Infinitely improvable upper bounds in the theory of polarons

An infinite convergent sequence of improving non-increasing upper bounds to the low-lying branch of the slowmoving “physical” Frohlich polaron ground-state energy spectral curve, adjacent to the ground state energy of the polaron at rest, was obtained by means of generalized variational method. The...

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Date:	2010
Main Author:	Soldatov, A.V.
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Language:	English
Published:	Інститут фізики конденсованих систем НАН України 2010
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Cite this:	Infinitely improvable upper bounds in the theory of polarons / A.V. Soldatov // Condensed Matter Physics. — 2010. — Т. 13, № 4. — С. 43701:1-9. — Бібліогр.: 16 назв. — англ.

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spelling	nasplib_isofts_kiev_ua-123456789-321302025-02-09T17:34:29Z Infinitely improvable upper bounds in the theory of polarons Нескінченнопокращені верхні границі в теорії поляронів Soldatov, A.V. An infinite convergent sequence of improving non-increasing upper bounds to the low-lying branch of the slowmoving “physical” Frohlich polaron ground-state energy spectral curve, adjacent to the ground state energy of the polaron at rest, was obtained by means of generalized variational method. The proposed approach is especially well-suited for massive analytical and numerical computations of experimentally measurable properties of realistic polarons, such as inverse effective mass tensor and excitation gap, and can be elaborated even further, without major alterations, to allow for treatment of multitudinous polaron-like models, those describing polarons of various sorts placed in external magnetic and electric fields among them. За допомогою узагальненого варіаційного методу одержано нескінченнозбіжну послідовність покращених незростаючих верхніх меж до низьколежачої гілки спектральної кривої "фізичного" полярона Фроліха в основному енергетичному стані, що повільно рухається, і яка є прилеглою до основного енергетичного стану полярона в спокої. Запропонований підхід особливо підходить для громіздких аналітичних і числових обчислень експериментально вимірюваних властивостей реалістичних поляронів, таких як тензор оберненої ефективної маси та щілина збуджень, та його можна розвинути навіть далі, без значних змін, для опису численних поляроноподібних моделей, зокрема таких, які описують полярони різних сортів, розміщених у зовнішніх магнітних та електричних полях. This work was supported by the RAS research program “Mathematical Methods in Nonlinear Dynamics” and by the RFBR grant No. 09–01–00086–a. 2010 Article Infinitely improvable upper bounds in the theory of polarons / A.V. Soldatov // Condensed Matter Physics. — 2010. — Т. 13, № 4. — С. 43701:1-9. — Бібліогр.: 16 назв. — англ. 1607-324X PACS: 71.38.-k, 71.38.Fp https://nasplib.isofts.kiev.ua/handle/123456789/32130 en Condensed Matter Physics application/pdf Інститут фізики конденсованих систем НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	An infinite convergent sequence of improving non-increasing upper bounds to the low-lying branch of the slowmoving “physical” Frohlich polaron ground-state energy spectral curve, adjacent to the ground state energy of the polaron at rest, was obtained by means of generalized variational method. The proposed approach is especially well-suited for massive analytical and numerical computations of experimentally measurable properties of realistic polarons, such as inverse effective mass tensor and excitation gap, and can be elaborated even further, without major alterations, to allow for treatment of multitudinous polaron-like models, those describing polarons of various sorts placed in external magnetic and electric fields among them.
format	Article
author	Soldatov, A.V.
spellingShingle	Soldatov, A.V. Infinitely improvable upper bounds in the theory of polarons Condensed Matter Physics
author_facet	Soldatov, A.V.
author_sort	Soldatov, A.V.
title	Infinitely improvable upper bounds in the theory of polarons
title_short	Infinitely improvable upper bounds in the theory of polarons
title_full	Infinitely improvable upper bounds in the theory of polarons
title_fullStr	Infinitely improvable upper bounds in the theory of polarons
title_full_unstemmed	Infinitely improvable upper bounds in the theory of polarons
title_sort	infinitely improvable upper bounds in the theory of polarons
publisher	Інститут фізики конденсованих систем НАН України
publishDate	2010
url	https://nasplib.isofts.kiev.ua/handle/123456789/32130
citation_txt	Infinitely improvable upper bounds in the theory of polarons / A.V. Soldatov // Condensed Matter Physics. — 2010. — Т. 13, № 4. — С. 43701:1-9. — Бібліогр.: 16 назв. — англ.
