Condensed Matter Physics 2009, Vol. 12, No 4, pp. 665–675
Method of intermediate problems in the Fr ¤ohlich
polaron model
A.V. Soldatov∗
V.A. Steklov Mathematical Institute, Department of Mechanics, 8 Gubkina Str., 119991 Moscow, Russia
Received June 13, 2009, in final form July 14, 2209
Method of intermediate problems in the theory of linear semi-bounded self-adjoint operators on rigged Hilbert
space was applied to the investigation of the ground state energy of the Fr ¤ohlich polaron model. It was shown
that various innite sequences of non-decreasing improvable lower bound estimates for the polaron ground
state energy can be derived for arbitrary values of the electron-phonon interaction constant. The proposed
approach allows for explicit numerical evaluation of the thus obtained lower bound estimates at all orders
and can be straightforwardly generalized for investigation of the low-lying branch of the slow-moving polaron
excitation energy spectral curve adjacent to the ground state energy of the polaron at rest. In conjunction
with numerous, already derived by multitudinous methods, well-known upper bound estimates for the energy
spectral curve of the Fr¤ohlich polaron as a function of the electron-phonon interaction constant and the polaron
total momentum, the aforesaid improvable lower bound estimates might provide one with virtually precise
magnitude for the energy of the slow-moving polaron.
Key words: polaron, Fr ¤ohlich polaron model, lower bound estimates, method of intermediate problems,
ground state energy
PACS: 71.38.-k, 71.38.Fp
1. The polaron concept and the Fro¨hlich polaron model
It is well known that 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”. Hence, polaron formation is a consequence
of dynamic electron-lattice interaction.
The concept of the polaron was introduced first by L.D. Landau in a very short paper [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, so that the
polaron behavior could be analyzed in the so-called adiabatic approximation. Subsequently, Landau
and Pekar [3] investigated the self-energy and the effective mass of the polaron for the adiabatic
or strong-coupling regime. Many other famous researchers, among them H. Fro¨hlich, R. Feynman
and N.N. Bogolyubov, contributed to the development of polaron theory later [4–6,8–10]. Since
its inception, the polaron concept remains of interest from at least two points of view, practical
∗E-mail: soldatov@mi.ras.ru
c© A.V. Soldatov 665
A.V. Soldatov
and theoretical as well: 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 Fro¨hlich polaron Hamiltonian intro-
duced by H. Fro¨hlich [6]
H = pˆ
2
2m + ~ω
∑
k
a+k ak +
∑
k
(
V ∗(k)a+k e−ikrˆ + h.c.
)
, (1)
where
V (k) = −i~ωk
(
4piα
V
√
~
2mω
)1/2
.
The operators pˆ and rˆ stand for the electron momentum and position coordinate quantum opera-
tors, satisfying the usual commutation relations
[pˆi, rˆj ] = −i~δij ,
and the operators a+k , ak, satisfying the usual commutation relations
[ak, a+k′ ] = δkk′ , [ak, ak′ ] = 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 =
{ 2pi
Lan1,
2pi
Lan2,
2pi
Lan3
}
, 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(2pi)3
∫
dk · · · = 1(2pi)3
∫ 2pi
0
dφ
∫ pi
0
dθ sin θ
∫ kD
0
dkk2 . . .
to be applied to all relevant expressions. Here kD = (6pi2)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 beginning that in this study we are preoccupied mainly 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 [7] and references therein. For the matter of convenience it is assumed further on that
~ = ω = m = 1.
2. Low-lying branch of the polaron energy spectrum
It is known that the polaron total momentum
Pˆ = pˆ +
∑
k
ka+k ak
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 Pˆ becomes a “c”-number by means of
the unitary transformation
H → H˜, H˜ = S−1HS, S = exp(−i
∑
k
kra+k ak),
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Method of intermediate problems in the Fr¤ohlich polaron model
so that
H˜ = 12(pˆ −
∑
k
ka+k ak)2 +
∑
k
a+k ak +
∑
k
(
V ∗(k)a+k + h.c.
