Spectrum of Two-Magnon non-Heisenberg Ferromagnetic Model of Arbitrary Spin with Impurity
We consider a two-magnon system in the isotropic non-Heisenberg ferromagnetic model of an arbitrary spin s on a n-dimensional lattice Zⁿ. We establish that the essential spectrum of the system consists of the union of at most four intervals. We obtain lower and upper estimates for the number of thre...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2013
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| Цитувати: | Spectrum of Two-Magnon non-Heisenberg Ferromagnetic Model of Arbitrary Spin with Impurity / S.M. Tashpulatov // Журнал математической физики, анализа, геометрии. — 2013. — Т. 9, № 2. — С. 239-265. — Бібліогр.: 11 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860114522821361664 |
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| author | Tashpulatov, S.M. |
| author_facet | Tashpulatov, S.M. |
| citation_txt | Spectrum of Two-Magnon non-Heisenberg Ferromagnetic Model of Arbitrary Spin with Impurity / S.M. Tashpulatov // Журнал математической физики, анализа, геометрии. — 2013. — Т. 9, № 2. — С. 239-265. — Бібліогр.: 11 назв. — англ. |
| collection | DSpace DC |
| container_title | Журнал математической физики, анализа, геометрии |
| description | We consider a two-magnon system in the isotropic non-Heisenberg ferromagnetic model of an arbitrary spin s on a n-dimensional lattice Zⁿ. We establish that the essential spectrum of the system consists of the union of at most four intervals. We obtain lower and upper estimates for the number of three-particle bound states, i.e., for the number of points of discrete spectrum of the system.
Рассмотрена двухмагнонная система в изотропной негейзенберговской ферромагнитной модели с произвольным значением спина s в n-мерной решетке Zⁿ. Установлено, что существенный спектр системы состоит из объединения не более чем четырех отрезков. Получены нижняя и верхняя оценки для количества точек дискретного спектра системы, т.е. для числа трехчастичных связанных состояний системы в n -мерной решетке Zⁿ.
|
| first_indexed | 2025-12-07T17:35:45Z |
| format | Article |
| fulltext |
Journal of Mathematical Physics, Analysis, Geometry
2013, vol. 9, No. 2, pp. 239–265
Spectrum of Two-Magnon non-Heisenberg
Ferromagnetic Model of Arbitrary Spin with Impurity
S.M. Tashpulatov
Institute of Nuclear Physics Academy of Sciences of Republic Uzbekistan
Tashkent, Ulugbek, Uzbekistan
E-mail: toshpul@mail.ru; sadullatashpulatov@yandex.ru
Received May 12, 2011, revised June 6, 2012
We consider a two-magnon system in the isotropic non-Heisenberg fer-
romagnetic model of an arbitrary spin s on a ν-dimensional lattice Zν . We
establish that the essential spectrum of the system consists of the union of
at most four intervals. We obtain lower and upper estimates for the num-
ber of three-particle bound states, i.e., for the number of points of discrete
spectrum of the system.
Key words: non-Heisenberg ferromagnet, essential spectrum, discrete
spectrum, three-particle discrete Schrödinger operator, compact operator,
finite-dimensional operator, lattice, spin.
Mathematics Subject Classification 2010: 46L60, 47L90, 70H06. 70F05,
81Q10, 45B05, 45C05, 47B25, 47G10, 34L05.
We consider a two-magnon system in the isotropic non-Heisenberg ferromag-
netic model of an arbitrary spin s with impurity on a ν-dimensional lattice Zν
and study the discrete and essential spectra of the system. The system consists
of three particles: two magnons and an impurity spin.
The Hamiltonian of the system has the form
Hreg = −
∑
m,τ
2s∑
n=1
Jn(Sz
mSz
m+τ − s2 +
1
2
(S+
mS−m+τ + S−mS+
m+τ ))
n
−
∑
τ
2s∑
n=1
(J0
n − Jn)(Sz
0Sz
τ − s2 +
1
2
(S+
0 S−τ + S−0 S+
τ ))n (1)
and acts on the symmetric Fock space H . Here Jn > 0 are the parameters of the
multipole exchange interaction between the nearest-neighbor atoms in the lattice
Zν , J0
n 6= 0 are the atom-impurity multipole exchange interaction parameters,
c© S.M. Tashpulatov, 2013
S.M. Tashpulatov
~Sm = (Sx
m; Sy
m; Sz
m) is the atomic spin operator of spin s at the lattice site m,
and τ = ±ej , j = 1, 2, . . . , ν, where ej are the unit coordinate vectors. Let ϕ0
denote the vacuum vector uniquely defined by the conditions S+
mϕ0 = 0 and
Sz
mϕ0 = sϕ0, where ||ϕ0|| = 1. We set S±m = Sx
m ± iSy
m, where S−m and S+
m
are the magnon creation and annihilation operators at the site m. The vector
S−mS−n ϕ0 describes the state of the system of two magnons located at the sites m
and n with spin s. The vectors { 1√
4s2+(4s2−4s)δm,n
S−mS−n ϕ0} form an orthonormal
system. Let H2 be the Hilbert space spanned by these vectors. The space is
called the two-magnon space of the operator H. We also denote the restriction of
H to H2 by H2.
Proposition 1. The space H2 is an invariant subspace of H. The operator
H2 = H/H2 is a bounded self-adjoint operator generating a bounded self-adjoint
operator H2 whose kernel in the momentum representation, i.e., in L2(T ν), is
given by the formula
(H̃2f)(x; y) = h(x; y)f(x; y)+
∫
T ν
h1(x; y; t)f(t; x+y−t)dt+D
∫
T ν
h2(x; s)f(s; y)ds
+E
∫
T ν
h3(y; t)f(x; t)dt +
∫
T ν
∫
T ν
h4(x; y; s; t)f(s; t)dsdt, (2)
where
h(x; y) = 8sA
ν∑
i=1
[1− cos
xk + yk
2
cos
xk − yk
2
]
and
h1(x; y; t) = −4s(2s− 1)B
×
ν∑
i=1
{1 + cos(xk + yk)− 2 cos
xk + yk
2
cos
xk − yk
2
} − 4C
ν∑
i=1
{cos
xk − yk
2
− cos
xk + yk
2
} cos(
xk + yk
2
− tk), x, y, t ∈ T ν , h2(x; s) =
ν∑
i=1
{1 + cos(xi − si)
− cos si − cosxi}, h3(y; t) =
ν∑
i=1
{1 + cos(yi − ti)− cos ti − cos yi},
and
h4(x; y; s; t) = F
ν∑
i=1
[1 + cos(xi + yi − si − ti) + cos(si + ti) + cos(xi + yi)
240 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2
Spectrum of Two-Magnon non-Heisenberg Ferromagnetic Model
− cos(xi−si−ti)−cos(yi−si−ti)−cosxi−cos yi]+Q
ν∑
i=1
[cos(xi−ti)+cos(yi−si)]
+M
ν∑
i=1
[cos(xi − si) + cos(yi − ti)] + N
ν∑
i=1
[cos si + cos ti + cos(xi + yi − si)
+ cos(xi + yi − ti)],
here
A = J1−2sJ2+(2s)2J3+. . .+(−1)2s+1J2s, B = J2−(6s−1)J3+(28s2−10s+1)J4−
(120s3−68s2+14s−1)J5+. . . , C = J1+(4s2−6s+1)J2−(24s3−32s2+10s−1)J3+
(112s4−160s3+72s2−14s+1)J4−(480s5−768s4+448s3−128s2+18s−1)J5+. . . ,
D = −2
∑2s
k=1(−2s)k(J0
k − Jk), E = D, F = (2s− 4s2)(J0
2 − J2) + (2s− 16s2 +
24s3)(J0
3 − J3) + . . . + . . . , Q = (−4s2 + 2s)(J0
2 − J2) + (−4s + 20s2− 24s3)(J0
3 −
J3) + . . . + . . . , M = 2[(J0
1 − J1) − (1 + 5s + 2s2)(J0
2 − J2) + (1 − 8s + 22s2 −
12s3)(J0
3 −J3)+ . . .+ . . .], N = −(J0
1 −J1)+ (1− 6s+4s2)(J0
2 −J2)− (1− 10s+
32s2 − 24s3)(J0
3 − J3) + . . . + . . . .].
In the isotropic non-Heisenberg ferromagnetic model of an arbitrary spin s
with impurity, the spectral properties of the above operator in the two-magnon
case are closely related to those of its two-particle subsystems. The initial system
is usually called a three-particle system, and the corresponding Hamiltonian is
called a three-particle operator. We first study the spectrum and the correspond-
ing eigenvectors, which we call the localized impurity states (LIS) of one-magnon
impurity systems, and the spectrum and the corresponding eigenvectors, which
we call the bound states (BS) of two-magnon systems.
1. One-Magnon Impurity States
The spectrum and the LIS in the one-magnon case of the isotropic non-
Heisenberg ferromagnetic model of arbitrary spin with impurity were studied
in [1].
The Hamiltonian of a one-magnon impurity system also has the form (1).
The vector S−mϕ0 describes the one magnon state of spin s located at the site
m. The vectors { 1√
2s
S−mϕ0} form an orthonormal system. Let H1 be the Hilbert
space spanned by these vectors. It is called the space of one-magnon states of
the operator H. Denote by H1 the restriction of the operator H to the space H1.
Proposition 2. The space H1 is an invariant subspace of the operator H.
The operator H1 = H/H1 is a bounded self-adjoint operator generating a bounded
self-adjoint operator H1 acting on the space l2(Zν) according to the formula
(H1f)(p) =
ν∑
k=1
(−1)k+1Jks
k
∑
p,τ
2k−1[2f(p)− f(p + τ)− f(p− τ)]
Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2 241
S.M. Tashpulatov
+
ν∑
k=1
(−1)k+1(J0
k − Jk)(2s)k
∑
p,τ
(f(τ)− f(0))(δp,τ − δp,0), (3)
where δk,j is the Kronecker symbol, and the summation over τ is over the nearest
neighbors. The operator H1 acts on the vector ψ = (2s)−1/2
∑
p f(p)S−p ϕ0 ∈ H1
by the formula
H1ψ =
∑
p
(H1f)(p)
1√
2s
S−p ϕ0. (4)
Proposition 2 is proved by using the well-known commutation relations for
the operators S+
m, S−p , and Sz
q : [S+
m, S−n ] = 2δm,nSz
m, [Sz
m, S±n ] = ±δm,nS±m.
