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NANOSYSTEMS: Khalkhuzhaev A.M., et al. Nanosystems:
PHYSICS, CHEMISTRY, MATHEMATICS Phys. Chem. Math., 2023,14 (5), 518-529.
http://nanojournal.ifmo.ru
Original article DOI 10.17586/2220-8054-2023-14-5-518-529
On the discrete spectrum of the Schrodinger operator using the 2+1 fermionic trimer on the lattice
Ahmad M. Khalkhuzhaev1, Islom A. Khujamiyorov2
1V. I. Romanovsky Institute of Mathematics of the Academy of Sciences of the Republic of Uzbekistan, Tashkent, Uzbekistan
2Samarkand State University, Samarkand, Uzbekistan Corresponding author: Ahmad M. Khalkhuzhaev, ahmad_x@mail.ru
PACS 02.30.Tb
Abstract We consider the three-particle discrete Schrodinger operator HMi7(K), K e T3, associated with the three-particle Hamiltonian (two of them are fermions with mass 1 and one of them is arbitrary with mass m = 1/y < 1), interacting via pair of repulsive contact potentials ^ > 0 on a three-dimensional lattice Z3. It is proved that there are critical values of mass ratios 7 = y1 and 7 = y2 such that if 7 e (0, y1), then the operator HMi7(0) has no eigenvalues. If 7 e (y1, y2), then the operator HMi7(0) has a unique eigenvalue; if 7 > y2, then the operator HMi7(0) has three eigenvalues lying to the right of the essential spectrum for all sufficiently large values of the interaction energy
Keywords Schrodinger operator, Hamiltonian, contact potential, fermion, eigenvalue, quasi-momentum, invariant subspace, Faddeev operator.
Acknowledgements We thank unknown referee for careful reading of the manuscript and useful comments.
For citation Khalkhuzhaev A.M.,Khujamiyorov I.A. On the discrete spectrum of the Schrodinger operator using the 2+1 fermionic trimer on the lattice. Nanosystems: Phys. Chem. Math., 2023,14 (5), 518-529.
1. Introduction
The study of few-body systems with contact interaction has a long history and a wide literature throughout the last eight decades, a concise retrospective may be found in [1]. The 2+1 fermionic system is an actual building block for the heteronuclear mixtures with inter-species contact interaction, see [2] for an outlook. For the 2+1 fermionic model, the rigorous construction of the Hamiltonian Ha for m> m*, together with the precise determination of m* and the proof of the self-adjointness and the semi-boundedness from below of Ha, was done in the work [3] by Correggi, Dell'Antonio, Finco, Michelangeli, and Teta, by means of quadratic form techniques for contact interactions [4]. In [5] the authors had qualified the main features of the spectrum of the Hamiltonian of point interaction for a three-dimensional quantum system consisting of three point-like particles, two identical fermions, plus a third particle of different species, with two-body interaction of zero range. For arbitrary magnitude of the interaction and arbitrary value of the mass parameter (the ratio between the mass of the third particle and that of each fermion) above the stability threshold, the essential spectrum is identified, the discrete spectrum is localized and its finiteness is proved. The existence or absence of bound states is proved in physically relevant regimes of masses.
Throughout physics, stable composite objects are usually formed by way of attractive forces, which allow the constituents to lower their energy by binding together. Repulsive forces separate particles in free space. However, in structured environment such as a periodic potential and in the absence of dissipation, stable composite objects can exist even for repulsive interactions that arise from the lattice band structure [6]. The Bose-Hubbard model which is used to describe the repulsive pairs is the theoretical basis for applications. The work [6] exemplifies the important correspondence between the Bose-Hubbard model [7], [8] and atoms in optical lattices, and helps pave the way for many more interesting developments and applications [9]. Stable repulsively bound objects should be viewed as a general phenomenon and their existence will be ubiquitous in cold atoms lattice physics. They give rise also to new potential composites with fermions [10] or Bose-Fermi mixtures [11], and can be formed in an analogous manner with more than two particles [12].