)
,
or
H˜ = 12(P −
∑
k
ka+k ak)2 +
∑
k
a+k ak +
∑
k
(
V ∗(k)a+k + h.c.
)
, (2)
in the pˆ – representation where Pˆ 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
H˜ → ˜˜H, ˜˜H = U−1H˜U, U = exp
{
−i
∑
k
(V (k)a+k − V ∗(k)ak)
}
,
provides us with the Hamiltonian
˜˜H = 12
(
P −
∑
k
k(a+k − V (k))(ak − V ∗(k))
)2
+
∑
k
a+k ak −
∑
k
|V (k)|2, (3)
which is just the sole Hamiltonian to be treated further on. The ultimate goal is to find the lowest
eigenvalue E(P ) of this Hamiltonian for a given total polaron momentum P. Then, for low-lying
polaron energy levels the function E(P ) might be represented as an expansion
E(P ) = Eg +
P 2
2meff
+O(P 4),
where Eg is the polaron ground state energy and the coefficient meff can be interpreted as the
polaron effective mass. Extensive work has already been done to evaluate E(P ) directly through
conventional perturbational calculations or to find upper bound estimates for E(P ) by means
of multitudinous variational methods. These approaches are beyond the scope of this work. It
is only worth noticing that, as a rule, perturbational schemes do not provide us with reliable
error bound estimates whilst the upper bounds can be relied upon only if they are supplemented
with corresponding lower bounds for the magnitude in question. To our knowledge, lower bounds
to the polaron energy spectrum have been receiving much less attention than the upper bounds
throughout very long history of polaron studies. Among the most remarkable contributions several
works by E.H. Lieb should be mentioned first of all [11,12] as well as the succeeding work by
D.M. Larsen [13], who improved the result of [11], though neither of the lower bounds to the
polaron ground state energy presented in these papers comes close to the best lowest upper bounds
available so far. Moreover, no algorithm has been proposed to improve these bounds step by step
in a regular way.
The purpose of the present research is to show that improvable lower bound estimates for the
low-lying branch of the polaron energy spectrum can be derived by the method of intermediate
problems in the theory of semi-bounded linear Hermitian operators on rigged Hilbert space. The
principal idea of the method goes back to H. Weyl [15] and A. Weinstein [16], and has been
elaborated since then by numerous contributors (see, for example, [17,18] and references therein)
with regard to problems of classical and quantum mechanics.
3. 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
Schro¨dinger equation
Hψ = Eψ,
667
A.V. Soldatov
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 spectrum energy levels of H are higher than the lowest
discrete spectrum energy levels of one’s interest. Let us assume, too, that these discrete eigenvalues
of H can be ordered in a nondecreasing sequence,
E1 6 E2 6 . . . , (4)
in which each degenerate eigenvalue, if any happens to be among others, appears the number of
times of its multiplicity. Eigenstates ψi, corresponding to the eigenvalues Ei, satisfy the equation
Hψi = Eiψi ,
and are assumed to be orthonormalized, so that
(ψi, ψj) = δij ,
where δij is Kronecker’s delta. It is further assumed that the Hamiltonian H can be decomposed
as
H = H0 +H ′, (5)
where H0 has 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)
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 (4) as the ones belonging to the total Hamiltonian H
E01 6 E02 6 . . . , (6)
with the degenerate eigenvalues appearing the number of times of their multiplicity. The corre-
sponding orthonormalized eigenstates ψ0i satisfy the equation
H0ψ0i = E0i ψ0i , (ψ0i , ψ0j ) = δij . (7)
Because H0 6 H in the sense of inequality
(ψ,H0ψ) 6 (ψ,Hψ)
for every ψ in the domain of H , it follows from the Weyl comparison theorem [15] that
E0i 6 Ei, (i = 1, 2, . . . ).
