CC BY
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NANOSYSTEMS: Muminov M.I., et al. Nanosystems:
PHYSICS, CHEMISTRY, MATHEMATICS Phys. Chem. Math., 2022, 13 (3), 237-244.
http://nanojournal.ifmo.ru
Original article DOI 10.17586/2220-8054-2022-13-3-237-244
Conditions for the existence of bound states of a two-particle Hamiltonian on a three-dimensional lattice
M. I. Muminov1,2'", A.M. Khurramov16, I.N. Bozorov1c 1Samarkand State University, Samarkand, 140104, Uzbekistan
2V. I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of sciences 100174, Tashkent, Uzbekistan [email protected], [email protected], [email protected] Corresponding author: M.I. Muminov, [email protected]
Abstract The Hamiltonian h of the system of two quantum particles moving on a 3-dimensional lattice interacting via some attractive potential is considered. Conditions for the existence of eigenvalues of the two-particle Schrodinger operator hM(k), k e t3, p e r, associated to the Hamiltonian h, are studied depending on the energy of the particle interaction ^ e r and total quasi-momentum k e t3 (t3 - three-dimensional torus). Keywords two-particle Hamiltonian, invariant subspace, unitary equivalent operator, virtual level, multiplicity of virtual level, eigenvalue.
Acknowledgements The authors express their kind gratitude to the referee for valuable remarks. The work was supported by the Fundamental Science Foundation of Uzbekistan, Grant No. FZ-20200929224. For citation Muminov M.I., Khurramov A.M., Bozorov I.N. Conditions for the existence of bound states of a two-particle Hamiltonian on a three-dimensional lattice. Nanosystems: Phys. Chem. Math., 2022, 13 (3), 237-244.
1. Introduction
To manipulate ultracold atoms and a unique setting for quantum simulations of interacting many-body systems, the coherent optical fields provide a strong tool because of their high-degree controllable parameters such as optical lattice geometry, dimension, particle mass, tunneling, two-body potentials, temperature etc. (See [1-4]). However, in such manipulations, due to diffraction, there is a fundamental limit for the length scale given by the wavelength of light [5] and therefore, the corresponding models are naturally restricted to a short-range case. The recent experimental and theoretical results show that integrating plasmonic systems with cold atoms, using optical potential fields formed from the near field scattering of light by an array of plasmonic nanoparticles, allows one to considerably increase the energy scales in the implementation of Hubbard models and engineer effective long-range interaction in many body dynamics [5-7].
In [8], the spectral properties of the two-particle operator depending on total quasi-momentum were investigated. In [9], the existence conditions and positiveness of eigenvalues of the two particle Hamiltonian with short range attractive perturbation was studied with respect to the quasi-momentum k and the virtual level at the lower edge of essential spectrum.
In [10], several numerical results for the bound state energies of one and two-particle systems was presented in two adjacent 3D layers, connected through a window. The authors investigated the relation between the shape of a window and energy levels, as well as number of eigenfunction's nodal domains.
In the recent work [11], the condition was obtained for the discrete two-particle Schrodinger operator with zerorange attractive potential to have an embedded eigenvalue in the essential spectrum depending on the dimension of the lattice. In [12], the discrete spectrum of the one-dimensional discrete Laplacian with short range attractive perturbation was studied.
In general, the Schrodinger operator h(k), k e td, associated to the Lattice Hamiltonian h of two arbitrary particles with some dispersion relation and short range potential interaction acts in L2 (td) as (see [13])
h(k) = h0(k) - v, k e td,
where h0(k) is a multiplication operator by (p) = — e(p) +—— e(p - k) and v is integral operator with kernel
mi m2
v(p, s) = v(p — s).