series	Condensed Matter Physics
work_keys_str_mv	AT soldatovav infinitelyimprovableupperboundsinthetheoryofpolarons AT soldatovav neskínčennopokraŝeníverhnígranicívteoríípolâronív
first_indexed	2025-11-28T19:05:53Z
last_indexed	2025-11-28T19:05:53Z
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fulltext	Condensed Matter Physics 2010, Vol. 13, No 4, 43701: 1–9 http://www.icmp.lviv.ua/journal Infinitely improvable upper bounds in the theory of polarons A.V. Soldatov∗ V.A. Steklov Mathematical Institute, Department of Mechanics, 8 Gubkina Str., 119991 Moscow, Russia Received July 5, 2010 An infinite convergent sequence of improving non-increasing upper bounds to the low-lying branch of the slow- moving “physical” Fröhlich polaron ground-state energy spectral curve, adjacent to the ground state energy of the polaron at rest, was obtained by means of generalized variational method. The proposed approach is especially well-suited for massive analytical and numerical computations of experimentally measurable prop- erties of realistic polarons, such as inverse effective mass tensor and excitation gap, and can be elaborated even further, without major alterations, to allow for treatment of multitudinous polaron-like models, those de- scribing polarons of various sorts placed in external magnetic and electric fields among them. Key words: Fröhlich polaron model, upper bound estimates, variational method, ground state energy PACS: 71.38.-k, 71.38.Fp 1. The polaron concept A local change in the electronic state in a crystal leads to the excitation of crystal lattice vibrations, i.e. the excitation of phonons. And vice versa, any local change in the state of the lattice ions alters the local electronic state. This situation is commonly referred to as an “electron– phonon interaction”. This interaction manifests itself even at the absolute zero of temperature, and results in a number of specific microscopic and macroscopic phenomena such as, for example, lattice polarization. When a conduction electron with band mass m moves through the crystal, this state of polarization can move together with it. This combined quantum state of the moving electron and the accompanying polarization may be considered as a quasiparticle with its own particular characteristics, such as effective mass, total momentum, energy, and maybe other quantum numbers describing the internal state of the quasiparticle in the presence of an external magnetic field or in the case of a very strong lattice polarization that causes self-localization of the electron in the polarization well with the appearance of discrete energy levels. Such a quasiparticle is usually called a “polaron state” or simply a “polaron”. The concept of the polaron was first introduced by L.D. Landau [1], followed by much more detailed work by S.I. Pekar [2] who investigated the most essential properties of stationary polaron in the limiting case of very intense electron-phonon interaction, in the so-called adiabatic approx- imation. Subsequently, Landau and Pekar [3] investigated the self-energy and the effective mass of the polaron for the adiabatic regime. Many other famous researchers have contributed to the development of polaron theory later on [4–9]. The polaron concept remains of interest from at least two points of view, practical and theoretical: it describes the physical properties of charge carriers in polar crystals and ionic semiconductors and, at the same time, represents a simple, but rich in content, field–theoretical model of a particle interacting with a scalar boson field. The model under consideration is the standard quantized Fröhlich polaron Hamiltonian intro- duced by H. Fröhlich [6] H = p̂ 2 2m + ~ω ∑ k b+ k bk + ∑ k Vk ( b+ k e−ikr̂ + bke ikr̂ ) , (1) ∗ E-mail: soldatov@mi.ras.ru c© A.V. Soldatov 43701-1 http://www.icmp.lviv.ua/journal A.V. Soldatov where Vk = ( 4πα V/r30 )1/2 