Systems of particles, with zero-range interactions between the pairs of particles, are investigated not only theoretically but also experimentally. Delta-like character of the interaction turns out to be realistic. This is a special case of the unitary regime, i.e. the case of negligible interaction range and huge, virtually infinite, scattering length. In this case, unitary gases posses a property of superfluidity [13], and they were intensively studied both experimentally and theoretically [14].
In this paper, we consider the Hamiltonian HMi7 for systems of three quantum particles (two of them are fermions with mass 1 and one of them is arbitrary with mass m = 1/7 < 1) with paired contact repulsive potentials ^ > 0 on a
© Khalkhuzhaev A.M.,Khujamiyorov I.A., 2023
three-dimensional lattice Z3. In the momentum representation, the total three-particle Hamiltonian expands into a direct operator integral (see. [15])
HM,7 = J ©HMi7 (K)dK.
T3
The fiber operator HMi7(K) = H0j7(K) + m(V + V2) parametrically depends on the total quasimomentum K e T3 = R3/(2nZ3). It is shown that the essential spectrum of the self-adjoint operator HMj7(K) consists of one or two segments, depending on the three-particle quasimomentum K e T3 and the interaction energy m > 0. Unlike the continuous case, the Schrodinger operator on a lattice can have eigenvalues at the right part of the essential spectrum as well.
The principal results of this paper are given for sufficiently big values of interaction energy m > 0, i.e., when the two-particle subsystems have bound states with positive energies: there are threshold values of the particle mass ratio Yi, Y2 such that if y e (0, Yi), then the operator HMj7(0) has no eigenvalues; if y e (yí, y2), then the operator HMj7(0) has a unique eigenvalue; if 7 > y2, then the operator HMi7(0) has three eigenvalues lying to the right of the essential spectrum. Existence of at least one eigenvalue of the three-particle discrete Schrodinger operator HM(K) = H0(K) - mV (m e R) for dimensions d =1, 2 was shown in [15] and [12], whose proofs are based on the unboundedness of the norm of the Faddeev operator T(K, z) at the lower bound of the essential spectrum z = inf(aess (HM(K))). If d > 3, then the operator T(K, z) is also bounded at the edges of the essential spectrum, i.e. in this case, methods for d =1,2 is not applicable.
In [16], the model operator H^s (see (2.6) paper in [16]), associated with three-particle discrete Schrodinger operator on a three-dimensional cubic lattice with pairwise zero-range attractive potentials, is studied, where the family of Friedrichs models with parameters ha(k), a = 1, 2, k e T3 is used. The existence of the critical value 7* of the parameter y is proved so that if two-particle subsystems have a resonance with zero energy and do not have bound states with negative energy, then H^s has an infinite number of eigenvalues, lying to the left of the essential spectrum for Y > y* « 13.607, and there is no Efimov's effect for y < Y*. The similar result holds for the operator we are considering HMi7(n), i.e., at y > Y* and fixed M = Mo(y), the operator HMi7(n) has an infinite number of eigenvalues to the right of the essential spectrum. "The two-particle branch" of the essential operator spectrum of HM 7(K) is shifted to with the order m if M ^ as a result of which an infinite number of eigenvalues of the operator are "absorbed" by the essential spectrum. Therefore, a natural question arises: whether there are eigenvalues of the operator HM 7(0), lying to the right of the essential spectrum for sufficiently large m, and if so, how many?
In this paper, we prove that the operator HM 7(0), 0 = (0,0,0), for y e (0, Yi) (Yi ~ 2, 937) has no eigenvalues, but for Yi < y < Y2 (y2 ~ 5,396) has a unique eigenvalue, and for y > Y2 has exactly three eigenvalues to the right of the essential spectrum for sufficiently large m. Physically, this shows the conditions for the system of two fermions (of mass 1), and an arbitrary particle (of mass m, m < 1) with pairwise repulsive interaction m, which is sufficiently large, to have no bound states, one bound state and three bound states, respectively.