The existence and absence of eigenvalues of the family h(k) depending on the energy of interaction and quasi-
33
momentum k were investigated in [14] and [15] for the cases e(p) = ^(1 — cos 2pj), v(p — s) = ^ pa cos(pa — sa)
i=1 a=1
© Muminov M.I., Khurramov A.M., Bozorov I.N., 2022
3 N 3
and e(p) = ^(1 - cos ), v(p - q) = ^ ^ cos - q4), respectively. The spectral properties of this operator i=1 l=1 i=1 h(k) for the one dimensional case was studied in [16]. The general case when the function e(p) satisfies some conditions
d
and v(p - s) = ^o + ^ Ma cos(pa - qa) was investigated in [17].
a=1
In [18], the Hamiltonian A > 0, describing the motion of one quantum particle on a three-dimensional lattice
in an external field was considered. The authors completely investigated the dependence of the number of eigenvalues of this operator on the interaction energy for ^ > 0 and A > 0. They showed that all eigenvalues arise either from the threshold virtual level (resonance) or from the threshold eigenvalues under a variation of the interaction energy.
In [19], the authors considered the two-particle Schrodinger operator H(k), (k e t3 = (-n, n]3 is the total quasi-momentum of a system of two particles) corresponding to the Hamiltonian of the two-particle system on the three-dimensional lattice z3. It was proved that the number N(k) = N(k(1), k(2), k(3)) of eigenvalues below the essential spectrum of the operator H(k) is a nondecreasing function in each k(i) e [0, n], i = 1,2, 3. Under some additional conditions on the potential v, the monotonicity of each eigenvalue zn(k) = zn(k(1), k(2), k(3)) of the operator H(k) in k(i) e [0, n] with other coordinates k being fixed was proved.
In this work we study the Hamiltonian h for a system of two particles on the lattice z3 interacting through attractive short-range potential V. We investigate the existence conditions of eigenvalues and bound states of the Hamiltonians hM(k), k e t3, associated to the Hamiltonian h. To study hM(k), we first construct the invariant subspaces H; C L2(t3), l = 1, 27 for the operator hM(k). Moreover, the investigation of spectral properties for hM(k) is reduced to study the operator hMjI(k) := hM(k) : H; ^ H;, l = 1,27. Further, eigenvalue problem for hMjI(k) is reduced to study of a compact equation of rank one, which allows one to analyze the spectrum of hMjI(k).
2. Statement of the main result
The two-particle Schrodinger operator hM(k), k e t3, ^ e r, associated to the Hamiltonian h for a system of two particles on the lattice z3 interacting via attractive short-range potential, is a self-adjoint operator which acts in L2 (t3)
as
hM(k) = h0(k) - ^v, k =(k1;k2,k3) e t3, ^ e r,
where h0 (k) is a multiplication operator by
1 1 3
Ek(p) = — e(p) +--e(p - k), e(p) = V71 - cos2pj),
m1 m2 z—'
i=1
with v being an integral operator with kernel
3 3 3
v(p - s) = 1 + cos(pa - Sa) + cos(pa - Sa) cos(p^ - Sp) + cos(pa - Sa), a=1 7=1 a=1
Note that by the Weyl theorem on the essential spectrum [20] the essential spectrum CTess(hM(k)) of the operator hM(k) coincides with the spectrum of the unperturbed operator h0(k)
^eSS(hM(k)) = a(ho(k)) = [m(k), M(k)],
where m(k) = min Ek(p), M(k) = max Ek(p). pe t3 pet3
Since v > 0 for ^ > 0,
sup (hM(k)/,/) < sup (ho(k)/,/) = M(k)(/,/), / e L2(t3).
1/1=1 11/1=1
Hence, hM(k) does not have eigenvalues lying to the right of the essential spectrum, i.e.,
a(hM(k)) n (M(k), to) = 0.
Similarly, for ^ < 0
/= 1(hM(k)/, /) > ^ m= 1(ho(k)/, /) = m(k)(/, /), / e L2(t3).
Therefore, hM(k) does not have eigenvalues lying to the left of the essential spectrum, i.e.,
a(hM(k)) n (-to, m(k)) = 0.
Let functions f; be defined as
3
f;(p) = H n;(pa), {n;(pa)} e {1,cospa, sinpa|, a{1,2,3}. (1)
This system consists of a 27 orthogonal functions The operator v can be expressed via the functions {<£>;(•)},
defined in (1), in the form
27
(vf )(p) = ][>!,/ Vl(p). (2)
1=1
Below, we describe the conditions for the existence of eigenvalues of hM(k).