1 k , α = e2 2~ωr0 ( 1 ε∞ − 1 ε0 ) , r0 = ( ~ 2mω )1/2 . The operators p̂ and r̂ stand for the electron momentum and position coordinate quantum opera- tors, satisfying the usual commutation relations [p̂i, r̂j ] = −i~δij , and the operators b+ k , bk, satisfying the usual commutation relations [bk, b + k′] = δkk′ , [bk, bk′ ] = 0, are Bose operators of creation and annihilation of longitudinal optical phonons of energy ~ω and wave vector k. It is assumed that the phonon wave vector runs over a very large but finite quasi- discrete set of values k = { 2π La n1, 2π La n2, 2π La n3 } , ni = 0,±1,±2, . . . ,±(L/2− 1),+L/2, i = 1, 2, 3, where a3 is the volume of the unit crystal cell and L3 is the number of these cells within the volume V of the crystal, L assumed to be even. The limit V → ∞ corresponds to the rule of the transition from the quasi-discrete to continuous spectrum lim V →∞ 1 V ∑ k · · · → 1 (2π)3 ∫ dk · · · = 1 (2π)3 2π ∫ 0 dφ π ∫ 0 dθ sin θ kD ∫ 0 dkk2 . . . to be applied to all relevant expressions. Here kD = (6π2)1/3/a is the Debye wave vector, a being the lattice constant. For any realistic, or “physical”, observable polaron, the value of kD is finite whilst the limit kD → ∞ corresponds to the so-called “field-theoretical” polaron model. It is important to emphasize from the very start that in this study we are mainly preoccupied with “physical” polaron model. Extensive useful discussion on various aspects of phenomenological polaron physics as well as on the derivation of physical quantum polaron model (1) and methods of its treatment can be found in [10] and refs. therein. For the matter of convenience it is assumed further on that ~ = ω = 1, m = 1/2. 2. Low-lying branch of the polaron energy spectrum It is known that the polaron total momentum P̂ = p̂+ ∑ k kb+ k bk is a constant of the motion and commutes with the Hamiltonian (1). Therefore, it is possible to transform the Hamiltonian to the representation in which P̂ becomes a “c”-number by means of the unitary transformation H → H̃, H̃ = S−1HS, S = exp ( −i ∑ k kr̂b+ k bk ) , so that H̃ = ( p̂− ∑ k kb+ k bk )2 + ∑ k b+ k bk + ∑ k Vk ( b+ k + bk ) , or H̃ = ( P− ∑ k kb+ k bk )2 + ∑ k b+ k bk + ∑ k Vk ( b+ k + bk ) , (2) 43701-2 Infinitely improvable upper bounds in the p̂-representation where P̂ becomes a quantum “c”-number P, the value of the polaron total momentum, and the Hamiltonian (2) no longer contains the electron coordinates. Another unitary transformation H̃ → H(f), H(f) = U−1H̃U, U = exp { ∑ k fk(b + k − bk) } , provides us with the Hamiltonian H(f) = ( P− ∑ k k(b+ k + fk)(bk + fk) )2 + ∑ k b+ k bk + ∑ k [fk + Vk](b + k + bk) + 2 ∑ k Vkfk + ∑ k f2 k , (3) or, in a much more convenient albeit equivalent form, H(f) = H0(f) +H1(f), where H0(f) = P 2 + ∑ k b+ k bk + ( ∑ k kb+ k bk )2 − α′, H1(f) = ∑ k [(1 + k2)fk + Vk](b + k + bk) + 2 ∑ km (k ·m)fkfmb + k bm + ∑ km (k ·m)fkfm(b+ k b+ m + bkbm) + 2 ∑ km (k ·m)fk(b + m bmbk + b+ k b+ m bm) − 2 ∑ k (P · k)(b+ k + fk)(bk + fk), and −α′ = 2 ∑ k Vkfk + ∑ k ( 1 + k2 ) f2 k + ( ∑ k f2 k k )2 , which is just the sole Hamiltonian to be treated further on. The ultimate goal is to find the lowest eigenvalue Eg(α,P, kD) of this Hamiltonian correspond- ing to the ground state energy of the slow-moving polaron for a given total polaron momentum P. Then, the function Eg(α,P, kD) could be expanded in powers of P as Eg(α,P, kD) = Eg(α, 0, kD) + P 2 2meff +O ( P 4 ) , where Eg(α, 0, kD) is the ground state energy of the polaron at rest and the coefficient meff can be interpreted as the polaron effective mass. In a general spatially anisotropic case, the so-called inverse mass tensor ( 1 meff ) ij = ∂2E(α,P, kD) ∂Pi∂Pj ∣ ∣ ∣ ∣ ∣ P=0 should be introduced instead of the scalar effective mass parameter meff . Another important goal would be to evaluate the so-called excitation gap, i.e. the distance between the ground and the