Applying the perturbation theory, one can show that the results obtained are preserved for small values K. Note that the problem of finding the number of eigenvalues of the operator HMi7 (K), which are more z (z > rmaXj7 (m, K)) reduces to the problem of finding the number of eigenvalues of the Faddeev-type operator AM 7 (K, z), which are more 1 (see. (4.2)). Sensitivity of the kernel of the integral operator AM 7(K, z) regarding change K leads to a change in the number of eigenvalues of the operator HM 7(K). Therefore, set the number of eigenvalues for all K e T3 is very difficult.
2. Statement of the problem and formulation of the main result
Let Z3 is a three-dimensional lattice, l2[(Z3)d], d = 2,3 is a Hilbert space of square integrable functions given on (Z3)d and l2'as [(Z3)d] c I2 [(Z3)d] is a subspace of antisymmetric functions with respect to permutation of the first two coordinates.
We consider a Hamiltonian of a system of three quantum particles (two of them are fermions with mass 1 and one of them is arbitrary with mass m = 1/y < 1) that interact through pairwise zero-range repulsive potentials on Z3. Without a loss of generality, we assume that the first two particles are fermions while the third one is a particle of a different nature.
The Hamiltonian of the system of two arbitrary free particles (a fermion and another particle) on Z3 in the coordinate representation is associated with the bounded self-adjoint operator h0j7 in l2[(Z3)2]:
1 Y
ho,Y = - 2 a ® I - 21 ® A,
where A is the lattice Laplacian, 1 is the unity operator in l2(Z3), and y = —.
m
The total Hamiltonian hM 7 of the system of two arbitrary particles with the zero-range repulsive potential acts in I2 [(Z3)2] and is a bounded perturbation of the free Hamiltonian h0j7 :
hMi7 = hoi7 + m^,
where ^ > 0, is the interaction energy of two repelling particles (a fermion and another particle), operator v describes the zero-range interaction of these particles
(VV>)(X2, X3) = ¿X2X3 X3)
and SX2X3 is the Kronecker symbol. In the space l2'as [(Z3)2], there is no two-particle zero-range interaction of fermions (see [15], [17]).
Similarly, the free Hamiltonian H0j7 of the system of three particles (two fermions and another particle) on lattice Z3 is specified in l2'as[(Z3)3] by the formula
11 Y
H0,Y = -^ A 0 1 0 1 - 21 0 A 0 1 - 210 10 A.
The total Hamiltonian HMi7 of the system of three particles with pairwise zero-range interactions is a bounded perturbation of the free Hamiltonian H0j7:
H Ml7 = H o,7 + m(V 1 + V 2),
where
("V 1^A)(X1, X2, X3) = ¿X2X3 ^A(X1, X2, X3)
and
("V2 V0(X1, X2, X3) = iX3X! V)(X1, X2, X3).
Let T3 is a three-dimensional torus and Uf [(T3)2] c U2 [(T3)3] be the Hilbert space of square integrable functions, defined on (T3)3 and antisymmetric with respect to permutation of the first two coordinates. Assume that dp is a unit measure in the torus T3, that is
J dp = 1.
T3
The study of spectra of the Hamiltonians hMi7 and HMi7 is reduced to studying the spectra of the family of operators
, k , k ^ ,, , , , K" ^ T3 fAín»
(k), k c ^ mld HM>Y(
.ta SchrodingorAperaf°r (
hMi7 (k) = hoi7 (k) + ^v,
where
hMi7(k), k e T3 and HMi7(K), K e T3, respectively (see [15], [18]).
The two-particle discrete Schrodinger operator hMi7 (k), k e T3 acts in L2(T3) by the formula
(ho,Y(k)f)(p) = Ek,7(p)f (p), Ek,7(p) = e(p) + 7^k - p),
3
e(p) = 3 - £(p), £(p) = ^>ospi, p =(P1,P2,P3) e T3, (2.1)
i=1
(vf )(p) = / f (s)ds.