Let us denote by n(^) the number of eigenvalues (with the multiplicities) lying outside the essential spectrum of the operator hM(k).
Remark that for the number n(^) of eigenvalues of hM(k), ^ > 0 (resp. ^ < 0) and k G t3, lying to the left (resp. to the right) of the essential spectrum the following estimate is true
0 < n(^) < 27.
Assumption 2.1. Assume that m = m1 = m2 and k G n, where n is a set of k = (k1; k2, k3) G t3 with ka =
n n
-^ or ka = 2 for some a G {1,2,3}.
We divide the set n into three subsets nn, n =1, 2, 3, defined as follows: nn contains elements k G n such that precisely n of their coordinates are equal to ±n/2.
Theorem 2.1. Let the Assumption 2.1 be fulfilled. Then the following statements are true
1. For any ^ > 0 (resp. ^ < 0) and k G n1, the operator (k) has at least 12 eigenvalues lying to the left (resp. to the right) of the essential spectrum.
2. For any ^ > 0 (resp. ^ < 0) and k G n2, the operator (k) has at least 18 eigenvalues lying to the left (resp. to the right) of the essential spectrum.
3. For any ^ > 0 (resp. ^ < 0) and k G n3, the operator hM(k) has 27 eigenvalues lying to the left (resp. to the right) of the essential spectrum.
We introduce the following subspaces l = 1,27, of L2(t3) as
nj _ ajeee nj _ o/eee nj _ o/eee nj _ o/eee nj _ o/eee nj _ o/eee nj _ o/eee
Hi = H000J H2 = Hn00J H3 = H0n0J H4 = H00n J H5 = H0nn J H6 = Hn0n J H7 = Hnn0i
o/ _ o/eee 0/ _ njoee nj _ njoee nj _ njoee nj _ njoee nj _ njeoe nj _ njeoe
H8 = J H9 = Hn00J H10 = Hnn0J Hii = Hn0nJ H12 = Hiii J H13 = H0n0J Hi4 = H0nnJ
eoe eoe ooe ooe eeo eeo eeo
h15 = Hnn0J H16 = j h17 = Hnn0J H18 = Hm J H19 = H00nJ H20 = H0nnJ H2i = Hn0nJ
eeo eoo oeo eoo oeo ooo
H22 = J H23 = H0nnJ H24 = Hn0nJ H25 = Hra, H26 = , H27 = ,
where o, e, 0 and n denote even, odd, n-even and n-odd notions of variable, respectively. For example Ho^r denotes a space of functions f (p) which are even with respect to each variables p1, p2 and odd with respect to p3, and n-even with respect to p1, and n-odd with respect to each variables p2, p3, i.e.,
H0nn = {f e L2(t3):
f(—pi,p2,p3) = f(pi,p2,p3), f(pi, —p2,p3) = f(pi,p2,p3), f(pi,p2, — p3) = —f(pi,p2,p3), f(pi + n,p2,p3) = f(pi,p2,p3), f(pi,p2 + n,p3) = —f(pi,p2,p3), f(pi,p2,p3 + n) = —f(pi,p2,p3)}.
Remark that the operator hM(k) is invariant with respect to H, l = 1, 27 (See Lemma 3.1). We denote by hMji(k) the restriction hM(k)|H| of hM(k) to Hi.
Note that G Hi, l = 1, 27. Therefore, the operator hMji(k), l =1,27 acts in Hi as
Vi(k) = ho(k) - ^vi,
where
(vi/)(p) = (^,/Mp), W G Hi, l =TT27.
Then we have
27
*(Mk))= U CT(Vi(k)).
i=1
Next, we study the operator hMji(k), l = 1, 27. We set
£i(k; z) = / Ti \' , ^ g Hi, l =1,27, z G c\[m(k), M(k)], (3)
Efc(s) - z
where
^2(s)ds
t3
~, n ( i i ri 2 T \
Ek(p) = 7 .--1---W -9 +--cos 2kj +--2 cos 2pJ .
f—i Vmi m2 y m2 mim-2 m2 /
If Assumption 2.1 is not fulfilled, then the integral (3) converges as z = m(k) (z = M(k)) (see Lemma 3.2 below).