first excited state of the slow-moving polaron Hamiltonian (3). It is worth noticing that P < 1 condition is to be imposed throughout all these calculations to avoid the creation of real phonons. Extensive work has already been done to evaluate E(α,P, kD) directly through conventional perturbational calculations or to find the upper bound estimates for its value by means of multi- tudinous variational methods. These approaches are beyond the scope of this work. It is only worth 43701-3 A.V. Soldatov mentioning that, as a rule, perturbational schemes do not provide us with reliable error bound es- timates whilst the quality of upper bounds derived by variational methods depends mostly on the choice of proper trial states in any particular case and, being this way, these bounds can- not be improved significantly, not to say infinitely, step by step, through any regular scheme of calculations. The purpose of the present research is to show that infinitely improvable upper bounds for the low-lying branch of the “physical” polaron energy spectrum E(α,P, kD) can be obtained by generalized variational method formulated for the first time in [12] and later in [14] in a different context. 3. “Physical” versus “field-theoretical” polaron Let us put fk = −Vk in (3), so that H(f) = ( P− ∑ k k(b+ k − Vk)(bk − Vk) )2 + ∑ k b+ k bk − ∑ k V 2 k . It is seen that Eg(α,P, kD) > − ∑ k V 2 k → −α 2kD π as V/r30 → ∞ (4) for arbitrary values of model parameters α,P and kD. The inequality (4) clearly shows the difference between the “physical” and “field-theoretical” polaron models. For the former, the ground state energy can only decrease no faster than linearly in α for any fixed P and kD, while the latter, as is well-known from numerous previous studies, allows for the quadratic in the electron-phonon interaction constant upper bound Eg(α, 0, kD) 6 − α2 3π , incompatible with (4) for large enough α. Nevertheless, the “field-theoretical” limit kD → ∞ has been routinely applied in quite a few polaron studies rather formally, mainly to facilitate analytical calculations of improper integrals arising along the way, without giving much thought to underlaying physics. Actually, for optical polarons observed so far in many practically important crystal substances, such as alkali halides, the value of dimensionless maximum wave vector kD is not large but rather manifestly small. For example, kD ≈ 0.3551 in units of 1/r0 for KCl, accompanied by kD ≈ 0.41 for KBr. It is also known that hardly any practical necessity exists to carry out calculations for α > 10 in the case of experimentally observable “physical” polarons. Hence, it seems perfectly justified and sensible to treat “physical” and “field-theoretical” polaron models in a different way from the very beginning. 4. Generalized variational method It was proved in [12] following the ideas outlined in [13], and also found later in [14] by a different approach, that for a quantum system represented by some Hamiltonian Ĥ and any normalized trial state \|ψ〉, such that 〈ψ\|ψ〉 = 1, Eg 6 min ( a (n) 1 , . . . , a(n)n ) 6 〈ψ\|Ĥ \|ψ〉, where the ordered by increase real numbers ( a (n) 1 , . . . , a (n) n ) are the roots of the n-th order poly- nomial equation Pn(x) = n ∑ i=0 Xix n−i = 0, 43701-4 Infinitely improvable upper bounds whereby X0 ≡ 1 and all the other coefficients Xi, 1 6 i 6 n are provided by the system of n linear equations MX+Y = 0, with Yi =M2n−i, Mij =M2n−(i+j), i, j = 1, 2, . . . n, and Mm = 〈 ψ\|Ĥm\|ψ 〉 . It is assumed that all moments Mm are finite. Moreover, it was proved that a limit exists Eg = lim n→∞ min ( a (n) 1 , . . . , a(n)n ) , and the following inequality holds min ( a (n+1) 1 , . . . , a (n+1) n+1 ) 6 min ( a (n) 1 , . . . , a(n)n ) . For example, at the first order Eg 6 a (1) 1 , a (1) 1 = 〈ψ\|Ĥ \|ψ〉, and at the second order Eg 6 min ( a (2) 1 , a (2) 2 ) = 〈ψ\|Ĥ \|ψ〉+ K3 2K2 − [ ( K3 