T3
The respective three-particle discrete Schrodinger operator HMi7(K) acts in [(T3)2] by the formula
HM,7 (K) = Ho,7 (K)+ + V2),
where
(Ho,7(K)f )(p, q) = Ek,7(p, q)f (p, q), Ek,7(p, q) = e(p) + e(q) + 7e(K - p - q). (V1f )(p, q) = J f (p, s)ds, (V2f)(p, q) = y f (s, q)ds.
T3 T3
Let us first introduce the following notation:
i ds f cos s1 ds TTT f cos2 s1 ds TTT f cos s1 cos s2ds
W = 1 ~T\, W1 = ---, Wn = ---, W12 = ----.
J e(s) J e(s) J e(s) J e(s)
T3 T3 T3 T3
The integral W is called the Watson integral and the other integrals W1, W11 and W12- Watson-type integrals (see, for example [20]).
The main result of the paper is the following theorem:
Theorem 2.1. Let
W1
Y1 = W11W + 2WW12 - 3W2 - 2,9368, Y2 = - 5,3985 (2.2)
(i) Assume that 7 e (0,y1). Then, there exists > 0 such that for any ^ > the operator HMi7 (0) has no eigenvalues lying to the above of the essential spectrum.
(ii) Assume that y € (71,72)- Then, there exists my > 0 such that for any m > MY the operator HMi7 (0) has a unique eigenvalue lying to the above of the essential spectrum.
(iii) Assume that 7 € (y2 , . Then, there exists my > 0 such that for any m > MY the operator HMi7 (0) have three eigenvalues to the above of the essential spectrum.
Remark 2.2. The number my takes on different values in the three cases of the Theorem 2.1.
3. On the spectrum of the two-particle operator hM 7 (k)
In this section, we study some facts related to the spectrum of the operator hM 7 (k).
Since v is compact, by Weyl's Theorem [19] for any k € T3, the essential spectrum aess(hM7(k)) of hM7(k) coincides with the spectrum of h0j7(k), i.e.,
^ess(h^,7 (k)) [Emin,7 (k) Emax,Y (k)L
where
3
Emin,Y(k) = min fki7(q) = 3(1 + 7) — V + 27coskj + y2,
q£T3 —'
i=1
3
Emax,7(k) = max fkj7(q) = 3(1 + 7) + V" \Jl + 27 cos k + y2 .
q£T3 —'
The following Lemma provides an implicit equation for eigenvalues of hM 7 (k) which is a simple application of the Fredholm determinants theory.
Lemma 3.1. The number z e C \ [£min,7 (k), fmax,7 (k)] is an eigenvalue of hMi7 (k) with multiplicity m if and only if z is a zero of the function
dq (3.1)
AMi7(k, z) = 1 - ^ J
z - Ek,7(q)
T3
with the multiplicity m.
The function Ami7(k, z) is called the Fredholm determinant associated to hMi7(k).
Note that, the function AM 7(k, z) is the Fredholm determinant of the operator I - ^vr0 7(k, z), where r0 7 (k, z) is the resolvent of the operator h0j7 (k) and v is the integral operator with the kernel v(q, q') = 1. Let us introduce first the following real number:
Mo(Y) = (1 + Y) W1 -
Note that this number means harmonic values of the kinetic energies of a fermion and another particle.
Lemma 3.2. Assume that m > m0(y)- Then for each k € T3 the operator hMi7 (k) has a unique simple eigenvalue zMi7 (k) above the essential spectrum.
Lemma 3.3. The eigenvalue zMi7 (k) = zMi7 (k1; k2, k3) is symmetric function with respect to permutation of the variables fcj, kj, even with respect to k € [—n, n], and decreasing with respect to k € [0, n], i = 1,2,3.
Proof. The proof of the lemma follows directly from the properties of the function AM 7 (k, z) and assertions of Lemma 3.1. ' □
Lemma 3.4. For any y > 0 and m > 3(1 + y), we have the following relations
9(1 + Y)2
M + 3(1 + y) < zMi7(n) < zMi7 (k) < zMi7(0) <m + 3(1+ y) +—-— -
M
Proof. The proof of the Lemma follows from Lemma (3.3) and properties of the function AM 7(k, z). □
Corollary 3.5. For any y > 0, the function zMi7 (k) has the following asymptotic expansions:
zM,7 (k)= m + 3(1 + y) + o( M)
as m ^ ro, uniformly k € T3.