We set
(m(k)) = w-mw) ■ (M <k» = iiiiTM(i)) ■ 1 = '■27
Let C(t3) be the Banach space of continuous (periodic) functions on t3 and Gi (z), 1 e {1, 2,..., 27} be the (Birman-Schwinger) integral operator with the kernel
z) = , z e (-TO,m(0)j U [M(0), m(0) = 0, M(0) = 6mi + m2 .
Eo(q) - z mim2
Definition 2.1. If number 1 is an eigenvalue of the operator G(0), (resp. G(M(0))) and the corresponding eigenfunc-tion ^ satisfies the condition
e Li(t3)\L2(t3), (resp. ■ _ ^ e Li(t3)\L2(t3^,
W v £o(-) - M(0)
then it means that the operator hMji (0) has a virtual level at the left edge (resp. at the right edge) of the essential spectrum.
Theorem 2.2. Suppose that Assumption 2.1 are not fulfilled. Then the following statements are true
1. For any 0 < ^ < ^(k) (resp. < ^ < 0) the operator hMji(k) has no eigenvalues lying to the left (resp. to the right) of the essential spectrum.
2. Let 0 < ^ = ^0(m(0)) (resp. ^(M(0)) = ^ < 0). If y>j(0) = 0, then hMij(0) has a virtual level at z = 0 (resp.
at z = 6 mi + m2), if ^ (0) = 0, then the number z = 0 (resp. z = 6 mi + m2) is an eigenvalue of h„ i (0).
mim2 mi m2
3. For any k e t3 and ^ > > 0 (resp. ^ < < 0), the operator hMji(k) has unique eigenvalue lying to the left (resp. to the right) of the essential spectrum.
Theorem 2.3. Let Assumption 2.1 be fulfilled. Then the following statements are true
1. For any ^ > 0 (resp. ^ < 0) and k e ni; there exist 1i, 12,..., 1i2 e {1, 2,... 27} such that the operator hMjii(k), i = 1,12 has a unique eigenvalue lying to the left (resp. to the right) of the essential spectrum.
2. For any ^ > 0 (resp. ^ < 0) and k e n2, there exist 1i, 12,..., 1i8 e {1, 2,... 27} such that the operator hMjii(k), i = 1,18 has a unique eigenvalue lying to the left (resp. to the right) of the essential spectrum.
3. For any ^ > 0 (resp. ^ < 0), k e n3 and 1 e {1, 2,... 27} the operator fcMji(k) has unique eigenvalue lying to the left (resp. to the right) of the essential spectrum.
^^ i m2
Remark 2.1. Note that Theorem 2.2, 2) shows that the number z = 0 (respectively, z = 6-) might be a
mi m2
virtual level or an eigenvalue or a virtual level and an eigenvalue for the operator hM(0). For the case ^ = ^°(m(0)) or ^ = ^8(m(0)), number z = 0 is a simple virtual level of hM(0) with
/i(P) = ^ e Li(t3)\L2(t3) or /8(p) = cospi cosP3 e Li(t3)\L2(t3).
For the case ^ = ^2(m(0)) = ^3(m(0)) = ^°(m(0)) or ^ = ^°°(m(0)) = ^6(m(0)) = ^°(m(0)), number z = 0 is a virtual level of (0) with multiplicity 3 with
/i+i(p) = C0^ e Li(t3)\L2(t3) or /4+i(p) = cospa,c°sP e Li(t3)\L2(t3), E0(p) E0(P)
{i, a, ft} = {1, 2, 3}.
3. Proof of the main results
Consider the operator hM(k) acting in L2(t3) by the formula
(k) = h0(k) - Mv,
where h0 (k) is the operator of multiplication by the function ¿k( ).