Lemma 1. The spectra of the operators H1 and H1 coincide.
P r o o f. Because H1 and H1 are bounded self-adjoint operators, it follows
that if λ ∈ σ(H1), then the Weyl criterion (see [2]) implies that there is a sequence
{ψn}∞n=1 such that ||ψn|| = 1 and
lim
n→∞ ||H1ψn − λψn|| = 0. (5)
We set ψn = (2s)−1/2
∑
p fn(p)S−p ϕ0.
Then
||H1ψn − λψn||2 = (H1ψn − λψn,H1ψn − λψn)
=
∑
p
||(H1fn(p)− λfn(p)||2( 1√
2s
S−p ϕ0,
1√
2s
S−p ϕ0) = ||H1Fn − λFn||2
×(
1
2s
S+
p S−p ϕ0, ϕ0) = ||(H1−λ)Fn||2( 1
2s
2sϕ0, ϕ0) = ||(H1−λ)Fn||2 → 0, n →∞.
Here Fn = (fn(p))p∈Zν and ||Fn||2 =
∑
p |fn(p)|2 = ||ψn||2 = 1. It follows that
λ ∈ σ(H1). Consequently, σ(H1) ⊂ σ(H1). Conversely, let λ ∈ σ(H1). Then, by
the Weyl criterion, there is a sequence {Fn}∞n=1 such that
||Fn|| =
√∑
p
|fn(p)|2 = 1 and ||(H1Fn − λFn|| → 0, n →∞. (6)
We conclude that ||ψn|| = ||Fn|| = 1 and ||H1Fn − λFn|| = ||H1ψn − λψn||.
Thus (6) and the Weyl criterion imply that λ ∈ σ(H1) and hence σ(H1) ⊂ σ(H1).
These two relations imply that σ(H1) = σ(H1).
The spectrum and the LIS of the operator H1 can be easily studied in its
quasimomentum representation. Denote by F the Fourier transformation
F : l2(Zν) → L2(T ν).
242 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2
Spectrum of Two-Magnon non-Heisenberg Ferromagnetic Model
Here T ν is the ν− dimensional torus endowed with the normalized Lebesgue
measure dλ : λ(T ν) = 1.
Proposition 3. The operator H̃1 = FH1F−1 acts on the space L2(T ν) by
the formula
(H̃1f)(x) = p(s)h(x)f(x) + q(s)
∫
T ν
h1(x; t)f(t)dt, (7)
where h(x) = ν −∑ν
i=1 cosxi, h1(x; t) = ν +
∑ν
i=1[cos(xi − ti) − cosxi − cos ti],
p(s) = −2
∑2s
k=1(−2s)kJk, q(s) = −2
∑2s
k=1(−2s)k(J0
k − Jk), t ∈ T ν .
To prove Proposition 3, the Fourier transform of (3) should be considered
directly.
It is clear that the continuous spectrum of the operator H̃1 is independent
of q(s)h1(x; t) and it fills the whole closed interval [mν ; Mν ], where mν =
minx∈T ν p(s)h(x), Mν = maxx∈T ν p(s)h(x).
Definition 1. An eigenfunction ϕ ∈ L2(T ν) of the operator H̃1 correspond-
ing to an eigenvalue z /∈ [mν ; Mν ] is called the LIS of the operator H̃1, and z is
called the energy of this state.
We consider the operator Kν(z) acting on the space H̃1 according to the
formula
(Kν(z)f)(x) =
∫
T ν
h1(x; t)
p(s)h(t)− z
f(t)dt, x, t ∈ T ν .
It is a compact operator in the space H̃1 for the values z lying outside the set
Gν = [mν ; Mν ].
Set
∆ν(z) = (1 + q(s)
∫
T ν
(1− cos t1)(ν −
∑ν
i=1 cos ti)dt
p(s)h(t)− z
)× (1 + q(s)
∫
T ν
sin2 t1dt
p(s)h(t)− z
)ν
×(1 +
q(s)
2
∫
T ν
(cos t1 − cos t2)2dt
p(s)h(t)− z
)ν−1, (8)
where dt = dt1dt2 . . . dtν .
Lemma 2. A number z0 /∈ [mν ; Mν ] is an eigenvalue of the operator H̃1 if
and only if it is a zero of the function ∆ν(z), i.e., ∆ν(z0) = 0.
P r o o f. In the case under consideration, the equation for the eigenvalues
is an integral equation with a degenerate kernel. Therefore it is equivalent to a
Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2 243
S.M. Tashpulatov
homogeneous linear system of algebraical equations. A homogeneous linear sys-
tem of algebraic equations has a nontrivial solution if and only if the determinant
of the system is zero. Taking into account that the function h(s1; s2; . . . ; sν) is
symmetric and carrying out the corresponding transformations, we present the
determinant of the system in the form ∆ν(z).
We denote a set of all pairs ω = (p(s); q(s)) by Ω and introduce the following
subsets in Ω for ν = 1 :
A1 = {ω : p(s) > 0,−p(s) ≤ q(s) < 0}, A2 = {ω : p(s) > 0, q(s) < −p(s)},
A3 = {ω : p(s) < 0, q(s) < p(s)}, A4 = {ω : p(s) > 0, p(s) < q(s)},
A5 = {ω : p(s) > 0, 0 < q(s) ≤ p(s)}, A6 = {ω : p(s) < 0, q(s) ≥ p(s)},
A7 = {ω : p(s) < 0, 0 < q(s) < −p(s)}, A8 = {ω : p(s) < 0, q(s) > −p(s)}.
We write
z1 = − [p(s) + q(s)][p(s)− 3q(s) +
√
D]
4q(s)
,
z2 =
[p(s) + q(s)]2
2q(s)
,
z3 = − [p(s) + q(s)][p(s)− 3q(s)−√D]
4q(s)
,
where D = [p(s) + q(s)][p(s) + 9q(s)].
The following theorem describes the variation of the energy spectrum of the
operator H̃1 in the one-dimensional case.
Theorem 1. (i) If ω ∈ A2
⋃
A3, (ω ∈ A4
⋃
A8), then the operator H̃1 has
exactly two LIS’s, ϕ1 and ϕ2, with the respective energies z1 and z2 (z2 and z3)
satisfying the inequalities z1 < z2 (z2 < z3) and zi < m1, i = 1, 2 (zj > M1,
j = 2, 3).
(ii) If ω ∈ A6 (ω ∈ A5), then the operator H̃1 has a single LIS ϕ with the
energy z = z1 (z = z3) satisfying the inequality z1 < m1 (z3 > M1).
(iii) If ω ∈ A1
⋃
A7, then the operator H̃1 has no LIS.
We sketch the proof of Theorem 1. In the one-dimensional case, the equation
∆1(z) = 0 is equivalent to the system of two equations,
1 + q(s)
∫
T
(1− cos t)2dt
p(s)h(t)− z
= 0, (9)
and
1 + q(s)
∫
T
sin2 tdt
p(s)h(t)− z
= 0. (10)
244 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2
Spectrum of Two-Magnon non-Heisenberg Ferromagnetic Model
In the one-dimensional case, the integrals in equations (9) and (10) can be
found explicitly for the values z /∈ G1 = [m1; M1]. We obtain:
(a) for z < m1:
1 +
q(s)
p(s)
+
zq(s)
p2(s)
+
z2q(s)
p2(s)
√
z[z − 2p(s)]
= 0, (11)
and
1 +
q(s)
p(s)
− zq(s)
p2(s)
+
zq(s)
p(s)
√
z[z − 2p(s)]
− zq(s)[z − 2p(s)]
p2(s)
√
z[z − 2p(s)]
= 0, (12)
(b) for z > M1:
1 +
q(s)
p(s)
+
zq(s)
p2(s)
− z2q(s)
p2(s)
√
z[z − 2p(s)]
= 0, (13)
and
1 +
q(s)
p(s)
− zq(s)
p2(s)
− zq(s)
p(s)
√
z[z − 2p(s)]
+
zq(s)[z − 2p(s)]
p2(s)
√
z[z − 2p(s)]
= 0. (14)
In turn, these equations are equivalent to the next equations:
(a) for z < m1:
{p2(s) + p(s)q(s) + zq(s)}
√
z[z − 2p(s)] + z2q(s) = 0, (11′)
and
{p2(s) + p(s)q(s)− zq(s)}
√
z[z − 2p(s)]− zq(s)[z − 2p(s)] = 0, (12′)
(b) for z > M1:
{p2(s) + p(s)q(s) + zq(s)}
√
z[z − 2p(s)]− z2q(s) = 0, (13′)
and
{p2(s) + p(s)q(s)− zq(s)}
√
z[z − 2p(s)] + zq(s)[z − 2p(s)] = 0. (14′)
Solving equation (11′), we find the root z = z1, and solving equation (12′), we
find the root z = z2. In turn, solving equation (13′), we find the root z = z3, and
solving equation (14′), we find the root z = z2. Whence the proof of Theorem 1
immediately follows in view of the existence of conditions for these solutions.
In the case of the dimension ν = 2, for the pairs ω, we introduce:
B1 = {ω : p(s) > 0,−p(s) ≤ q(s) < 0}, B2 = {ω : p(s) < 0, 0 < q(s) ≤ −p(s)},
Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2 245
S.M. Tashpulatov
B3 = {ω : p(s) > 0,−25
9 p(s) ≤ q(s) < −p(s)}, B4 = {ω : p(s) < 0,
25
9
p(s) ≤
q(s) < 0}, B5 = {ω : p(s) > 0, 0 < q(s) <
25
9
p(s)}, B6 = {ω : p(s) < 0,
−p(s) ≤ q(s) < −25
9
p(s)}, B7 = {ω : p(s) > 0,−100
27
p(s) ≤ q(s) < −p(s)},
B8 = {ω : p(s) < 0,
100
27
p(s) ≤ q(s) <
25
9
p(s)}, B9 = {ω : p(s) > 0,
25
9
p(s) ≤
q(s) <
100
27
p(s)}, B10 = {ω : p(s) < 0,−25
9
p(s) ≤ q(s) < −100
27
p(s)}, B11 = {ω :
p(s) > 0, q(s) ≤ −100
27
p(s)}, B12 = {ω : p(s) < 0, q(s) ≤ 100
27
p(s)}, B13 = {ω :
p(s) > 0, q(s) ≥ 100
27
p(s)}, B14 = {ω : p(s) < 0, q(s) > −100
27
p(s)}.