Therefore, the eigenvalues of H0 already provide a rough lower bound to the eigenvalues of H . The
HamiltonianH0 is called the base Hamiltonian as usual. It is worth noticing that the decomposition
(5) 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 Hamil-
tonian H from below by a non-decreasing sequence of the so-called truncated intermediate Hamil-
tonians 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, . . . ),
H l,k 6 H l,k+1 6 H, (l, k = 1, 2, . . . ). (8)
Therefore, the Hamiltonians H l,k increase whatever index k or l is increased and thus must give
improvable lower bounds for the lowest eigenvalues of the original Hamiltonian H . It was shown
[19] 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, . . . ). (9)
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Method of intermediate problems in the Fr¤ohlich polaron model
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
(ψ0i , ψ)E0i ψ0i +E0l+1[ψ −
l
∑
i=1
(ψ0i , ψ)ψ0i ], (l = 1, 2, . . . ), (10)
or, alternatively, in Dirac’s more transparent bra and ket notation
H l,0 =
l
∑
i=1
E0i |E0i 〉〈E0i | +E0l+1[Iˆ −
l
∑
i=1
|E0i 〉〈E0i | ], (l = 1, 2, . . . ),
where Iˆ stands for the identity operator. Truncations of H0 satisfy the inequalities
H l,0 6 H l+1,0 6 H0, (l = 1, 2, . . . ), (11)
which were proved in general case in [20].
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 linearly independent vectors p1, p2, . . . , pk:
P kφ =
k
∑
i=1
αipi , (12)
where constants αi must satisfy the equations
[pj , P kφ] = [pj , φ] =
k
∑
i=1
αi[pj , pi], (j = 1, 2, . . . , k). (13)
Here an auxiliary inner product with respect to the metric operator H ′ was introduced as
[ψ, φ] = (ψ,H ′φ) =
∫
ψ∗H ′φdτ
for every pair of vectors ψ, φ for which H ′ψ and H ′φ are defined. These vectors pi are to be
normalizable in the sense of the original inner product (6), i. e.
(pi, pi) = Ni, Ni < +∞, (i = 1, 2, . . . , k)
but neither their explicit normalization nor orthonormalization are 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, . . . ),
which in terms of original inner product reads as
0 6 (φ,H ′P kφ) 6 (φ,H ′P k+1φ) 6 (φ,H ′φ), (k = 1, 2, . . . ). (14)
From equations (12), (13) it follows that
H ′P kφ =
k
∑
i,j=1
(H ′pi, φ)bijH ′pj ,
where bij are the elements of the matrix inverse to the matrix with terms [pj , pi]. As a consequence
of equation (14)
H ′P k 6 H ′P k+1 6 H ′, (k = 1, 2, . . . ), (15)
669
A.V. Soldatov
and the intermediate truncated Hamiltonians Hk defined as
Hk = H0 +H ′P k, (k = 1, 2, . . . )
satisfy inequalities
Hk 6 Hk+1 6 H
by construction if inequalities (15) are taken into account. According to equations (11) and (15)
H l,k 6 H l,k+1 6 H l,0 +H ′ 6 H, (l, k = 1, 2, . . . ).
Therefore, the lowest ordered eigenvalues E l,ki of H l,k must satisfy the parallel inequalities
El,ki 6 E
l+1,k
i 6 Eki 6 Ei, (i, l, k = 1, 2, . . . ), (16)
and
El,ki 6 E
l,k+1
i 6 Ei, (i, l, k = 1, 2, . . . ), (17)
thus providing improvable lower bounds for the original eigenvalues Ei of the Hamiltonian H .
As was proved in [20], the so-constructed Hamiltonian H l,k can have no continuous spectrum
and must have E0l+1 as an eigenvalue of infinite multiplicity. Therefore, only those eigenvalues
of H l,k that are smaller or equal to E0l+1 can be considered as lower bound estimates for the
eigenvalues of the initial Hamiltonian H .
The truncation procedure (10) 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 + H˜ ′, (l = 1, 2, . . . ),
where the difference H˜ ′ = 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, . . . )
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 (10).