The operator hM(k) is unitary equivalent to the operator hM(k) (See Lemma 2 in [15]). The equivalence is performed by the unitary operator U : ¿2(t3) ^ ¿2(t3) as hM(k) = U-ihM(k)U, where
(U/)(p) = /(p - 10(k)),
+ — cos 2kj
0(k) = (0i(ki),02(k2),03(k3)), g»(fei) = arccos ; mi m2 , i = 1,2,3.
A +--— cos 2kj + A
mi m1m2 ^ m2
Lemma 3.1. 1. The following equality holds
27
L2(t3 ) = 0 Hl. (4)
i=i
2. The operator hM(k) is invariant with respect to H;, l = 1, 27, i.e., hM(k) : H; ^ H;.
Proof. 1. For brevity, let us introduce some notations: o, e, 0 and n denote even, odd, n-even and n-odd notions of variables, respectively. For example, He denotes a space of functions f (p) which are even with respect to variable pi, similarly, Hge denotes a space of functions f (p) which are even with respect to each variables p1, p2 and n-even with respect to p1.
We represent f G L2 (t3) as
f (P1,P2,P3) = f e(Pl ,P2 ,P3) + f °(P1,P2,P3),
where
fe(p1,p2,p3) = f(P1,P2,P3)+2f(-P1,P2,P3) g He,
f>1,P2,P3) = f (P1'P2'P3) -2f (-P1,P2,P3) G H°.
It is clear that L2 (t3) = He © Ho.
Similarly, we represent the functions fe and fo as
fe(P1,P2,P3) = f ee(P1,P2,P3) + f e0(P1,P2,P3)
and
f °(P1,P2,P3) = f 0e(P1,P2,P3) + f00(P1,P2,P3),
where
fee(p1,P2,P3) = f (P1'P2'P3)+2f (P1, -P2,P3) G Hee,
fe°(p1,P2,P3) = fe(P1,P2,P3) -2fe(P1, -P2,P3) G H",
f°e(p1,P2,P3) = f0(P1,P2,P3)+2f0(P1, -P2,P3) G H°e,
f00(p1,p2,p3) = f0(P1,P2,P3) -2f0(P1, -P2,P3) G Hoo. Then He = Hee © He0 and H0 = H0e © H00.
Arguing similarly step by step we obtain the equality of the direct sum of subspaces
L2 (t3) = Heee © Hee0 © He0e © H0ee © He00 © H0e0 © H00e © H000. (5)
Each subspace in (5) is represented via the direct sum of subspaces defined as combination of n-even and n-odd functions
eee eee eee eee eee eee eee eee eee
H = H000 © Hn00 © H0n0 © H00n © H0nn © Hn0n © Hnn0 © Hm j
njeeo njeeo H — H00n m Heeo m Heeo m Heeo , Heoe eoe _ H0n0 eoe eoe eoe m H0nn m Hnn0 m
Hoee njoee ^ njoee _ Hn00 m Hnn0 m Hoee m Hoee Heoo _ Heoo m Heoo ,
Hoeo _ Hoeo m Hoeo Hn0n m ' Hooe ooe _ Hnn0 m Hooe , Hooo _ Hooo .
Substituting these equalities into (5), we obtain (4). 2. By (2), the operator v can be expressed via v; as
27
(vf)(p) = J>;f)(p), (v;f)(p) = (^;,f)^;(p), ^ G H;, l =1,27.
l=i
Since {^¡(•)}> l = 1, 27 is an orthogonal system in L2(t3), v : H; ^ H;. One can see that £fc(p)^>;(p) G H;. Hence, M*0 : H; ^ H;. □
The following lemma is proven in [14] Lemma 3.2. Suppose Assumption 2.1 does not hold. Then the integral
Ek(s) — m(k)
t3
converges for any ^ G C(t3).
Lemma 3.3. A number z, z G c \ [m(k), M(k)], is an eigenvalue of hM,;(k) iff A;(p, k; z) = 0, where
A;(p, k; z) = 1 — p£;(k; z). (6)
Proof of lemma 3.3. Let z e c \ [m(k),M(k)] be an eigenvalue of hMji(k) and /i, 1 = 1, 27 be the corresponding eigenfunction, i.e., the equation
hM,i (k)/i = z/i
has a nontrivial solution /i. Then
/i = ^ro(k,z)v^ /i, 1 = 1727, (7)
where r0(k,z) is a multiplication operator by the function -—1-. Denote
£fc(p) - z
Vi = (^i,/i). (8)
Then equation (7) can be represented as
/i(p) = Vi. (9)
Ek (P) - z
By substituting (9) in (8), we obtain the following equation
Vi = M i 1 = 1727.