The next theorem describes the variation of the energy spectrum of the op-
erator H̃1 in the two-dimensional case.
Theorem 2. (i) If ω ∈ B1
⋃
B2, then the operator H̃1 has no LIS.
(ii) If ω ∈ B3
⋃
B4 (ω ∈ B5
⋃
B6), then the operator H̃1 has a single LIS
ϕ with the energy z1 (z2), where z1 < m2 (z2 > M2). The energy level is of
multiplicity one.
(iii) If ω ∈ B7
⋃
B8 (ω ∈ B9
⋃
B10) then the operator H̃1 has exactly two
LIS’s, ϕ1 and ϕ2, with the respective energies z1 and z2 (z3 and z4), where zi <
m2, i = 1, 2 (zj > M2, j = 3, 4). The energy levels are of multiplicity one.
(iv) If ω ∈ B11
⋃
B12 (ω ∈ B13
⋃
B14), then the operator H̃1 has three LIS’s,
ϕ1, ϕ2 and ϕ3, with the respective energies z1, z2 and z3 (z4, z5 and z6), where
zi < m2, i = 1, 2, 3 (zj > M2, j = 4, 5, 6). The energy levels z1 and z3 (z4 and
z6) are of multiplicity one, while z2 (z5) is of multiplicity two.
P r o o f. The functions
ϕ(z) =
∫
T 2
(1− cos t1)(2− cos t1 − cos t2)dt
p(s)h(t)− z
, ψ(z) =
∫
T 2
sin2 t1dt
p(s)h(t)− z
,
θ(z) =
∫
T 2
(cos t1 − cos t2)2dt
p(s)h(t)− z
are the monotone increasing functions of z for z /∈ [m2; M2]. Their values can be
exactly calculated at the points z = m2 and z = M2. For z < m2 and p(s) > 0, the
function ϕ(z) increases from 0 to (p(s))−1, the function ψ(z) increases from 0 to
9(25p(s))−1, and the function θ(z) increases from 0 to 27(50p(s))−1. For z > M2
and p(s) > 0, these functions increase from −∞ to 0, from −9(25p(s))−1 to 0, and
from −27(50p(s))−1 to 0, respectively. If p(s) < 0 and z < m2, then they increase
from 0 to ∞, from 0 to −9(25p(s))−1, and from 0 to −27(50p(s))−1, respectively.
For p(s) < 0 and z > M2, the functions ϕ(z), ψ(z), and θ(z) increase from (p(s))−1
246 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2
Spectrum of Two-Magnon non-Heisenberg Ferromagnetic Model
to 0, from 9(25p(s))−1 to 0, and from 27(50p(s))−1 to 0. Investigating the equation
∆2(z) = 0 outside the domain of the continuous spectrum, we immediately prove
the assertion of Theorem 2.
In the case ν = 3, we introduce the notation:
a =
∫
T 3
sin2 s1 ds1ds2ds3
3− cos s1 − cos s2 − cos s3
=
∫
T 3
sin2 s1 ds1ds2ds3
3 + cos s1 + cos s2 + cos s3
,
b =
∫
T 3
(cos s1 − cos s2)2 ds1ds2ds3
3− cos s1 − cos s2 − cos s3
=
∫
T 3
(cos s1 − cos s2)2 ds1ds2ds3
3 + cos s1 + cos s2 + cos s3
.
As it is seen, we have 0 < a < b < 1 and 2a < b. We now consider the following
subsets in Ω for the case ν = 3 :
Q1 = {ω : p(s) > 0, −p(s) < q(s) < 0}, Q2 = {ω : p(s) > 0, 0 < q(s) <
p(s)
3
},
Q3 = {ω : p(s) < 0,
p(s)
3
< q(s) < 0}, Q4 = {ω : p(s) < 0, 0 < q(s) < −p(s)},
Q5 = {ω : p(s) > 0, −2p(s)
b
< q(s) ≤ −p(s), Q6 = {ω : p(s) < 0,
2p(s)
b
< q(s) ≤ p(s)
3
}, Q7 = {ω : p(s) > 0,
p(s)
3
< q(s) ≤ 2p(s)
b
},
Q8 = {ω : p(s) < 0, −p(s) < q(s) ≤ −2p(s)
b
}, Q9 = {ω : p(s) > 0,
−p(s)
a
≤ q(s) < −2p(s)
b
}, Q10 = {ω : p(s) < 0,
p(s)
a
< q(s) ≤ 2p(s)
b
},
Q11 = {ω : p(s) > 0,
2p(s)
b
≤ q(s) <
p(s)
a
}, Q12 = {ω : p(s) < 0,
−2p(s)
b
≤ q(s) < −p(s)
a
}, Q13 = {ω : p(s) > 0, q(s) ≤ −p(s)
a
},
Q14 = {ω : p(s) < 0, q(s) ≤ p(s)
a
}, Q15 = {ω : p(s) > 0,
p(s)
a
≤ q(s)},
Q16 = {ω : p(s) < 0, −p(s)
a
≤ q(s)}.
Theorem 3. (i) If ω ∈ Q1
⋃
Q2
⋃
Q3
⋃
Q4, then the operator H̃1 has no
LIS.
(ii) If ω ∈ Q5
⋃
Q6 (ω ∈ Q7
⋃
Q8), then the operator H̃1 has a single LIS ϕ
with the energy z < m3 (z > M3). The energy level is of multiplicity one.
Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2 247
S.M. Tashpulatov
(iii) If ω ∈ Q9
⋃
Q10 (ω ∈ Q11
⋃
Q12), then the operator H̃1 has two LIS’s,
ϕ1 and ϕ2, with the energy levels z1 and z2 (z3 and z4), where zi < m3, i = 1, 2
(zj > M3, j = 3, 4). Furthermore, the energy level z1 (z3) is of multiplicity one,
while z2 (z4) is of multiplicity two.
(iv) If ω ∈ Q13
⋃
Q14 (ω ∈ Q15
⋃
Q16), then the operator H̃1 has exactly three
LIS’s, ϕ1, ϕ2 and ϕ3, with the energies z1, z2 and z3 (z4, z5 and z6) satisfying the
inequalities zi < m3, i = 1, 2, 3 (zj > M3, j = 4, 5, 6). Moreover, the energy
level z1 (z4) is of multiplicity one, z2 (z5) is of multiplicity two, and z3 (z6) is of
multiplicity three.
Theorem 3 is proved basing on the monotonicity of the functions
ϕ(z) =
∫
T 3
(1− cos t1)(3− cos t1 − cos t2 − cos t3)dt
p(s)h(t)− z
, ψ(z) =
∫
T 3
sin2 t1dt
p(s)h(t)− z
,
θ(z) =
∫
T 3
(cos t1 − cos t2)2dt
p(s)h(t)− z
for z /∈ [m3;M3]. Further we will use the values of the Watson integral [3]. It
should be taken into account that the measure is normalized in the case under
consideration.
It can be similarly proved that in the ν− dimensional lattice, the system
has at most three types of LIS’s (not counting the degeneracy multiplicities of
their energy levels) with the energies zi /∈ [mν ; Mν ]. Furthermore, for i = 1, 2, 3,
the corresponding energy levels are of multiplicity one, of multiplicity ν and of
multiplicity (ν − 1). The domains of these LIS’s can also be found.
We now consider the case p(s) ≡ 0. If p(s) ≡ 0 and Jn 6= 0, n = 1, 2, . . . , 2s,
then the function ∆ν(z) = 0 takes the form ∆ν(z) = detA× detB, where A =
a1 b1 b1 · · · b1
a2 b2 0 · · · 0
a2 0 b2 · · · 0
...
...
...
...
...
a2 0 0 · · · b2
is a (ν+1)×(ν+1) matrix, B=
b2 0 0 · · · 0
0 b2 0 · · · 0
0 0 b2 · · · 0
...
...
...
...
...
0 0 0 · · · b2
is a diagonal ν × ν matrix. Here
a1 = 1− νq(s)
2z
, a2 =
q(s)
2z
, b1 =
q(s)
z
, b2 = 1− q(s)
2z
.
Theorem 4. If p(s) ≡ 0, and Jn 6= 0, n = 1, 2, . . . , 2s, then the operator
H̃1 has exactly two LIS’s (not counting the multiplicities of degeneration of their
248 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2
Spectrum of Two-Magnon non-Heisenberg Ferromagnetic Model
energy levels), ϕ1 and ϕ2, with the energies z1 = q(s)
2 and z2 = 2ν+1
2 q(s). The
energy z1 is of multiplicity (2ν − 1)−, while z2 is of multiplicity one. Moreover,
zi < mν , i = 1, 2, (zi > Mν , i = 1, 2), if q(s) < 0 (q(s) > 0).
P r o o f. The equation ∆ν(z) = 0 is equivalent to the system of two
equations,
b2ν−1
2 = 0 (15)
and
a1b2 − νa2b1 = 0. (16)
Equation (15) has a root equal to z = q(s)
2 , and it is clear that its multiplicity
is 2ν−1, while equation (16) has a solution z = z2. Consequently, for the arbitrary
values of ν, the system has at most three types of LIS’s.
2. Two-Magnon States
The Hamiltonian of a two-magnon system has the form
H ′ = −
∑
m,τ
2s∑
n=1
Jn(~Sm
~Sm+τ )n, (17)
where Jn > 0 are the parameters of the multipole exchange interaction be-
tween the nearest-neighbor atoms in the lattice. Hamiltonian (17) acts on the
symmetric Fock space H . The vector S−mS−n ϕ0 describes the state of a sys-
tem of two magnons with spin s located at the sites m and n. The vectors
{ 1√
4s2+(4s2−4s)δm,n
S−mS−n ϕ0} form an orthonormal system. Denote the Hilbert
space spanned by these vectors by H2. It is called the space of two-magnon
states of the operator H ′. By H ′
2, we denote the restriction of the operator H ′ to
H2 : H ′
2 = H ′/H2 .