(3.2)
4. Essential spectrum of a three-particle operator HMi7 (K).
For any K e T3, recalling that
Emin,7(K) = min Ek,y(p, q), EmaXj7(K) = max Ek,y(p, q),
p,qGT3 p,qGT3
Tmin,Y(m, K) = min {zMl7(K - p) + e(p)}, Tmax,7(m, K) = max{zMi7(K - p) + e(p)},
p£T3 p£T3
where zMi7 (p) is an eigenvalue of the operator hMi7(p) and the essential spectrum of HMi7(K) coincides with the union of two segment:
^eSS(HMj7 (K)) = [Emin,7 (K),£max,7 (K)] U [Tmin,7 (M, K),Tmax,Y (M, K)]. (4.1)
The proof of a similar assertion is given in the paper [18]. Note that [rmin,7 (m, K), rmaXj7 (m, K)] and [Emin,7 (K), Emax,7 (K)] are called the "two-particle branch" and the "three-particle branch" of the essential spectrum of HMi7(K), respectively.
For fixed 7,7 > 0, we study the discrete spectrum of the operator HMi7(0), 0 = (0,0,0) for sufficiently large M > 0. It follows from Lemma 2.4 and the structure of the essential spectrum that (see (4.1)), that the two-particle branch [Tmin,Y (m, 0), TmaXj7 (m, 0)] of the essential spectrum shifts with order m at m ^ In what follows we always assume z > inf aess(HMi7(0)) = rmaXj7(m, 0).
Discrete spectrum of a three-particle operator HMi7 (0).
First, we show that the operator HMi7(K) has no eigenvalues below the essential spectrum.
Lemma 4.1. Assume that K e T3. Then for any m > 0 and 7 > 0 the operator HMi7 (K) has no eigenvalues below the essential spectrum.
Proof. Since the operator V = V + V2 is positive by the minimax principle we can conclude that
^№,7(K)/,/) = ufnytHo.Y(K)/,/) + m(V/, /)] > |/in£1(Ho,7(K)/,/) = Emin,Y(K), leading to a(HM,7(K)) n (-to, Emin,7(K)) = 0. □
For any z > rmaXj7(m, 0), we define the self-adjoint compact operator of the form
(Am,7 (zW(p) = , -M /-(4.2)
V AM,7
(p, z) (z — e0,y(p, s)) V AM,7(s, z)
T3
defined in
-0(s)ds
D(AMi7(z))= jv e L2(T3): J
v/Ami7 (s, z)
T3
where
Ami7(p, z) := Ami7(-p, z - e(p)), (4.3)
and the function Ami7(.,.) is given by formula (3.1).
The operator Ami7(z) is called the Faddeev-type operator corresponding to the operator HMi7(0) (see Remark 4.3 and [21], [22]).
Hence, we found the equivalent equation for the eigenfunctions of the three-particle operator HMi7 (0).
Lemma 4.2. The number z > rmax,7 (m, 0) is an eigenvalue of the operator HMi7 (0) if and only if the number 1 is an eigenvalue of the operator Ami7 (z).
Proof. Let z > rmax,7 (m, 0) is the eigenvalue of the operator HMi7(0) and / is the respective eigenfunction, i.e., the equation
Eo,7(p, q)f (p, q)+ mJ /(p, s)ds + mJ /(s, q)ds = zf (p, q) (4.4)
T3 T3
has a nonzero solution / e Lif [(T3)2]. Introducing the notation
f (p) = (Vif )(p, q) = J /(p, s)ds, (4.5)
T3
from (4.4) for z > rmax,7 (m, 0), we have
/ (p, q) = M-^E-f^. (4.6)
z - Eo,7(p, q)
Since the function f is antisymmetric, the function ^ given by formula (4.5), belongs to the space L2 (T3) and satisfies the condition
J ¥>(p) dp = 0.