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). It was proved in [19] that those eigenvalues of the
Hamiltonian (9) different from E0l+1 (and also from E01 , . . . , E0l , should there be any eigenvalues of
this kind in some special cases ) are the roots of the equation
det











(pj , H ′pi) +
l
∑
ν=1
(ψ0ν , H ′pi)(H ′pj , ψ0ν)
E0ν −E
+
(H ′pj , H ′pi) −
l
∑
ν=1
(ψ0ν , H ′pi)(H ′pj , ψ0ν)
E0l+1 −E











= 0. (18)
Each solution of (18) provides n linear independent eigenfunctions of H l,k where n is the nullity
of the coefficient matrix in (18). If the number of such eigenfunctions is less than k + l then it
is necessary to verify if some of the eigenvalues E01 , E02 , . . . , E0l of H l,0 are also the eigenvalues of
H l,k. A verification algorithm for the test of this assumption was outlined in [19].
From the consideration above it is clear that H l,k may possess at most l + k eigenvalues
different from E0l+1. Let us notice that E0l+1 is an eigenvalue of H l,k of infinite multiplicity by
construction and each of its corresponding eigenfunctions is orthogonal to all other eigenfunctions of
the Hamiltonian (9). To obtain the lower bounds for the eigenvalues Ei of the original Hamiltonian
670
Method of intermediate problems in the Fr¤ohlich polaron model
H , the so-obtained eigenvalues of H l,k lying below E0l+1 are to be ordered in a nondecreasing
sequence
El,k1 6 El,k2 6 · · · 6 El,kt , t 6 l + k, (19)
in which each eigenvalue is repeated according to its multiplicity. Then the lower bounds result
from the fundamental inequalities (16) and (17) as
El,ki 6 Ei, (i = 1, 2, . . . , t), E0l+1 6 Ei, (i = t+ 1, t+ 2, . . . ). (20)
4. Intermediate problems for the Fro¨hlich polaron model
For the particular Hamiltonian (2) the Hamiltonians H0 and H ′ in (5) can be identified with
H0 =
∑
k
a+k ak −
∑
k
|V (k)|2 − ε (21)
and
H ′ = 12
(
P −
∑
k
k(a+k − V (k))(ak − V ∗(k))
)2
+ ε, (22)
respectively. Here an arbitrary positive parameter ε was introduced formally and identically to
ensure strict positivity of the Hamiltonian H ′ as required by the positivity condition (7). Later this
parameter can be chosen arbitrarily small or may even be employed appropriately as a variational
parameter. Because by construction ˜˜H = H0 +H ′ ∀ ε, then Eg > E0g +E′g, where Eg, E0g and E′g
are the ground-state energies of ˜˜H , H0 and H ′ respectively. Since E ′g = ε, we have
Eg > −
∑
k
|V (k)|2 → −α
√
2kD
pi as V → ∞. (23)
Inequality (23) clearly shows the above-mentioned difference between the so-called “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 kD while the latter, as is well-known, allows for the asymptotic
behavior quadratic in the electron-phonon interaction constant, e.g.
Eg 6 −
α2
3pi ,
incompatible with (23) for large enough α. The eigenvalues of H0 are equidistant
E01 = −ε−
∑
k
|V (k)|2, E02 = 1 − ε−
∑
k
|V (k)|2, . . . , E0n = (n− 1) − ε−
∑
k
|V (k)|2,
and highly degenerate except for the ground state, which is the phonon vacuum state |0〉, but the
multiplicity of all of them remains finite until the limiting transition V → ∞ is carried out. The
corresponding normalized eigenstates of H0 are
ψ1 = |0〉, ψ2(k) = a+k |0〉 ≡ |k〉, . . . ,
bgψn(k1, . . . ,kn−1) ≡ |k1, . . . ,kn−1〉 ∼ a+k1a
+
k2 . . . a
+
kn−1 |0〉.