2K2 )2 +K2 ]1/2 , a (2) 1 = 〈ψ\|Ĥ \|ψ〉+ K3 2K2 − [ ( K3 2K2 )2 +K2 ]1/2 , a (2) 2 = 〈ψ\|Ĥ \|ψ〉+ K3 2K2 + [ ( K3 2K2 )2 +K2 ]1/2 , (5) where K2 and K3 are the cumulants K2 = 〈 ψ\|(Ĥ − 〈ψ\|Ĥ \|ψ〉)2\|ψ 〉 , K3 = 〈 ψ\|(Ĥ − 〈ψ\|Ĥ \|ψ〉)3\|ψ 〉 . It is obvious that the second order upper bound (5) would lie below the first order upper bound for most physically relevant quantum models and most reasonable choices of the trial state \|ψ〉. Moreover, if 〈ψ\|Eg〉 6= 0 , then limn→∞ min ( a (n) 1 , . . . , a (n) n ) = Eg. Furthermore, an excitation gap, should there happen to be any discernable one in the spectrum, can be approximated at the n-th order by the difference Gn = a (n) 2 − a (n) 1 . 5. Infinitely improvable upper bounds for “physical” polaro n at rest For P = 0, canonically transformed Fröhlich polaron model (3) can be written down as H(f) = ∑ k b+ k bk + ( ∑ k kb+ k bk )2 − α′ + ∑ k [( 1 + k2 ) fk + Vk ] (b+ k + bk) + 2 ∑ km (k ·m)fkfmb + k bm + ∑ km (k ·m)fkfm(b+ k b+ m + bkbm) + 2 ∑ km (k ·m)fk(b + m bmbk + b+ k b+ m bm). 43701-5 A.V. Soldatov Let us choose phonon vacuum state \|0〉 as a trial state \|ψ〉 for H(f), so that inequality Eg(α, 0, kD) 6 〈0\|H(f)\|0〉 = 2 ∑ k Vkfk + ∑ k (1 + k2)f2 k holds, the right-hand side of which is minimized by fk = −Vk/(1 + k2), and, eventually, Eg(α, 0, kD) 6 −α · 2 π arctg(kD), Eg(α) 6 −α if kD → ∞. (6) The bound (6) is precisely the upper bound derived in [11] for finite kD. In order to better calculate the upper bounds at higher orders of generalized variational method it is only necessary to calculate moments 〈0\|Hm(f)\|0〉 for sufficiently large integer exponents m. This can be easily accomplished by means of the Wick theorem. The resulting multitudinous products of integrals of the kind kD ∫ 0 kpdk (1 + k2)q , p, q − non-negative integers, (7) can be evaluated analytically wherever necessary as well as all the concomitant integrals over the angular variables of the corresponding wave vectors. At the second order variational approximation (5) Eg(α, 0, kD) 6 min ( a (2) 1 , a (2) 2 ) = −α 2 π arctg(kD) + K3 2K2 − [ ( K3 2K2 )2 +K2 ]1/2 , K2 = α2 2 3π2 ( arctg(kD)− kD 1 + k2D )2 , K3 = α2 4 3π2 ( arctg(kD)− kD 1 + k2D )2 + α2 8 3π2 ( kD − 3 2 arctg(kD) + 1 2 kD 1 + k2D )( arctg(kD)− kD 1 + k2D ) + α3 8 9π3 ( arctg(kD)− kD 1 + k2D )3 , (8) lim α→∞ Eg(α, 0, kD) α 6 − 2 π arctg(kD)− 2 3π ( arctg(kD)− kD 1 + k2D ) , while lim kD→∞ Eg(α, 0, kD) 6 −α. The excitation gap is approximated at the second order as G2(α, 0, kD) = a (2) 2 − a (2) 1 = 2 [ ( K3 2K2 )2 +K2 ]1/2 with lim α→0 G2(α, 0, kD) = 0 and lim kD→∞ G2(α, 0, kD) = ∞ correspondingly. 43701-6 Infinitely improvable upper bounds 6. Infinitely improvable upper bounds for slow-moving “phys ical” polaron The same trial state \|0〉 can be employed in general case P 6= 0 leading to inequality Eg(α,P, kD) 6 〈0\|H(f)\|0〉 = P 2 + 2 ∑ k Vkfk + ∑ k ( 1 + k2 ) f2 k − 2 ∑ k (P · k)f2 k + ( ∑ k f2 kk )2 , the right-hand side of which is minimized by fk = −Vk/ [ 1− 2k ·P(1− η) + k2 ] , where η is defined self-consistently by the equation ηP = ∑ k f2 k k = ∑ k V 2 k k/ [ 1− 2k ·P(1 − η) + k2 ] , or, alternatively, by ηP 2 = ∑ k V 2 k k ·P/ [ 1− 2k ·P(1− η) + k2 ] , as was done in [11]. A compromise choice fk = −Vk/ [ 1− 2k ·P+ k2 ] , eliminating all terms linear in Bose operators b+ k , bk in (3), is equally possible too, while the simplest choice fk = −Vk/(1 + k2) seems to be the choice of preference, especially if the persistent no-real-phonons condition \|P\| < 1 is kept in mind and accounted for, because the technicalities of calculation of arbitrary order moments 〈0\|Hm(f)\|0〉 for this choice are exactly the same as they were in the case P = 0, i.e. based on the Wick theorem exclusively and without involvement of any integrations over wave vectors more complicated and laborious than the integration (7). 