We find the action of operator (17) on the space l2(Zν ×Zν), i.e., the coordi-
nate representation for the spin values s = 1, s = 3/2, s = 2, s = 5/2, and obtain
the momentum representation of these operators in the space L2(T ν × T ν). Fi-
nally, we generalize these formulas for the arbitrary values of s. The operator H̃ ′
2
in the momentum representation acts on the space H̃2 according to the formula
(H̃ ′
2f)(x; y) = h(x; y)f(x; y) +
∫
T ν
h1(x; y; t)f(t;x + y − t)dt, (18)
where
h(x; y) = A
ν∑
i=1
[1− cos
xi + yi
2
cos
xi − yi
2
]
Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2 249
S.M. Tashpulatov
and
h1(x; y; t) = B
ν∑
i=1
[1− 2 cos
xi + yi
2
cos
xi − yi
2
+ cos(xi + yi)]
−C
ν∑
i=1
[cos
xi − yi
2
− cos
xi + yi
2
] cos(
xi + yi
2
− ti), x, y, t ∈ T ν .
Here
A =
8(J1 − 2J2), if s = 1,
12(J1 − 3J2 + 9J3), if s = 3/2,
16(J1 − 4J2 + 16J3 − 64J4), if s = 2,
20(J1 − 5J2 + 25J3 − 125J4 + 625J5), if s = 5/2,
B =
−4J2, if s = 1,
−12(J2 − 8J3), if s = 3/2,
−24(J2 − 11J3 + 93J4), if s = 2,
−40(J2 − 15J3 + 151J4 − 1484J5), if s = 5/2,
C =
−4(J1 − J2), if s = 1,
−4(J1 + J2 − 23J3), if s = 3/2,
−4(J1 + 5J2 − 83J3 + 773J4), if s = 2,
−4(J1 + 11J2 − 199J3 + 2291J4 − 23119J5), if s = 5/2.
Proposition 4. The space H2 is invariant with respect to the operator H ′.
The operator H ′
2 = H ′/H2 is a bounded self-adjoint operator generating a bounded
self-adjoint operator H ′
2 acting on the space l2(Zν×Zν). The operator H ′
2 in the
momentum representation in the space L2(T ν×T ν) acts according to the formula
(H̃ ′
2f)(x; y) = h(x; y)f(x; y) +
∫
T ν
h1(x; y; s)f(s; x + y − s)ds, (19)
where
h(x; y) = 8sA
ν∑
k=1
[1− cos
xk + yk
2
cos
xk − yk
2
],
h1(x; y; t) = −4s(2s− 1)B
ν∑
k=1
{1 + cos(xk + yk)− 2 cos
xk + yk
2
cos
xk − yk
2
}
−4C
ν∑
k=1
{cos
xk − yk
2
− cos
xk + yk
2
} cos(
xk + yk
2
− tk), x, y, t ∈ T ν ,
here A = J1 − 2sJ2 + (2s)2J3 + . . . + (−1)2s+1J2s, B = J2 − (6s− 1)J3 + (28s2 −
10s+1)J4− (120s3−68s2 +14s− 1)J5 + . . . , C = J1 +(4s2− 6s+1)J2− (24s3−
250 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2
Spectrum of Two-Magnon non-Heisenberg Ferromagnetic Model
32s2 +10s− 1)J3 +(112s4− 160s3 +72s2− 14s+1)J4− (480s5− 768s4 +448s3−
128s2 + 18s− 1)J5 + . . . .
The spectra and bound states of the energy operator of two-magnon systems
in the isotropic non-Heisenberg ferromagnetic model of arbitrary spin s with
impurity were studied in [4]. We consider the manifolds ΓΛ = {(x; y) : x+y = Λ}.
The following fact is important for further studying of the spectrum of the
operator H̃ ′
2.
Let the total quasi-momentum of the system x + y = Λ be fixed. By L2(ΓΛ),
we denote the space of functions that are square integrable over the manifold
ΓΛ = {(x; y) : x + y = Λ}. It is known [5] that the operators H̃ ′
2 and the space
H̃2 can be decomposed into the direct integrals H̃ ′
2 =
⊕∫
T ν H̃ ′
2ΛdΛ, H̃2 =⊕ ∫
T ν H̃2ΛdΛ of the operators H̃ ′
2Λ and the space H̃2Λ such that the spaces
H̃2Λ are invariant under H̃ ′
2Λ, and the operator H̃ ′
2Λ acts on the space H̃2Λ as
(H̃ ′
2ΛfΛ)(x) = hΛ(x)fΛ(x)−
∫
T ν
h1Λ(x; t)fΛ(t)dt,
where hΛ(x) = h(x; Λ− x), h1Λ(x; t) = h1(x; Λ− x; t) and fΛ(x) = f(x; Λ− x).
It is known that the continuous spectrum of the operator H̃ ′
2 is independent
of the functions h1Λ(x; t) and it consists of the intervals GΛ = [mΛ; MΛ], where
mΛ = minxhΛ(x),MΛ = maxxhΛ(x).
The eigenfunction ϕΛ ∈ L2(T ν) of the operator H̃ ′
2 corresponding to an eigen-
value zΛ /∈ GΛ is called the bound state of the operator H̃ ′
2, and zΛ is called the
energy of this BS.
Denote the 2s−th (J1; J2; . . . ; J2s) by P and introduce the following subsets
of the 2s−th P for ν = 1 :
Q1 = {P : A < 0, B < 0, C < 0}, Q2 = {P : A > 0, B > 0, C > 0},
Q3 = {P : A > 0, B > 0, C < 0}, Q4 = {P : A < 0, B < 0, C > 0},
Q5 = {P : A < 0, B > 0, C < 0}, Q6 = {P : A > 0, B < 0, C > 0},
Q7 = {P : B = 0, A = C > 0}, Q8 = {P : B = 0, A = C < 0}.
Let ∆ν
Λ(z) = detD, where
D =
d1,1 d1,2 d1,3 · · · d1,ν+1
d2,1 d2,2 d2,3 · · · d2,ν+1
d3,1 d3,2 d3,3 · · · d3,ν+1
...
...
...
...
dν,1 dν,2 dν,3 · · · dν,ν+1
dν+1,1 dν+1,2 dν+1,3 · · · dν+1,ν+1
,
Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2 251
S.M. Tashpulatov
and
d1,1 = 1− 4s(2s− 1)B
∫
T ν
gΛ(s)ds
hΛ(s)− z
,
d1,k+1 = −4C
∫
T ν
fΛk
(sk)ds
hΛ(s)− z
, k = 1, ν,
dk+1,1 = −4s(2s− 1)B
∫
Tν
ηΛk
(sk)gΛ(s)ds
hΛ(s)− z
, k = 1, ν,
dk+1,k+1 = 1− 4C
∫
T ν
ηΛk
(sk)fΛk
(sk)ds
hΛ(s)− z
, k = 1, ν,
dk+1,i+1 = −4C
∫
T ν
ηΛk
(sk)fΛi(si)ds
hΛ(s)− z
, k = 1, ν, i = 1, ν, k 6= i.
In these formulas
gΛ(s) =
ν∑
k=1
[1 + cos Λk − 2 cos
Λk
2
cos(
Λk
2
− sk)],
fΛk
(sk) = cos(
Λk
2
− sk)− cos
Λk
2
, k = 1, ν, ηΛk
(sk) = cos(
Λk
2
− sk), k = 1, ν.
Lemma 3. A number z = z0 /∈ GΛ is an eigenvalue of the operator H̃ ′
2Λ if
and only if it is a zero of the function ∆ν
Λ(z), i.e., ∆ν
Λ(z0) = 0.
The proof of Lemma 3 is similar to that of Lemma 2.
In the case when ν = 1, the change of the energy spectrum is described by
the theorems below.
Theorem 5. 1. Let P ∈ Q1 and Λ ∈]0;π[ (Λ ∈]π; 2π[).
a) If C 6= 2s(2s − 1)B, then the operator H̃ ′
2 has two BS’s, ϕ1 and ϕ2, with
the energy levels z1 < mΛ and z2 > MΛ.
b) If C = 2s(2s−1)B, then the operator H̃ ′
2 has only one BS ϕ with the energy
level z < mΛ.
2. Let P ∈ Q2 and Λ ∈]0;π[ (Λ ∈]π; 2π[).
a) If 2sA < C < 2s(2s − 1)B, cos Λ
2 > C
2s(2s−1)B , (C > 2s(2s − 1)B, A <
(2s − 1)B), then the operator H̃ ′
2 has three BS’s, ϕi, i = 1, 2, 3; with the energy
values zk < mΛ, k = 1, 2; and z3 > MΛ.
b) If C < 2sA < 2s(2s − 1)B, cos Λ
2 > C
2s(2s−1)B , (C > 2s(2s − 1)B, A =
(2s− 1)B), then the operator H̃ ′
2 has two BS’s, ϕi, i = 1, 2, corresponding to the
252 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2
Spectrum of Two-Magnon non-Heisenberg Ferromagnetic Model
energy values z1 < mΛ and z2 > MΛ. In this case the third BS vanishes because
it is absorbed by the continuous spectrum.
c) If C < 2s(2s − 1)B < (2s − 1)A, cos Λ
2 > C
2s(2s−1)B , (C > 2s(2s − 1)B,
A > (2s − 1)B), then the operator H̃ ′
2 has only one BS ϕ with the energy value
z > MΛ.
d) If C = 2s(2s − 1)B, then the operator H̃ ′
2 has only one BS ϕ with the
energy value z < mΛ.
e) If C > 2s(2s−1)B (C < 2s(2s−1)B), then the operator H̃ ′
2 has two BS’s,
ϕ1, ϕ2, corresponding to the energy values z1 < mΛ, z2 > MΛ.
3. Let P ∈ Q3 and Λ ∈]0;π[ (Λ ∈]π; 2π[).
a) If C ≥ −2s(2s− 1)B, then the operator H̃ ′
2 has two BS’s, ϕ1 and ϕ2, with
the energy values z1 and z2, where z1 < mΛ, and z2 > MΛ.
b) If C < 2s(2s − 1)B, then the operator H̃ ′
2 has only one BS ϕ with the
energy value z < mΛ.