These groups of eigenstates span subspaces with zero, one and n−1 phonons respectively. Now, let
us design a sequence of the truncated Hamiltonians H l,0 for the polaron model. For our purposes
it would be convenient to choose them as
gH1,0 = E01 |0〉〈0| +E02 [Iˆ − |0〉〈0| ], . . . , (24)
Hn,0 = E01 |0〉〈0| + · · · +E0n
˜∑
k1,...,kn−1
|k1, . . . ,kn−1〉〈k1, . . . ,kn−1|
+E0n+1

Iˆ − |0〉〈0| −
∑
k
|k〉〈k| − · · · − ˜
∑
k1,...,kn−1
|k1, . . . ,kn−1〉〈k1, . . . ,kn−1|

 , (25)
671
A.V. Soldatov
so that ˜˜H = H0 +H ′ = H1,0 +H ′1 = · · · = Hn,0 +H ′n , (26)
where
H ′n = H0 −Hn,0 +
1
2
(
P −
∑
k
k(a+k − V (k))(ak − V ∗(k))
)2
+ ε. (27)
Summation ˜
∑
k1,...,kn
in taken over the states |k1, . . . ,kn〉 in such a way that all the equivalent
phonon states distinct from each other exclusively by permutation of quantum numbers k1, . . . ,kn
are counted only once. Because H0 −Hn,0 are non-negative by construction, all the Hamiltonians
(27) are positively definite. Hence, any of them may serve as a new H ′ in the intermediate problems
formalism. Substituting our choice (24) and (27) for H l,0 and H ′ in equation (18) and taking
recourse again to conventional Dirac’s bra and ket notation throughout, we arrive at
det
{
〈pj |H ′1|pi〉 +
〈0|H ′1|pi〉〈pj |H ′1|0〉
E01 −E
+ 〈pj |(H
′
1) 2|pi〉 − 〈0|H ′1|pi〉〈pj |H ′1|0〉
E02 −E
}
= 0,
i, j = 1, 2, . . . , k. (28)
This equation is a source of multitudinous lower bounds to the ground state energy of ˜˜H lying
below the onset of the continuous spectrum. The proximity of these lower bounds to the true value
is stipulated by the number of the trial states {|pi〉, i = 1, . . . , k} and by the particular choice of
these states. At the same time, for any number and choice of the trial states the resulting equation
(28) will always be polynomial in E. It is very important to stress that neither part of the original
Hamiltonian (3) has been omitted so far upon deriving the principle equation (28) and no other
approximations have been made. As a consequence, the precision of the lower bounds (28) depends
totally on the particular choice of the states {|pi〉, i = 1, . . . , k} and the size of this set of states.
The larger the size k of the set, the larger the subspace spanned by this set would be providing
that {|pi〉, i = 1, . . . , k} are linearly independent. Of course, it is worth noticing that the problem
of convergence of the lower bounds to the precise values of the discrete spectrum has not been
solved yet for the method of intermediate problems in general though some important encouraging
partial results have been obtained for some classes of self-adjoint linear operators on Hilbert space.
Therefore, it is only natural to believe that for large enough set of the trial states (28) would
yield reliable lower bounds for the polaron ground state energy. Anyway, it is already seen from
the formalism outlined above that the method of intermediate problems delivers improvable lower
bounds and, regarding the choice of the states {|pi〉, i = 1, . . . , k}, seems to be flexible enough to
achieve sufficiently good for all practical purposes lower bounds already for a set of trial states of
moderate size k. As to the particular choice of these trial states, it is nearly obvious even from the
purely technical point of view that the natural choice for the first trial state |p1〉 would be
|p1〉 = |0〉, (29)
so that equation (28) reads
〈0|H ′|0〉 + 〈0|H
′|0〉〈0|H ′|0〉
E01 −E
+ 〈0|(H
′) 2|0〉 − 〈0|H ′|0〉〈0|H ′|0〉
E02 − E
= 0. (30)
Despite simplicity of this equation, the stemming from its lower bound improves the trivial lower
bound (23) by default due to the contribution from the positive definite operator (22) exclusively.
The rest H0 −H1,0 of the operator (27) cannot contribute to (30) at all because, in general case,
(H0 −Hn,0)|k1, . . . ,km〉 = 0, if m 6 n.