T3
Substituting the expression (4.6) into (4.5), we obtain that the equation
— M J
ds \ f <^>(s)ds
= —M
z — Eoi7 (p, s) J J z — Eoi7(p, s)
T3 T3
has a nonzero solution ^ € L2(T3). Hence, using notation (3.1) and (4.3), we make sure that ^ € L2(T3) is the solution of the equation
f I ^(s)ds ^(p) = a^tycptz^ 7 z — Eo,y(p, s)- (4J)
T3
If we set ^(p) = (p, z) y(p), from (4.7) we have
—M / -0(s)ds
V>(p) =
y/A M,Y(p, z) ^ (z — E0,7(p, s))
T3
i.e., A =1 is the eigenvalue of the operator Ami7(z) and
f -0(s)ds
J v^y(s,z)
T3
0.
Suppose that, for some z > rmax,7 (m, 0) the number 1 is the eigenvalue of the operator Ami7 (z),and ^ € D(Ami7 (z)) is the corresponding eigenfunction. Then, the function f is given by formula (4.6), where y(p) = ^(p) ^/ami7(p, z), belongs to the space L!f [(T3)2] and satisfies the equality (4.4). □
Remark 4.3. a) Note that the relation between eigenfunctions f and respectively, of HMi7(0) and Ami7(z) corresponding to the eigenvalues z and 1 is
(Ami7(p, z))-1/2 ^(p) — (Am,7(q, z))-1/2 ^(q)
f(p, q) = m-' M'7V
at t
b) A limit operator
z — Eoi7(p, q)
Therefore, we can say that the operator Amj7(z) is the Faddeev-type operator.
lim, nam,t (z) = ,7 (Tmax,Y (m 0))
is a compact self-adjoint operator in L2 (T3).
For the bounded self-adjoint operator B, acting in the Hilbert space H and for some A € R define a number n[A, B]
by
n[A, B] := maxjdim Hb (A): Hb (A) C H; (B^) > A, v € Hb (A), |M| = 1}.
If some point of the essential spectrum of the operator B is greater than A then n[A, B] equals infinity, if n[A, B] is finite, it equals to the number of eigenvalues of the operator B, that are greater than A (see. example Lemma Glazman [23]).
The known Birman-Schwinger principle (see. [15]) leads to the following lemma.
Lemma 4.4. Let m > mo(y). Then, for any z > rmax,7(m, 0) the equality holds
n[z,HMi7(0)] = n[1, Ami7(z)].
5. On the spectrum of the operator AM 7 (z).
It is well-known that the three-particle branch [Emin,7(0), Emax,7(0)] of the essential spectrum of the operator HM 7(0) is independent of the parameter m > 0, and the two-particle branch [rmin,7(m, 0), rmax,7(m, 0)] of the essential spectrum shifts to when m ^ Therefore, in what follows, we assume that m is large enough and z >
(m, 0).
1 x2
Using the equality --= 1 — x + --, (x = —1), and given notation (2.1), we have
1 + x 1 + x
1 = 1 f1 — (e(p)+e(s)+yc(p + s)) + z (y; p, s) y (51)
z — eoi7 (p, s) a(Y,z) y a(Y, z) a(Y, z)
where
( ) 6 3 d Z( ) (e(p)+e(s)+7e(p + s))2
a(7, z) = z - 6 - 3y and Z(y; p, s) = --^—-r-—.