7 Efc(s) - z
t3
If z e c \ [m(k), M(k)] is an eigenvalue of the operator fcMji(k) then Ai(^, k; z) = 0. Conversely, let A1(m, k; z) = 0 with z e c \ [m(k), M(k)], i.e.,
1 - (k; z) = 0.
Then the function
Vi(p) = _ (p)— Ek (p) - z
is an eigenfunction of the operator hMji(k) corresponding to the eigenvalue z e c \ [m(k), M (k)].
Lemma 3.3 gives the following result.
Corollary 3.1. A number z, z e c \ [m(k), M(k)], is an eigenvalue of fcM(k) iff A(m, k; z) = 0, where
□
27
A(m, k; z) = JjAi(M,k; z).
i=i
Further we prove the main results for m > 0. The case m < 0 will be proven in a similar way. Proof of Theorem 2.2. By Lemma 3.2, the integral
f ^2(s)ds J Efc(s) - m(k)
t3
converges for any ^ e Hi, 1 = 1,27.
1. The function Ai(^,k; •) is monotonically decreasing for z e (-ro,m(k)) (m e (0, ro)) for any fixed m > 0
(z < m(k)). Then we have
A1(m, k; z) > A1(m, k; m(k)) > Ai(M0(k),k; m(k)) = 0 for all m e (0, m0(k)).
According to Lemma 3.3, the operator hMji (k), 1 = 1, 27 has no eigenvalues lying to the left of the essential spectrum.
2. Let z = 0 and m = M0(m(0)), 0 = (0,0,0) e t3. Then
Ai(M?(m(0)), 0;0) = 1 - M0(m(0)) / = 0.
J E0(s) - m(0)
t3
Then the function
/i(p)=. ^-, 1 = 1727.
<?0(p) - m(0) '
is a solution of the equation hMji(0)/i = 0. Indeed,
Vi (0)/i = ^i(p)(1 - MV(0)) / - ^ (s)ds) =0.
^ J ¿-0(s) - m(0)/
t3
Note that from the equation (1), we have
33
^i(0) = JJ ni(0) = 0, 1 = 178, ^ (0) = n ni(0) = 0, 1 = 9727.
Therefore,
f G Li(t3)\L2(t3), l = 1, 8 f G L2(t3), l = 9, 27.
This yields that (0) has a virtual level at z = 0 for any l = 1, 8 and z = 0 is an eigenvalue of (0) for any l = 9,27.
3. Let p > p0(k). Then
lim A;(p, k; z) = A;(p, k; m(k)) = 1--< 0.
z^m(fc) P0(k)
Note that
lim A;(p, k; z) = 1.
z^ — w
Then from the continuity and monotonicity of A;(p, k; •) in (-to, m(k)), we have that there exists unique z; G
(-to, m(k)) such that
A;(p, k; z;) =0.
According to Lemma 3.3, the operator (k) has unique eigenvalue lying to the left of the essential spectrum.
□
n
Proof of Theorem 2.3. 1. We prove theorem for the case k G n1, k1 = ± . Then the function £fc (•) does not depend of
p1 is expressed as
3
_ 6 3 ^ 1 _
¿k (p) =--V — ^2 + 2 cos 2kj cos 2pj.
mm
i=2
We separate the functions £;(k; •) with ^;(p1,0,0) = 0. There are 12 such functions and after integrating them with respect to s1 they can be represented as
a(k; z) = 2n / -, &(k; z) = £9(k; z) = n / -,
J £fc (s) - z J £k(s) - z
t2 t2
i cos2 sjds f cos2 sjds
?i+1(k;z) = 2n / —, Ci+3(k;z) = Ci+s(k;z) = n —, i= 2 з,
J ¿fc(s) - z J ¿fc(s) - z
t2 t2
, . ¡' cos2 s2 cos2 s3ds ^ . ^ ¡' cos2 s2 cos2 s3ds
&(k; z) = 2n -^-, £8(k; z) = ^(k; z) = n -^-.