7. Summary It was shown that ground-state energy function Eg(α,P, kD) of the slow-moving “physical” Fröhlich polaron can be approximated from above by infinite convergent sequence of upper bounds applicable for arbitrary values of the electron-phonon interaction strength α, polaron total mo- mentum P and limiting wave vector kD. These bounds are provided by the generalized variational method. Then, various experimentally observable polaron characteristics of practical interest can be derived from these bounds. The proposed algorithm for the construction of the upper bounds is well suited for implementation by means of modern programming and computational environments destined for seamless fusion of analytical and numerical computation within the same application, such as, for example, Mathematica or Maple. The usage of the parallel computing techniques is advisable and would be highly advantageous, too, due to the intrinsic nature of the algorithm heavily relying on the Wick theorem and recursion relations for massive analytic integrations over wave vectors. The proposed approach is in no way limited to the Fröhlich polaron model considered above. It is rather universal and, being so, applicable without any major alterations to a broad range of polaron models of all sorts, including those ones concerned with manifestations of various polaron- like phenomena in quantum systems of lowered dimensions, such as quantum wells, wires and dots, with or without external electric and/or magnetic fields. It would be highly desirable to complement convergent non-increasing upper bounds to the ground state energy Eg(α,P, kD) with a sequence of infinitely improving non-decreasing lower bounds derived, for example, by means of the method of intermediate problems in the theory 43701-7 A.V. Soldatov of linear semi-bounded self-adjoint operators on rigged Hilbert space, because, under favorable conditions, this combined set of two-side bounds might provide us with virtually precise magnitude for the energy of the slow-moving polaron. An algorithm for the construction of the lower bounds of this kind was proposed in [15, 16]. It is essential to stress once again that the method of the upper bounds, as presented above, was shown to be applicable to the so-called “physical” polaron model where kD is finite, which property has been consistently employed throughout all calculations. Nevertheless, this does not mean at all that no generalization of the method to the case of the “field-theoretical” polaron model is possible. Actually, the main formal obstacle to such a generalization stems from the fact that the moments 〈0\|Hm(f)\|0〉 and, consequently, the cumulants, of higher orders diverge for kD → ∞, as illustrated, for example, by equation (8) showing linear divergence in kD of the third order cumulant K3. The appearance of divergencies of this type is not the flaw of the method itself. It is rather incurred by improper choice of the phonon vacuum state \|0〉 as the trial state \|ψ〉, which choice is well-suited for extensive “physical” polaron calculations but inappropriate for the analysis of the properties of “field-theoretical” polaron. A research on upper bounds to the “field-theoretical” polaron ground state energy derived by generalized variational method with properly designed trial states will be published elsewhere. A plain-style approach to the “field-theoretical” polaron problem is always feasible strictly within the framework of the “physical” polaron studies already undertaken, which would mean limiting the range of the interaction constant magnitude 0 < α < αmax and choosing large enough kD and large enough order n of the variational approximation scheme. Then, the ground state energy function Eg(α, kD,P) of the “physical” polaron and, consequently, its upper bounds, will approximate the corresponding function Eg(α, kD → ∞,P) of the “field-theoretical” polaron from above for this restricted domain