4. Let P ∈ Q4 and Λ ∈]0;π[ (Λ ∈]π; 2π[).
a) If 2sA−2s(2s−1)B−C > 0, cos Λ
2 > C
2sA−2s(2s−1)B−C (cos Λ
2 6= C
2s(2s−1)B ),
then the operator H̃ ′
2 has three (two) BS’s, ϕi, i = 1, 2, 3 (ϕj , j = 1, 2) correspon-
ding to the energy values zk < mΛ, k = 1, 2; z3 > MΛ(z1 < mΛ, z2 > MΛ).
b) If 2sA− 2s(2s− 1)B − C > 0, − C
2s(2s−1)B < cos Λ
2 < C
2sA−2s(2s−1)B−C or
2sA − 2s(2s − 1)B − C < 0 (cos Λ
2 = C
2s(2s−1)B ), then the operator H̃ ′
2 has only
one BS ϕ with the energy value z > MΛ.
5. Let P ∈ Q5 and Λ ∈]0;π[ (Λ ∈]π; 2π[).
a) If cos Λ
2 > − C
2s(2s−1)B , C ≥ 2sA (cos Λ
2 < C
2s(2s−1)B , C ≥ 2sA), then the
operator H̃ ′
2 has three BS’s, ϕ1, ϕ2 and ϕ3, corresponding to the energy values
zi < mΛ, i = 1, 2; and z3 > MΛ.
b) If C < 2sA, 2sA − 2s(2s − 1)B − C < 0, cos Λ
2 > C
2sA−2s(2s−1)B−C (C <
2sA, 2sA − 2s(2s − 1)B − C < 0, cos Λ
2 < − C
2sA−2s(2s−1)B−C ), then the operator
H̃ ′
2 has three BS’s, ϕ1, ϕ2 and ϕ3, corresponding to the energy values zi < mΛ, i =
1, 2; and z3 > MΛ.
c) If C < 2sA, 2sA − 2s(2s − 1)B − C < 0, − C
2s(2s−1)B < cosΛ
2 < C
2s(2s−1)B
(C < 2sA, 2sA − 2s(2s − 1)B − C < 0, − C
2sA−2s(2s−1)B−C ≤ cos Λ
2 < C
2s(2s−1)B )
or C < 2sA, 2sA− 2s(2s− 1)B−C ≥ 0 (C > 2sA, 2sA− 2s(2s− 1)B−C ≥ 0),
then the operator H̃ ′
2 has only one BS ϕ with the energy value z > MΛ.
d) If cos Λ
2 = − C
2s(2s−1)B , C ≥ 2sA (cosΛ
2 = C
2s(2s−1)B , C ≥ 2sA), then
the operator H̃ ′
2 has two BS’s, ϕ1 and ϕ2, with the energy values z1 < mΛ and
z2 > MΛ.
e) If cos Λ
2 < − C
2s(2s−1)B (cos Λ
2 > C
2s(2s−1)B ), then the operator H̃ ′
2 has two
BS’s, ϕ1 and ϕ2, with the energy values z1 < mΛ and z2 > MΛ.
Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2 253
S.M. Tashpulatov
f) If cos Λ
2 = − C
2s(2s−1)B , C < 2sA (cos Λ
2 > C
2s(2s−1)B , C < 2sA), then the
operator H̃ ′
2 has only one BS ϕ with the energy value z > MΛ.
6. Let P ∈ Q6 and Λ ∈]0;π[ (Λ ∈]π; 2π[).
a) If cos Λ
2 < − C
2s(2s−1)B (cos Λ
2 > C
2s(2s−1)B ), then the operator H̃ ′
2 has two
BS’s, ϕ1 and ϕ2, with the energy values z1 < mΛ, and z2 > MΛ.
b) If cos Λ
2 ≥ − C
2s(2s−1)B (cos Λ
2 ≤ C
2s(2s−1)B ), then the operator H̃ ′
2 has only
one BS ϕ with the energy value z < mΛ.
7. Let P ∈ Q7
⋃
Q8 and Λ 6= 0.
Then the operator H̃ ′
2 has two BS’s, ϕ1 and ϕ2, with the energy values z1 <
mΛ, and z2 > MΛ.
In the case where ν = 1 and Λ = 0, the change of the energy spectrum is
described by the following theorems.
Theorem 6. Let Λ = 0. a) If P ∈ Q1, C > 2s(2s − 1)B, then the operator
H̃ ′
2 has two BS’s, ϕ1 and ϕ2, with the energy values z1 < mΛ, and z2 > MΛ.
b) If P ∈ Q1, C ≤ 2s(2s− 1)B, then the operator H̃ ′
2 has only one BS ϕ with
the energy value z < mΛ.
2.a) If P ∈ Q2, 2sA < C < 2s(2s−1)B, then the operator H̃ ′
2 has three BS’s,
ϕi, i = 1, 2, 3; with the energy values zj < mΛ, j = 1, 2; and z3 > MΛ.
b) If P ∈ Q2, C ≤ 2sA, C < 2s(2s−1)B or P ∈ Q2, 2sA < 2s(2s−1)B < C,
then the operator H̃ ′
2 has two BS’s, ϕi, i = 1, 2 with the energy values z1 < mΛ
and z2 > MΛ.
c) If P ∈ Q2, C = 2s(2s− 1)B > 2sA, then the operator H̃ ′
2 has only one BS
ϕ with the energy value z < mΛ.
d) If P ∈ Q2, C = 2sA ≥ 2s(2s − 1)B or P ∈ Q2, 2s(2s − 1)B < 2sA < C,
then the operator H̃ ′
2 has only one BS ϕ with the energy value z > MΛ.
e) If P ∈ Q2, C = 2s(2s − 1)B < 2sA or P ∈ Q2, 2s(2s − 1)B < 2sA < C,
then the operator H̃ ′
2 has no BS.
3.a) If P ∈ Q3, C < −2s(2s− 1)B, A ≥ (2s− 1)B, then the operator H̃ ′
2 has
two BS’s, ϕi, i = 1, 2, with the energy values z1 < mΛ and z2 > MΛ.
b) If P ∈ Q3, A < (2s − 1)B, then the operator H̃ ′
2 has only one BS ϕ with
the energy value z > MΛ.
c) If P ∈ Q3, C ≥ −2s(2s − 1)B, A ≥ (2s − 1)B, then the operator H̃ ′
2 has
only one BS ϕ with the energy value z < mΛ.
4.a) If P ∈ Q4, C > −2s(2s − 1)B, then the operator H̃ ′
2 has two BS’s, ϕ1
and ϕ2, with the energy values zi < mΛ, i = 1, 2.
b) If P ∈ Q4, C < −2s(2s − 1)B, then the operator H̃ ′
2 has only one BS ϕ
with the energy value z < mΛ.
c) If P ∈ Q4, C = −2s(2s− 1)B, then the operator H̃ ′
2 has no BS.
5.a) If P ∈ Q5, −2s(2s − 1)B < C < 2sA, C > sA − s(2s − 1)B, then the
operator H̃ ′
2 has two BS’s, ϕ1 and ϕ2, with the energy values zi < mΛ, i = 1, 2.
254 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2
Spectrum of Two-Magnon non-Heisenberg Ferromagnetic Model
b) If P ∈ Q5, −2s(2s − 1)B < C < 2sA, C ≤ sA − s(2s − 1)B or P ∈ Q5,
C = −2s(2s− 1)B < 2sA, then the operator H̃ ′
2 has no BS.
c) If P ∈ Q5, C = −2s(2s− 1)B ≥ 2sA or P ∈ Q5, C < −2s(2s− 1)B, then
the operator H̃ ′
2 has only one BS ϕ with the energy value z < mΛ.
6.a) If P ∈ Q6, 2sA ≤ C < −2s(2s−1)B, then the operator H̃ ′
2 has two BS’s,
ϕ1 and ϕ2, with the energy values zi > MΛ, i = 1, 2.
b) If P ∈ Q6, C = 2sA > −2s(2s − 1)B or P ∈ Q6, C < −2s(2s − 1)B,
C < 2sA, then the operator H̃ ′
2 has no BS.
c) If P ∈ Q6, C = −2s(2s − 1)B < 2sA or P ∈ Q6, C > −2s(2s − 1)B,
C 6= 2sA, then the operator H̃ ′
2 has only one BS ϕ with the energy value z > MΛ.
7. If P ∈ Q7 (P ∈ Q8), then the operator H̃ ′
2 has only one BS ϕ with the
energy value z > MΛ (z < mΛ).
A sketch of the proofs of Theorems 5, 6 is given below. In the case under con-
sideration, the equation for eigenvalues is an integral equation with a degenerate
kernel. It is therefore equivalent to a system of the linear homogeneous algebraic
equations. The system is known to have a nontrivial solution if and only if its
determinant is equal to zero. In this case, the equation ∆ν
Λ(z) = 0 is therefore
equivalent to the equation stating that the determinant of the system is zero. In
the case where ν = 1, the determinant has the form
∆1
Λ(z) = detD,
where
D =
(
d1,1 d1,2
d2,1 d2,2
)
.
Here
d1,1 = 1− 4s(2s− 1)B
∫
T
gΛ(s)ds
hΛ(s)− z
, d1,2 = −4C
∫
T
fΛ(s)ds
hΛ(s)− z
,
d2,1 = −4s(2s− 1)
∫
T
ηΛ(s)gΛ(s)ds
hΛ(s)− z
, d2,2 = 1− 4C
∫
T
ηΛ(s)fΛ(s)ds
hΛ(s)− z
,
gΛ(s) = 1 + cos Λ− 2 cos
Λ
2
cos(
Λ
2
− s), fΛ(s) = cos(
Λ
2
− s)− cos
Λ
2
,
ηΛ(s) = cos(
Λ
2
− s).
Expressing all integrals in the equation ∆1
Λ(z) = 0 via the integral
J?(z) =
∫
T
dt
hΛ(t)− z
,
Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2 255
S.M. Tashpulatov
we can see that the equation ∆1
Λ(z) = 0 is equivalent to the equation
{C(z−8sA)2 +8sA[2s(2s−1)B +C]cos2 Λ
2
(z−8sA)+128s3(2s−1)A2B cos4
Λ
2
}
×J?(z) = −C(z − 8sA) + 8sA[2sA− C − 2s(2s− 1)B] cos2
Λ
2
. (20)
Because
1
hΛ(t)− z
is a continuous function for z /∈ [mΛ;MΛ] and
[J?(z)]′ =
∫
T
1
[hΛ(t)− z]2
> 0,
the function J?(z) is an increasing function of z for z /∈ [mΛ; MΛ]. Moreover,
J?(z) → 0 as z → −∞, J?(z) → +∞ as z → mΛ − 0, J?(z) → −∞ as z →
MΛ + 0, and J?(z) → 0 as z → +∞. Analyzing equation (20) outside the set
GΛ = [mΛ;MΛ], we get the proof of Theorems 5, 6.