Also, it is easy to see that
〈0|H ′n|k1, . . . ,km〉 = 〈0|H ′|k1, . . . ,km〉 ∀m and 〈0|H ′|k1, . . . ,km〉 = 0 for m > 3.
672
Method of intermediate problems in the Fr¤ohlich polaron model
Therefore, for the n-th order partitioning (26) of the original Hamiltonian starting from n > 3 and
the choice (29),
det









〈0|H ′|0〉 + 〈0|H
′|0〉〈0|H ′|0〉
E01−E
+
∑
k
〈k|H ′|0〉〈0|H ′|k〉
E02−E
+
˜∑
k1,k2
〈k1,k2|H ′|0〉〈0|H ′|k1,k2〉
E03−E
+
〈0|(H ′) 2|0〉−〈0|H ′|0〉〈0|H ′|0〉−
∑
k
〈k|H ′|0〉〈0|H ′|k〉− ˜
∑
k1,k2
〈k1,k2|H ′|0〉〈0|H ′|k1,k2〉
E0n+1−E








= 0.
(31)
The last term in (31) can be omitted in the limit n→ ∞, En+1 → ∞, and the resulting equation
provides the best lower bound available for the particular choice (29) of |p1〉.
It is also seen from the structure of (18) that, in general, neither particular choice of only one
single trial state |p1〉 can produce lower bound to the ground state energy higher than the second
eigenvalue E02 of the Hamiltonian H0 unless, of course, 〈0|H ′|p1〉 = 0. To obtain the higher lying
(and thus better) lower bounds one has to extend the set of trial states {|pi〉} and, consequently, to
enlarge the subspace on which H ′ is projected. In principle, there are many ways of choosing these
additional trial functions but it would be prudent to choose them so as to ensure explicit analytical
calculability of all matrix elements in equation (18) first of all. Then, it would be desirable to project
H ′ onto the subspace formed by those trial states which have already been used in polaron studies
successfully, for example, the states employed in variational approaches aimed at derivation of the
best available upper bounds to the polaron energy spectrum. Among them, the Lee-Law-Pines
variational function
|ψ(0)LLP〉 = U−1 · ULLP(f (0)(k))|0〉 = U−1 · exp
{
∑
k
(a+k f (0)(k) − akf (0)∗(k))
}
|0〉,
introduced in [14], meets these requirements and can be incorporated into the set of trial states
as |p2〉. Here the function f (0)(k) might subsequently be chosen to maximize the lower bound
(18), which task seems to be more complicated than the one of the upper bound minimization
undertaken in [14]. Or, alternatively, the value of f(k) can be taken exactly the same as in [14]. In
the same fashion, generalized Lee-Law-Pines variational states of the type
|ψ(1)LLP〉 = U−1 · ULLP(f (1)(k))
(
d(1)0 |0〉 +
∑
k
d(1)1 (k)|k〉
)
, . . . , (32)
|ψ(n)LLP〉 = U−1 ·ULLP(f (n)(k))

d(n)0 |0〉+
∑
k
d(n)1 (k)|k〉+. . .+
˜∑
k1,...,kn
d(n)n (k1, . . . ,kn)|k1, . . . ,kn〉

 ,
introduced by D.M. Larsen [13], can be employed as |p3〉, |p4〉, . . . , |pn+2〉 trial states in (18) in
order to increase the dimensionality of the trial subspace.