of α. An alternative option going along the same pattern of reasoning would be to majorize initial Hamiltonian (3) from above by modifying the electron- phonon interaction as Vk → Vke −εk2 , ε > 0. This measure ensures finiteness of all moments and cumulants under the passage to the limit kD → ∞ which could be undertaken now immediately at the beginning of the calculations. Then, to obtain reasonable upper bounds to the ground state energy of the “field-theoretical” polaron for any value of α belonging to the restricted domain 0 < α < αmax, one has to choose small enough ε and carry out calculations at large enough order n of the generalized variational method. Exemplary calculations of this kind will be published elsewhere too. Acknowledgements This work was supported by the RAS research program “Mathematical Methods in Nonlinear Dynamics” and by the RFBR grant No. 09–01–00086–a. References 1. Landau L.D., Phys. Z. Sowietunion, 1933, 3, 664. [English translation: Collected Papers. Gordon and Breach, New York, 1965.] 2. Pekar S.I., Sov. Phys. JETP, 1946, 16, 341; Pekar S.I., Research in Electron Theory of Crystals. Moscow, Gostekhizdat, 1951. [English translation: Research in Electron Theory of Crystals. US AEC Report AEC–tr–5575.] 3. Landau L.D., Pekar S.I., Sov. Phys. JETP, 1948, 18, 419. 4. Fröhlich H., Pelzer H., Zienau S., Philos. Mag., 1950, 41, 221. 5. Fröhlich H., Proc. Roy. Soc. A, 1937, 160, 230. 6. Fröhlich H., Adv. Phys., 1954, 3, 325. 7. Feynman R.P., Phys. Rev., 1955, 97, 660; Feynman R.P., Statistical Mechanics. Benjamin, Reading, MA, 1972. 8. Bogolubov N.N., Ukrainian Math. J., 1950, 2, No. 2, 3. 9. Bogolubov N.N., Bogolubov N.N. (Jr.), Aspects of the Polaron Theory. JINR Communications P–17– 81–65, Dubna, 1981; Bogolubov N.N., Bogolubov N.N. (Jr.), Aspects of Polaron Theory. Equilibrium 43701-8 Infinitely improvable upper bounds and Nonequilibrium Problems. World Scientific Publishing Company, New Jersey, london, Singapore, Beijing, Shanghai, Hong Kong, Taipei, Chennai, 2008. 10. Davydov A.S., Solid State Theory. Nauka, Moscow, 1976. 11. Lee T.D., Low F., Pines D., Phys. Rev., 1953, 90, 297. 12. Soldatov A.V., Int. J. Mod. Phys. B, 1995, 9, No. 22, 2899. 13. Peeters F.M., Devreese J.T., J. Phys. A: Math. Gen., 1984, 17, 625. 14. Kireev A.N., Int. J. Mod. Phys. B, 1997, 11, No. 10, 1235. 15. Soldatov A.V., Condens. Matter Phys., 2009, 12, No. 4, 665. 16. Soldatov A.V., Phys. of Part. and Nucl., 2010, 41, No. 7, 1079. Нескiнченнопокращенi верхнi границi в теорiї поляронiв А.В. Солдатов Математичний iнститут iм. В.А. Стєклова, Вiддiл механiки, Москва, Росiя За допомогою узагальненого варiацiйного методу було отримано нескiнченнозбiжну послiдовнiсть покращених незростаючих верхнiх границь до низьколежачої гiлки спектральної кривої “фiзичного” полярона Фролiха в основному енергетичному станi, що повiльно рухається, i яка є прилеглою до основного енергетичного стану полярона в спокої. Запропонований пiдхiд особливо пiдходить для громiздких аналiтичних i числових обчислень експериментально вимiрюваних властивостей реалiстичних поляронiв, таких як тензор оберненої ефективної маси i щiлина збуджень, вiн може бути розвинутий навiть далi, без значних змiн, для опису численних поляроноподiбних моделей, зокрема таких, якi описують полярони рiзних сортiв, розмiщених у зовнiшнiх магнiтних та електричних полях. Ключовi слова: модель полярона Фролiха, оцiнки верхнiх границь, варiацiйний метод, енергiя основного стану 43701-9 The polaron concept Low-lying branch of the polaron energy spectrum ``Physical'' versus ``field-theoretical'' polaron Generalized variational method Infinitely improvable upper bounds for ``physical'' polaron at rest Infinitely improvable upper bounds for slow-moving ``physical'' polaron Summary

Infinitely improvable upper bounds in the theory of polarons

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