The energy spectrum of the operator H̃ ′
2 in the case where ν = 2 for the total
quasi-momentum of the form Λ = (Λ1; Λ2) = (Λ0; Λ0) is described below. It is
easy to see that if the parameters Jn, n = 1, 2s and Λ0 satisfy the conditions of
Theorems 5, 6, then the statements of the theorems are true. Only one additional
BS ϕ̃ appears, whose energy value is z̃, because z̃ < mΛ (z̃ > MΛ) if C > 0 (C <
0). If C = 0, the operator H̃ ′
2 does not have an additional BS.
The proof of this statement is based on the fact that if ν = 2 and Λ = (Λ0; Λ0),
then the function ∆ν
Λ(z) has the form
∆ν
Λ(z) = [1− 2C
∫
T 2
[cos(Λ0
2 − t1)− cos(Λ0
2 − t2)]2dt1dt2
hΛ(t1; t2)− z
]ΨΛ(z), (21)
where
ΨΛ(z) = {1− 4s(2s− 1)B
∫
T 2
gΛ(t)
hΛ(t1; t2)− z
dt1dt2}[1− 4C
×
∫
T 2
fΛ(t1)ηΛ(t1; t2)
hΛ(t1; t− 2)− z
dt1dt2]− 32s(2s− 1)BC
∫
T 2
ξΛ(t1)
hΛ(t1; t2)− z
dt1dt2
×
∫
T 2
fΛ(t1)gΛ(t)
hΛ(t1; t2)− z
dt1dt2, t ∈ T 2, Λ ∈ T ν .
Here gΛ(t) = 2+2 cosΛ0−2 cos Λ0
2 [cos(Λ0
2 − t1)+cos(Λ0
2 − t2)], fΛ(t1) = cos(Λ0
2 −
t1), ηΛ(t1; t2) = cos(Λ0
2 −t1)+cos(Λ0
2 −t2)−2 cos Λ0
2 , ξΛ(t1) = cos(Λ0
2 −t1)−cos Λ0
2 .
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Spectrum of Two-Magnon non-Heisenberg Ferromagnetic Model
Therefore the equation ∆ν
Λ(z) = 0 holds if either the equation
1− 2C
∫
T 2
[cos(Λ0
2 − t1)− cos(Λ0
2 − t1)]2dt1dt2
hΛ(t1; t2)− z
= 0 (22)
or
ΨΛ(z) = 0 (23)
holds.
It is easy to see that equation (22) has the unique solution z̃ < mΛ if C > 0;
if C < 0, then this solution satisfies the condition z̃ > MΛ. If C = 0, equation
(22) has no solution. Expressing the integrals in (23) via the integral
J?(z) =
∫
T 2
dt1dt2
hΛ(t1; t2)− z
,
we obtain
ηΛ(z)J?(z) = ξΛ(z),
where
ηΛ(z) = C(z − 16sA)2 + 16sA[2s(2s− 1)B + C]
× cos2
Λ0
2
(z − 16sA) + 512s3(2s− 1)A2B cos4
Λ0
2
,
and
ξΛ(z) = −C(z − 16sA) + 16sA[2sA− C − 2s(2s− 1)B] cos2
Λ0
2
.
In its turn, for ηΛ(z) 6= 0, the above last equation is equivalent to the equation
J?(z) =
ξΛ(z)
ηΛ(z)
. (24)
Analyzing equation (24) outside the set GΛ and taking into account that the
function J?(z) is monotonic for z /∈ [mΛ; MΛ], we obtain the statements similar
to those of Theorems 5, 6.
For all other quasi-momenta, Λ = (Λ1; Λ2), Λ1 6= Λ2, there exist the sets
Gj , j = 0, 5, of the parameters Jn, n = 1, 2s and Λ such that in every set Gj
the operator H̃ ′
2 has exactly j BS’s (taking the multiplicity of energy levels into
account) with the corresponding energy values zk, k = 1, 5, and zk /∈ GΛ.
Indeed, in this case, for ν = 2, the function ∆ν
Λ(z) has the form
∆ν
Λ(z) = detD,
Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2 257
S.M. Tashpulatov
where
D =
d1,1 d1,2 d1,3
d2,1 d2,2 d2,3
d3,1 d3,2 d3,3
.
Here
d1,1 = 1−4s(2s−1)B
∫
T 2
gΛ(s)ds1ds2
hΛ(s)− z
, d1,k+1 = −4C
∫
T 2
fΛk
(sk)
hΛ(s)− z
ds1ds2, k = 1, 2,
dk+1,1 = −4s(2s− 1)B
∫
T 2
ζΛk
(sk)gΛ(s)ds1ds2
hΛ(s)− z
, k = 1, 2,
dk+1,k+1 = 1− 4C
∫
T 2
ζΛk
(sk)fΛk
(sk)ds1ds2
hΛ(s)− z
, k = 1, 2,
dk+1,j+1 = −4C
∫
T 2
ζΛk
(sk)fΛj (sj)ds1ds2
hΛ(s)− z
, k = 1, 2, j = 1, 2, k 6= j.
In these formulas
gΛ(s) =
2∑
k=1
[1 + cos Λk − 2 cos
Λk
2
cos(
Λk
2
− sk)],
fΛk
(sk) = cos(
Λk
2
− sk)− cos
Λk
2
, k = 1, 2,
ζΛk
(sk) = cos(
Λk
2
− sk), k = 1, 2.
Expressing all integrals in the equation ∆ν
Λ(z) = 0 via J?(z) and performing
some algebraic transformations, we can reduce it to the form
θΛ(z)J?(z) = χΛ(z), (25)
where θΛ(z) is the fifth-order polynomial in z, and χΛ(z) is the lower-order poly-
nomial in z. Analyzing equation (25) outside the set GΛ and taking into account
that the function J?(z) with z /∈ [mΛ;MΛ] is monotonic, we can easily verify that
the equation has no more than five solutions outside the set GΛ.
For an arbitrary ν ≥ 3 and Λ = (Λ1; Λ2; . . . ; Λν) = (Λ0; Λ0; Λ0; . . . ; Λ0) ∈ T ν ,
the change of the energy spectrum of the operator H̃ ′
2 is similar to that observed
in the case of ν = 1. In this case, if the parameters J1, J2, . . . , J2s and Λ0 satisfy
the conditions of Theorems 5, 6, then there exist the statements of these theorems
that are true. In this situation, the operator H̃ ′
2 with C 6= 0 has only one
258 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2
Spectrum of Two-Magnon non-Heisenberg Ferromagnetic Model
additional BS with the energy z. Moreover, the energy level of this additional BS
z degenerates ν − 1 times, and z < mΛ (z > MΛ) if C > 0 (C < 0). For all
other values of the total quasi-momentum Λ, the operator H̃ ′
2 has at most 2ν +1
BS’s (taking the multiplicity of the energy levels into account) with the energy
values lying outside the set GΛ.
The proof of these statements is based on finding zeros of the function ∆ν
Λ(z).
Expressing all integrals in ∆ν
Λ(z) via J?(z), we can bring the equation ∆ν
Λ(z) = 0
to the form
J?(z) =
CΛ(z)
DΛ(z)
, (26)
where DΛ(z) is the (2ν +1) th-order polynomial in z, and CΛ(z) is also a polyno-
mial in z whose order (with respect to DΛ(z)) is lower. The analyzing of equation
(26) outside the set GΛ leads to the proof of the above statements.
Theorem 7. Let A = 0 and ν be arbitrary. Then the operator H̃ ′
2 has two
BS’s, ϕ1 and ϕ2, (not taking the multiplicity of energy levels into account) with
the energy values z1 = −2C−8s(2s−1)B
∑ν
i=1 cos2 Λi
2 and z2 = −2C. Moreover,
z1 is not degenerate, while z2 is degenerative ν − 1 times, and zi /∈ GΛ, i = 1, 2,
for all Λ ∈ T ν , i.e., the energy values of these BS’s lie outside the continuous
spectrum domain of the operator tildeH ′
2Λ. When B = 0, this BS’s vanishes
because it is incorporated into the continuous spectrum.
P r o o f. If A = 0, then hΛ(s) ≡ 0, and
∆ν
Λ(z) = (1 +
2C
z
)ν−1{[1 +
8s(2s− 1)B
∑ν
k=1 cos2 Λk
2
z
](1 +
2C
z
)
−16s(2s− 1)BC
∑ν
k=1 cos2 Λk
2
z2
}.
Solving the equation ∆ν
Λ(z) = 0, we prove the theorem.
Note. In the theorem, the zero-order degeneracy corresponds to the case
where there is no BS.
Let π̃ = (π; π; . . . ; π) ∈ T ν .
Theorem 8. Let Λ = π̃, Λ, π̃ ∈ T ν and C 6= 0. Then the operator H̃ ′
2 has
only one BS ϕ with the energy value z = 8sAν − 2C, and this energy level is of
multiplicity ν. In addition, if C > 0, then z < mΛ, and if C < 0, then z > MΛ.
When C = 0, this BS vanishes because it is absorbed by the continuous spectrum.
The proof is based on the equality hΛ(x) = 8sAν with Λ = π̃ and also on the
corresponding form of the function ∆ν
Λ(z) = (1− 2C
8sAν−z )ν with Λ = π̃.
Theorem 9. Let C = 0, and ν be an arbitrary number. Then the operator
H̃ ′
2 has at most one BS, the corresponding energy level is of multiplicity one, and
z /∈ GΛ.
Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2 259
S.M. Tashpulatov
P r o o f. If C = 0, the relations
h1Λ(x; t) = −4s(2s− 1)B
ν∑
k=1
[1 + cosΛk − 2 cos
Λk
2
cos(
Λk
2
− xk)],
∆ν
Λ(z) = 1− 4s(2s− 1)B
∫
T ν
gΛ(s)ds
hΛ(s)− z
,
where
gΛ(s) =
ν∑
k=1
[1+cos Λk−2 cos
Λk
2
cos(
Λk
2
−sk)], Λ∈T ν, s∈T ν, ds = ds1ds2 . . . dsν ,
hold. Using the form of the determinant ∆ν
Λ(z) and solving the corresponding
equation, we get the proof of Theorem 9.
Besides, the qualitative pictures of the change of the energy spectrum of
operator H̃ ′
2 in the cases for s = 1/2 and s > 1/2 are shown to be different. We
also show that the energy spectrum of the system is the same either for integer
and half-integer values of s or for odd and even values of s.
3. Structure of Essential Spectrum of Three-Particle System
We first determine the structure of the essential spectrum of a three-particle
system consisting of two magnons and an impurity spin, and then estimate the
number of thee-particle BS’s in the system. Comparing formulas (2) and (7)
and using the tensor products of the Hilbert spaces and the tensor products of
the operators in Hilbert spaces [6], we can verify that the operator H̃2 can be
represented in the form H̃2 = H̃1
⊗
E + E
⊗
H̃1 + K1 + K2, where E is the unit
operator in H̃1, and K1 and K2 are the integral operators
(K1f)(x; y) =
∫
T ν
h1(x; y; t)f(t; x + y − t)dt,
(K2f)(x; y) =
∫
T ν
∫
T ν
h4(x; y; s; t)f(s; t)dsdt.
The kernels of these operators have the forms
h1(x; y; t) = −4s(2s− 1)B
ν∑
i=1
{1 + cos(xk + yk)− 2 cos
xk + yk
2
cos
xk − yk
2
}
−4C
ν∑
i=1
{cos
xk − yk
2
− cos
xk + yk
2
} cos(
xk + yk
2
− tk), x, y, t ∈ T ν ,
260 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2
Spectrum of Two-Magnon non-Heisenberg Ferromagnetic Model
and
h4(x; y; s; t) = F
ν∑
i=1
[1 + cos(xi + yi − si − ti) + cos(si + ti) + cos(xi + yi)
− cos(xi−si−ti)−cos(yi−si−ti)−cosxi−cos yi]+Q
ν∑
i=1
[cos(xi−ti)+cos(yi−si)]
+M
ν∑
i=1
[cos(xi − si) + cos(yi − ti)] + N
ν∑
i=1
[cos si + cos ti + cos(xi + yi − si)
+ cos(xi + yi − ti)],
here B = J2− (6s− 1)J3 + (28s2− 10s + 1)J4− (120s3− 68s2 + 14s− 1)J5 + . . . ,
C = J1 + (4s2 − 6s + 1)J2 − (24s3 − 32s2 + 10s − 1)J3 + (112s4 − 160s3 +
72s2 − 14s + 1)J4 − (480s5 − 768s4 + 448s3 − 128s2 + 18s − 1)J5 + . . . , F =
(2s − 4s2)(J0
2 − J2) + (2s − 16s2 + 24s3)(J0
3 − J3) + . . . + . . . , Q = (−4s2 +
2s)(J0
2 − J2) + (−4s + 20s2 − 24s3)(J0
3 − J3) + . . . + . . . , M = 2[(J0
1 − J1) −
(1 + 5s + 2s2)(J0
2 − J2) + (1 − 8s + 22s2 − 12s3)(J0
3 − J3) + . . . + . . .], N =
−(J0
1 −J1)+ (1−6s+4s2)(J0
2 −J2)− (1−10s+32s2−24s3)(J0
3 −J3)+ . . .+ . . ..
As we have already mentioned, for the fixed total quasi-momentum x + y =
Λ of the two-magnon subsystem, the operator H ′
2 and the space H2 can be
decomposed into direct integrals H̃ ′
2 =
⊕ ∫
T ν H̃ ′
2ΛdΛ, H̃2 =
⊕∫
T ν H̃2ΛdΛ, such
that the operators K1Λ become compact after the decomposition.
It can be seen from the expressions for the kernels of K1 and K2 that K1Λ
and K2 are finite-rank operators, i.e., finite-dimensional operators. Therefore,
the essential spectra of H̃2 and H̃1
⊗
E + E
⊗
H̃1 coincide. A simple verifica-
tion shows that the spectrum of H̃1 is independent of Λ, i.e., of λ and µ. The
spectrum of A
⊗
E +E
⊗
B, where A and B are densely defined bounded linear
operators, was studied in [6-8]. In these papers there were also given the explicit
formulas expressing σess(A
⊗
E + E
⊗
B) and σdisc(A
⊗
E + E
⊗
B) in terms
of σ(A), σdisc(A), σ(B), and σdisc(B):
σdisc(A
⊗
E + E
⊗
B) = {(σ(A)\σess(A)) + (σ(B)\σess(B))}\{(σess(A)
+σ(B))
⋃
(σ(A) + σess(B))},
σess(A
⊗
E + E
⊗
B) = (σess(A) + σ(B))
⋃
(σ(A) + σess(B)).
It is clear that σ(A
⊗
E + E
⊗
B) = {λ + µ : λ ∈ σ(A), µ ∈ σ(B)}.
It can be seen from the results of [1] that the spectrum of H̃1 consists of the
continuous spectrum and at most three eigenvalues of multiplicity one, multiplic-
ity (ν − 1), and multiplicity ν.
Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2 261
S.M. Tashpulatov
First we prove the theorem on the finite-dimensional perturbations of bounded
linear operators in Banach spaces.
Theorem 10. Let A and B be the linear bounded self-adjoint operators with
the difference of the self-adjoint operator with finite rank m. Then σess(A) =
σess(B), and at most m eigenvalues appear (taking into account their degeneration
multiplicities).
P r o o f. Let C = A − B. As C is a self-adjoint operator of rank m, the
function C(A− z)−1 is analytical and it has the value of the operator of rank at
most m in C\σ(A). It is meromorphic in C\σess(A) with finite-rank residues at
points in σdisc(A). If z /∈ σ(A), then (B − z)−1 exists if and only if there exists
(1 − C(A − z)−1)−1. We can conclude that in every component of C\σ(A) the
operator (1 − C(A − z)−1)−1 is somewhere reversible. The components C\σ(A)
and C\σess.(A) coincide because of the discreteness of σdisc(A). By the Fredholm
meromorphic theorem, the operator (1 − C(A − z)−1)−1 exists on C\σess(A)
everywhere, but the discrete set D′ where it has finite rank residues. Here D′ =
σdisc(A)
⋃
D′′, where D′′ consists of no more than m points, since the operator
C(A− z)−1 can have an eigenvalue equal to 1 with multiplicity no more than m.
It follows that the operator B can have only a discrete spectrum in C\σess(A)
such that σess(B) ⊂ σess(A).
Every component of C\σess(B) has the points lying neither in σ(A) nor in
σ(B). As C is a self-adjoint operator of rank m, the function C(B − z)−1 is ana-
lytical and has the values of the operator of rank no more than m in C\σ(B). It is
meromorphic in C\σess(B) with the finite rank residues at the points of σdisc(B).
If z /∈ σ(B), then (A−z)−1 exists if and only if there exists (1+C(B−z)−1)−1. One
can conclude that in every component of C\σ(B), the operator (1+C(B−z)−1)−1
is somewhere reversible. The components C\σ(B) and C\σess(B) coincide be-
cause of the discreteness σdisc(B). By the Fredholm meromorphic theorem, the
operator (1+C(B−z)−1)−1 exists in C\σess(B) everywhere except the discrete set
D1 where it has finite-rank residues. Here D1 = σdisc(B)
⋃
D2, where D2 consists
of at most m points, since the operator C(B− z)−1 can have an eigenvalue equal
to −1 with the multiplicities at most m. Hence the operator A can have only
a discrete spectrum in C\σess(B) such that σess(A) ⊂ σess(B). Consequently,
σess(A) = σess(B). And we can conclude that when there are perturbations of
self-adjoint operators with rank m, the essential spectrum of the operator ex-
ists, and at most m eigenvalues appear (taking into account their degeneration
multiplicities).
Notice that the problems on the finite rank perturbations for the compact
operators were considered in [9–11].
The theorems below describe the structure of the essential spectrum
of H̃1
⊗
E + E
⊗
H̃1 and give lower and upper estimations for N, the number
262 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2
Spectrum of Two-Magnon non-Heisenberg Ferromagnetic Model
of points of discrete spectrum of the operator H̃2.
Theorem 11. If ν = 1 and ω ∈ A1
⋃
A7, then the essential spectrum of
the operator H̃2 consists of a single interval σess.(H̃2) = [0; 4p(s)] or σess(H̃2) =
[4p(s); 0], and the relation 0 ≤ N ≤ 12 holds for the number N of three-particle
BBs.
Theorem 12. If ν = 1 and ω ∈ A6 or ω ∈ A5, then the essential spec-
trum of the operator H̃2 consists of the union of two intervals, σess(H̃2) =
[0; 4p(s)]
⋃
[z1; z1 + 2p(s)] or σess(H̃2) = [4p(s); 0]
⋃
[z1; z1 + 2p(s)], and the re-
lation 1 ≤ N ≤ 13 holds for the number N of the three-particle operator.
Theorem 13. If ν = 1 and ω ∈ A2
⋃
A3 or ω ∈ A4
⋃
A8, then the essential
spectrum of the operator H̃2 consists of the union of three intervals, σess(H̃2) =
[0; 4p(s)]
⋃
[z1; z1 + 2p(s)]
⋃
[z2; z2 + 2p(s)], or σess(H̃2) = [4p(s); 0]
⋃
[z1; z1 +
2p(s)]
⋃
[z2; z2 + 2p(s)], and the relation 3 ≤ N ≤ 15 holds for the number N
of the three-particle operator.
Theorem 14. If ν = 2 and ω ∈ B1
⋃
B2, then the essential spectrum of
the operator H̃2 consists of a single interval σess(H̃2) = [0; 8p(s)], or σess(H̃2) =
[8p(s); 0], and the relation 0 ≤ N ≤ 22 holds for the number N of the three-particle
operator.