5. Summary
It was shown that there are many ways of applying the method of intermediate problems to the
investigation of the physical Fro¨hlich polaron model for which kD is finite. Each way relies on its
own choice of the original Hamiltonian partitioning (26), complemented with the customized choice
of trial variational states {|pi〉, i = 1, . . . , k}, and, therefore, yields its own sequence of improvable
lower bounds to the ground state energy of the polaron at rest as well as to the adjacent low-lying
673
A.V. Soldatov
branch of the excitation energy spectral curve of the moving polaron. Convergence of the thus
obtained lower bounds to the respective true values has not been proved in general yet and, at the
present stage, can be judged upon in the course of practical numerical calculations only. The sets
{|pi〉, i = 1, . . . , k} can be chosen to make analytical calculation of all requisite matrix elements
feasible. At the same time, any realistic calculations will inadvertently involve a lot of routine
work. Because of this, the choice (32) for the variational states looks especially attractive as it
allows for relatively easy implementation of the matrix element calculation algorithm by means of
contemporary programming tools for analytical computation such as Mathematica or Maple.
It is essential to stress once again here that the method of intermediate problems, as it was
presented above, was shown to be applicable to the so-called “physical” polaron model where kD
is finite, which property has been employed consistently 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 Hamiltonian (21) becomes unbounded from below for kD → ∞. In other words, in the
partitioning (21), (22) of the original Hamiltonian (3) the Hamiltonian (22) is “too positive” whilst
the Hamiltonian (21) is, in some sense, “not positive enough”, which makes the whole partitioning
too unbalanced for successful application of the method in the limiting case kD → ∞. Fortunately,
much more balanced, albeit not so simple, partitioning schemes exist for the Fro¨hlich polaron
model, for example, the partitionings derived in [11,12], which surely allow for treatment of the
“field-theoretical” polaron model. The analysis of this and other balanced partitioning schemes with
respect to the method of intermediate problems will be presented somewhere else later. Moreover,
the particular approach to the intermediate problem construction employed in this study is not
unique. Other implementations of the general idea of the method of intermediate problems may
happen to be free from deficiencies of the presented approach and better suited for the investigation
of the “field-theoretical” polaron model.
The method outlined here is in no way limited to polaron studies. Actually, it might be applied
to many similar quantum models employed in solid state physics provided that these models possess
a gap in the spectrum of the free field in the free Hamiltonian H0 like it was the case here with the
free phonon field. It is also desirable for the rest part of H0, related to the free “matter” part of
the model, to possess at least several discrete energy states lying below the onset of the respective
continuous spectrum of the “matter” part. If this is not the case, a canonical transformation,
eliminating the “matter” variables from the “field-matter” interaction in similarity to (3), should
exist thus making the approach applicable.
Acknowledgements
This work is supported by the RAS research program “Fundamental Problems of Nonlinear
Dynamics” and by the RFBR grant No. 09–01–00086–a.
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Method of intermediate problems in the Fr¤ohlich polaron model
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Метод промiжних задач у моделi полярона Фрьолiха
В.А. Солдатов
Математичний iнститут iм. В.А. Стєклова, вул. Губкiна 8, Москва 119991, Росiя
Отримано 13 червня 2009 р., в остаточному виглядi – 14 липня 2009 р.
Метод промiжних задач в теорiї лiнiйних напiвобмежених самоспряжених операторiв у пристосова-
ному просторi Гiльберта використовується для дослiдження енергiї основного стану в моделi поля-
рона Фрьолiха. Показано, що для довiльних значень константи електрон-фононної взаємодiї можуть
бути отриманi рiзнi безмежнi неспаднi послiдовностi оцiнок знизу для енергiї основного стану поля-
рона. Запропонований пiдхiд дозволяє отримати точнi числовi оцiнки знизу у всiх порядках i може
бути узагальнений для вивчення нижньої гiлки енергетичного спектру збуджених станiв повiльного
полярона, що межує з енергiєю основного стану нерухомого полярона. В комбiнацiї з iншими добре
вiдомими пiдходами до оцiнки зверху залежностей енергетичного спектру полярона Фрьолiха вiд
константи електрон-фононної взаємодiї i повного iмпульсу полярона вищезгаданi оцiнки з уточне-
нням знизу можуть забезпечити знаходження практично точних значень для енергiї полярона, що
повiльно рухається.
Ключовi слова: полярон, модель полярона Фрьолiха, оцiнки знизу, метод промiжних задач,
енергiя основного стану
PACS: 71.38.-k, 71.38.Fp
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