowledgements This work has been supported by the RF support program “Support for Leading Scientific Schools”, grant No. 1758.2003.1, by the RAS research program “Mathematical Problems of Non- linear Dynamics” and by the RFBR grant No. 05–02–16663–a. References 1. Weyl H., Journ. für die reine und angewandte Math., 1912, 141, 1. 2. Weinstein A., Stenger W. Intermediate Problems for Eigenvalues. Academic Press, New York and London, 1972. 3. Gould S.H. Variational Methods for Eigenvalue Problems: An Introduction to the Weinstein Method of Intermediate Problems, 2nd ed. Univ. of Toronto Press, 1966. 4. Suzuki Y., Varga K. Stochastic Variational Approach to Quantum Mechanical Few-Body Problems. Springer-Verlag, Berlin, 1998. 5. Soldatov A.V., Int. J. Mod. Phys. B., 2004, 18, 211. 6. Soldatov A.V., Mod. Phys. Lett. B., 2005, 19, 361. 7. Bessis N., Bessis G., Joulakian B., J. Phys. A., 1982, 15, 3679. 8. Lai C.S., J. Phys. A, 1983, 16, L181. 9. Crandall R.E., Phys. A, 1983, 16, L395. 10. Cohen M., Phys. A, 1984, 17, L101. 11. Adamowski J., Sobkowicz M., Szafran B., Bednarek S., Phys. B, 2000, 62, 4234. 12. Buck B., Friedrich H., Wheatley C. Nucl. Phys. A, 1977, 275, 246. 13. Fock V., Z. Phys., 1928, 47, 446. 14. Bazley N.W., Phys. Rev., 1960, 120, 144. 15. Bazley N.W., Fox D.W., Phys. Rev., 1961, 124, 483. 16. Bazley N.W., Fox D.W., Rev. Mod. Phys., 1963, 35, 712. 17. Bazley N.W., Fox D.W., J. Research Natl. Bur. Stand., 1961, 65 B, No. 2. 18. Varga K., Suzuki Y., Phys. Rev. C., 1995, 52, 2885. �� i� � � � � � ��i � � �� ,��i�� i� � �� .�.��1, . .!�"��#$��, %��.,1 �.&.'()*i+i+2 1 I,-./.0. 12.312./4,/5 i1. 6.7.8.9:;<=2,=0;. >0?:i,2 8, @<-:=2 119991, A<-iB 2 I,-./.0. .3<C3./4,<D EiF/:/ i1. @.@.G<H<;I?<=2,=0;. @3.C<;<Hi4,2, 14–?, 03143 J/D=, K:C2D,2 LMNOPQRS 18 TOURV 2005 N. W?H<=<CI9.X-B F2-.<-<=,i-.X 13.<Y0 ZC<1i[,/\ F2Y24 Y< Y<-;iY[3,,B =;2-,/\ F,243,X 3,3CHiD .2=;2-,/\ -.2,i= :=2,.<=<D .<4:/ (J]), -E<C1<=2,<D 0.C/10I4/1 >20-<=/1 Z<.3, î2;<1 = ZC/-0.,<--.i F<=,i_,X<H< 12H,i.,<H< Z<;B. G0Y04/ H;2Y:/1 ,2 HC2,/ î J] .2 12I4/ -:i,43,i H;/?/,0 .2 C2Yi0-=F291<YiD, ^35 Z<.3, î2; 1<[30.C/10=2./;/_3 -:i,43,3 4/-;< ,2Y;/_:<=/\ 3;3:.C<,i=, E<C10I4/43C3F ^3 C32;i-./4,0 1<Y3;X J] F H;2Y:<I HC2,/^3I C<FYi;0 1i[ J] .2 DD <.<43,,B1. 7CH013,.09.X-B,`< 13.<Y ZC<1i[,/\ F2Y24, `< F2?3FZ3409 F?i[,i Z<:C2`3,i F,243,,B Y;B ,/[,X<D HC2,/ î =;2-,/\ F,243,X ;i,i5,/\ ,2Zi=F=’BF2,/\ aC1i.<=/\ <Z3C2.<Ci= 0 >i;X?3C.<=<10 ZC<-.<Ci, 1<[3 ?0./ Z<9Y,2- ,/5 F :;2-/4,/1 =2Ci2 î5,/1 13.<Y<1 A3;3B-Ai.^2 .2 -.<\2-./4,/1 =2Ci2 î5,/1 13.<Y<1, Y2I4/ =C3F0;X.2.i 3E3:./=,/5 F2-i? Y;B 2,2;i./4,/\ ./ 4/-;<=/\ Y<-;iY[3,X 3,3CH3./4,<H< -Z3:.C0 .2 =;2--,/\ -.2,i= H20-<=/\ :=2,.<=/\ .<4<:, `< 0.C/10I.X =iY 12;<H< Y< -3C3Y,X<H< 4/-;2 ,2Y;/_:<=/\3;3:.C<,i=, F :<,.C<;X<=2,<I .2 F2Y2,<I .<4,i-.I. bcdefgi hcfgi: jklmnoki nopjq, krlsmi tmlpummv, krlsmi snlmq, kuwxmv nl mqymv zwlmq{i PACS: 73.21.La, 85.35.Be, 75.75.+a, 03.65.Ge, 02.30.Tb 159 160

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

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