J £fc(s) - z J £fc(s) - z
t2 t2
Since (¿fc(p) - m(k)) = O(p2) as |p| ^ 0, the last equations give
lim A;(p, k; z) = -to.
z^m(fc)
According to the continuity and monotonicity of A; (p, k; •) in (-to, m(k)) and
lim A;(p, k; z) = 1,
z^ — w
there exists unique z; G (-to, m(k)) such that
A;(p, k; z;)=0, l = 1712.
n
The cases k = ± ^, i = 2, 3 can be considered in a similar way.
n
2. We prove theorem for the case k G n2 with k1 = k2 = ± ^. The function £fc( ) does not depend of p1, p2 and is expressed as
_ 6 1 1_
Ek (p) =---^2 + 2 cos 2k3 cos2p3.
mm
Then there exist 18 functions £;(k; •), l = 1,18 with ^;(p1 ,p2,0) = 0. These functions are represented via integrals with respect to s3 and contain a numerator function ^5;(s3) with (0) = 0. Since (5fc (p3) - m(k)) = O(p3) as p3 ^ 0, the last equations give
lim £;(k; z) =+to, lim A;(p, k; z) = -to, l = 1,18.
z^m(fc) z^m(fc)
Hence there exists unique z; G (-to, m(k)) such that
A;(p, k; z;) = 0, l = 1,18. The remaining cases with k G n2 are proved in a similar way.
3. The case k G n3 can also be considered by similar discussions as in parts 1) and 2). □
Theorem 2.3 leads to Theorem 2.1.
4. Conclusion
We investigate the existence conditions for eigenvalues of the two-particle Schrodinger operator hM(k), k e t3, M e r corresponding to the Hamiltonian of the two-particle system on the three-dimensional lattice, where hM(k) is considered as a perturbation of free Hamiltonian h0(k) by the potential operator mv with rank 27.
To study spectral properties of hM(k), we first constructed the invariant subspaces Hi C L2(t3), 1 = 1,27 for the operator hM(k). Moreover, investigation of the spectral properties of the operator hM(k) is reduced to the study of the operator hMji(k) := hM(k) : Hi ^ Hi, 1 = 1, 27. Further, eigenvalue problem for hMji(k)/ = z/, z / o-ess(hM(k)) is reduced to the study of an integral operator pG;(z) of rank one. This allowed us to analyze the eigenvalue problem of hMji(k) for any Me r.
Particularly, if k = 0, then there exist the numbers M°(m(0)) > 0 and m°(M(0)) < 0,1 = 1,27 such that
(i) for any m with 0 < m < M°(0) (resp. m°(0) < m < 0) the operator hMii(0) has no eigenvalues lying to the left (resp. to the right) of the essential spectrum;
(ii) for m = M0(m(0)) (resp. m°(M(0)) = m), if <^(0) = 0, then hMji(0) has a virtual level at z = 0 (resp. at
z = 6mi + m2), if <i (0) = 0, then the number z = 0 (resp. z = 6 mi + m2) is an eigenvalue of h„ i (0); mi m2 mim2
(iii) for any m, M > M0(0) > 0 (resp. m < M°(0) < 0), the operator hMji (0) has unique eigenvalue lying to the left (resp. to the right) of the essential spectrum.
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Submitted 17 May 2022; accepted 26 May 2022
Information about the authors:
M. I. Muminov - Samarkand State University, University blv., 15, Samarkand, 140104, Uzbekistan; V. I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of sciences 100174, Tashkent, Uzbekistan; [email protected] A. M. Khurramov - Samarkand State University, University blv., 15, Samarkand, 140104, Uzbekistan; [email protected] I. N. Bozorov - Samarkand State University, University blv., 15, Samarkand, 140104, Uzbekistan; [email protected] Conflict of interest: the authors declare no conflict of interest.