Theorem 15. If ν = 2 and ω ∈ B3
⋃
B4 or ω ∈ B5
⋃
B6, then the essential
spectrum of the operator H̃2 consists of the union of two intervals, σess(H̃2) =
[0; 8p(s)]
⋃
[z1; z1 + 4p(s)], or σess(H̃2) = [8p(s); 0]
⋃
[z1; z1 + 4p(s)], and the rela-
tion 1 ≤ N ≤ 23 holds for the number N of the three-particle operator.
Theorem 16. If ν = 2 and ω ∈ B7
⋃
B8 or ω ∈ B9
⋃
B10, then the essential
spectrum of the operator H̃2 consists of the union of three intervals, σess(H̃2) =
[0; 8p(s)]
⋃
[z1; z1 + 4p(s)]
⋃
[z2; z2 + 4p(s)], or σess(H̃2) = [8p(s); 0]
⋃
[z1; z1 +
4p(s)]
⋃
[z2; z2 + 4p(s)], and the relation 3 ≤ N ≤ 25 holds for the number N
of the three-particle operator.
Theorem 17. If ν = 2 and ω ∈ B11
⋃
B12 or ω ∈ B13
⋃
B14, then the essen-
tial spectrum of the operator H̃2 consists of the union of four intervals, σess(H̃2) =
[0; 8p(s)]
⋃
[z1; z1+4p(s)]
⋃
[z2; z2+4p(s)]
⋃
[z3; z3+4p(s)], or σess(H̃2) = [8p(s); 0]
⋃
[z1; z1+
4p(s)]
⋃
[z2; z2 + 4p(s)]
⋃
[z3; z3 + 4p(s)], and the relation 6 ≤ N ≤ 28 holds for
the number N of the three-particle operator.
Theorem 18. If ν = 3 and ω ∈ Q1
⋃
Q2
⋃
Q3
⋃
Q4, then the essential spec-
trum of the operator H̃2 consists of a single interval σess(H̃2) = [0; 12p(s)] or
σess(H̃2) = [12p(s); 0], and the relation 0 ≤ N ≤ 32 holds for the number N of
three-particle BBs.
Theorem 19. If ν = 3 and ω ∈ Q5
⋃
Q6 or ω ∈ Q7
⋃
Q8, then the essential
spectrum of the operator H̃2 consists of the union of two intervals, σess(H̃2) =
Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2 263
S.M. Tashpulatov
[0; 12p(s)]
⋃
[z1; z1 + 6p(s)], or σess(H̃2) = [12p(s); 0]
⋃
[z1; z1 + 6p(s)], and the
relation 1 ≤ N ≤ 33 holds for the number N of the three-particle operator.
Theorem 20. If ν = 3 and ω ∈ Q9
⋃
Q10 or ω ∈ Q11
⋃
Q12, then the essential
spectrum of the operator H̃2 consists of the union of three intervals, σess(H̃2) =
[0; 12p(s)]
⋃
[z1; z1 + 6p(s)]
⋃
[z2; z2 + 6p(s)], or σess(H̃2) = [12p(s); 0]
⋃
[z1; z1 +
6p(s)]
⋃
[z2; z2 + 6p(s)], and the relation 3 ≤ N ≤ 35 holds for the number N of
the three-particle operator.
Theorem 21. If ν = 3 and ω ∈ Q13
⋃
Q14 or ω ∈ Q15
⋃
Q16, then the
essential spectrum of the operator H̃2 consists of the union of four intervals,
σess(H̃2) = [0; 12p(s)]
⋃
[z1; z1 + 6p(s)]
⋃
[z2; z2 + 6p(s)]
⋃
[z3; z3 + 6p(s)], or
σess(H̃2) = [12p(s); 0]
⋃
[z1; z1 + 6p(s)]
⋃
[z2; z2 + 6p(s)]
⋃
[z3; z3 + 6p(s)], and the
relation 6 ≤ N ≤ 38 holds for the number N of the three-particle operator.
P r o o f. The proofs of Theorems 11-21 are similar. Therefore we prove
one of the theorems. As an example, we prove Theorem 21. From Theorem 3
(in statement (iv)) from [1], it is seen that for ω ∈ Q13
⋃
Q14 (ω ∈ Q15
⋃
Q16)
the operator H̃1 has exactly three LIS’s, ϕ1, ϕ2 and ϕ3, with the energies z1, z2
and z3 (z4, z5 and z6) satisfying the inequalities zi < m3, i = 1, 2, 3 (zj > M3,
j = 4, 5, 6). Moreover, the level z1 (z4) is of multiplicity one, the level z2 (z5) is
of multiplicity two and the level z3 (z6) is of multiplicity three.
The continuous spectrum of the operator H̃1 consists of the interval [0; 6p(s)]
or [6p(s); 0]. Therefore, the essential spectrum of the operator H̃2 consists of a set
[0; 6p(s)] + {[0; 6p(s)], z1, z2, z3}, i.e., σess(H̃2) = [0; 12p(s)]
⋃
[z1; z1 + 6p(s)]
⋃
[z2;
z2 +6p(s)]
⋃
[z3; z3 +6p(s)]. The numbers 2z1, 2z2, 2z3, z1 + z2, z1 + z3, z2 + z3 are
the eigenvalues of the operator H̃1
⊗
E +E
⊗
H̃1 and are outside the domain of
the essential spectrum of H̃1
⊗
E + E
⊗
H̃1. It is clear that the multiplicity of
their eigenvalues is at most 3× 3 = 9. Consequently, these six eigenvalues of the
operator H̃1
⊗
E + E
⊗
H̃1 belong to the discrete spectrum of the considering
three-particle operator.
Then, the operator K1Λ in the three-dimensional case is the seven-rank ope-
rator, while the rank of the operator K2 is equal to 25. Consequently, as follows
from Theorem 10, the number N of the points of discrete spectrum of the three-
particle operator is not less than 6 and not more than 6 + 7 + 25 = 38.
Theorem 22. Let ν be an arbitrary number, p(s) ≡ 0, and Jn 6= 0, n =
1, 2, . . . , 2s. Then the essential spectrum of the operator H̃2 consists of three
points, σess(H̃2) = {0; q(s)
2 ; 2ν+1
2 q(s)}, and the relation 3 ≤ N ≤ 10ν +5 holds for
the number N of the points of discrete spectrum of the three-particle operator.
P r o o f. When ν is an arbitrary number, p(s) ≡ 0, and Jn 6= 0, n =
1, 2, . . . , 2s, by Theorem 4 from [1], the operator H̃1 has two eigenvalues equal to
z1 = q(s)
2 and z2 = 2ν+1
2 q(s), where z1 is of multiplicity (2ν − 1), while z2 is of
264 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2
Spectrum of Two-Magnon non-Heisenberg Ferromagnetic Model
multiplicity one. The essential (continuous) spectrum of the operator H̃1 consists
of a single point 0. Therefore, σess(H̃2) = {0; q(s)
2 ; 2ν+1
2 q(s)}, and the points
q(s); (2ν+1)q(s); (ν+1)q(s) are the eigenvalues of the operator H̃1
⊗
E+E
⊗
H̃1.
Now, taking into account that the operators K1Λ and K2 are of ranks 2ν +1 and
8ν + 1, respectively, we immediately obtain the proof of Theorem 22.
It should be noticed that if h(x; y) is an arbitrary 2π-periodic continuous
function, h2(x; s) = h3(x; s) is an arbitrary degenerated 2π-periodic continuous
kernel, and h1(x; y; t) and h4(x; y; s; t) are also arbitrary degenerated 2π-periodic
continuous kernels, i.e., the operators K1Λ and K2 are arbitrary finite-dimensional
operators, then the analogous results are true.
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Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 2 265
|
| id | nasplib_isofts_kiev_ua-123456789-106748 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1812-9471 |
| language | English |
| last_indexed | 2025-12-07T17:35:45Z |
| publishDate | 2013 |
| publisher | Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| record_format | dspace |
| spelling | Tashpulatov, S.M. 2016-10-04T17:20:53Z 2016-10-04T17:20:53Z 2013 Spectrum of Two-Magnon non-Heisenberg Ferromagnetic Model of Arbitrary Spin with Impurity / S.M. Tashpulatov // Журнал математической физики, анализа, геометрии. — 2013. — Т. 9, № 2. — С. 239-265. — Бібліогр.: 11 назв. — англ. 1812-9471 https://nasplib.isofts.kiev.ua/handle/123456789/106748 We consider a two-magnon system in the isotropic non-Heisenberg ferromagnetic model of an arbitrary spin s on a n-dimensional lattice Zⁿ. We establish that the essential spectrum of the system consists of the union of at most four intervals. We obtain lower and upper estimates for the number of three-particle bound states, i.e., for the number of points of discrete spectrum of the system. Рассмотрена двухмагнонная система в изотропной негейзенберговской ферромагнитной модели с произвольным значением спина s в n-мерной решетке Zⁿ. Установлено, что существенный спектр системы состоит из объединения не более чем четырех отрезков. Получены нижняя и верхняя оценки для количества точек дискретного спектра системы, т.е. для числа трехчастичных связанных состояний системы в n -мерной решетке Zⁿ. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Журнал математической физики, анализа, геометрии Spectrum of Two-Magnon non-Heisenberg Ferromagnetic Model of Arbitrary Spin with Impurity Article published earlier |
| spellingShingle | Spectrum of Two-Magnon non-Heisenberg Ferromagnetic Model of Arbitrary Spin with Impurity Tashpulatov, S.M. |
| title | Spectrum of Two-Magnon non-Heisenberg Ferromagnetic Model of Arbitrary Spin with Impurity |
| title_full | Spectrum of Two-Magnon non-Heisenberg Ferromagnetic Model of Arbitrary Spin with Impurity |
| title_fullStr | Spectrum of Two-Magnon non-Heisenberg Ferromagnetic Model of Arbitrary Spin with Impurity |
| title_full_unstemmed | Spectrum of Two-Magnon non-Heisenberg Ferromagnetic Model of Arbitrary Spin with Impurity |
| title_short | Spectrum of Two-Magnon non-Heisenberg Ferromagnetic Model of Arbitrary Spin with Impurity |
| title_sort | spectrum of two-magnon non-heisenberg ferromagnetic model of arbitrary spin with impurity |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/106748 |
| work_keys_str_mv | AT tashpulatovsm spectrumoftwomagnonnonheisenbergferromagneticmodelofarbitraryspinwithimpurity |