z - Eoi7(p, s)
Taking into account the equality (5.1), we represent the operator Amiy(z) as a sum
Ami7(z) = A« (z) + A^ (z), (5.2)
where
M [ (C(p) + £(s) + 7^(p + s) - a(7, z))0(s)ds
(41Y (z)0)(p)
a2(Y,z)7 v/AMj7 (p,zyA^,7 (s,z)
T3
(A(2) (z)0)(p) -__L f z(y;p,s)0(s)ds
(z)0)(p) = a2(7,z) y ^A;;^^A;;^•
In what follows, it is shown that the norm of the operator A^Y (z) tends to zero as m ^ (see Lemma 5.5). Therefore, let us establish the existence of eigenvalues of the operator A^Y(z) which are greater 1 for large enough m > 0. Let us find the invariant subspaces with respect to A^Y(z). The Hilbert space L2(T3) can be represented as a direct sum
L2(T3) = L°(T3) © L2(T3),
where
L2(T3) = {0 e L2(T3) : 0(-p) = -0(p)}, L2(T3) = {0 e L2(T3) : 0(-p) = V(p)}. Lemma 5.1. The subspaces L2(T3) and L2(T3) are invariant under the operators Amiy(z), A^Y(z) and A^Y(z). Proof. From the definitions Amiy(p, z) and e(p) it follows that
Am,y(-p, z) = Am,y(p, z - e(-p)) = Am,y(-p, z - e(p)) = Am,y(p, z). (5.3)
If 0 e L2(T3), then making the change of variable s = -q, given equalities E0,Y (-p, -q) = E0,Y(p, q) and (5.3), we get
^(-p) = (amy (z№)(-p) =--, M /-0(s)ds = =
v Am,y(-p, z^ (z - Eo,y(-p, s)VAM,7(s, z)
T3
" r ^(q)dq ,
= np).
"\/A;,7(p,z) 7 (z - Eojy(-p, -q)VA;,7(q, z)
T3
Therefore, the subspace L2(T3) is invariant under A;,Y(z). Since the operator A;,Y(z) is self-adjoint, orthogonal complement L2(T3) of subspaces L2(T3) is also invariant under the operator A;,Y(z). The other statements are proved similarly. □
Denote by P2 and P2 the space projection operators in L2(T3) into subspaces L2(T3) and L2(T3), respectively. For 0 G L2(T3), the following equalities are true
(P20)(p) = 2[0(p) - 0(-p)], (Pe0)(p) = 1[0(p) + 0(-p)].
From the invariance of subspaces L2(T3) and ¿2(T3) with respect to the operator A^Y(z), it follows that the projectors P2 and P2 are permutable with operator a;1y (z), i.e.,
P oA;iy (z) = A;iy (Z)P o, P2A;iy (z) = A;,y (Z)P 2.
Denote by A;1y2)(z) the operator restriction A;; (z) to subspace L2(T3). Then by definition of the operator A;1y2)(z) =
P2A;1Y(z)P2 = A;iY(z)P2 it follows that for any 0 G ¿2(T3), it occurs that
(A;iy2)(Z)0)(p) =--M7 y fsinpsins0(s)ds•
«2(7,zVA;y(p>z) •/ VA;,y(s> z)
T3
By analogous reasoning, one can verify that the restriction A;1y2)(z) = A;,Y (z) - A;1y2)(z) of the operator A;,Y (z) to the subspace L2(T3 ) has the form:
(A;iY2)(z)0)(p) = —- f I ( y (cos Pi + cos
;,Y a2(7,zVA;,Y(p,z) J Vi=t
T3
, ^(s)ds
+Y cospj cos Sj — a(Y, z)) '
v/AMi7 (s,z)
Lemma 5.2. For any z > rmax,7(m, 0), the operator is negative, that is
(a^WV, ^ < 0 for all ^ € L2(T3). Proof. Indeed, for any ^ € L2(T3), we have
Y^ sinpj sin Sj ^(s)-0(p)dsdp
(t3)
^^ f sin sj^(s)ds f sinpi^(p)dp
a2(Y,z) j=l/3 VAM,7(s,z)7 VAm.yM
T3 T3
MY
t2 (
3
a2(7,z) j=1
E
2
sin pi^(p)dp
v/AMi7 (p,z)
T3
< 0.
Let $ be the one-dimensional subspace spanned by a function
c(z)
^Q(p)
v/AMj7 (p,z) ' where c(z) is the normalizing factor, that is
1 f ds
2(z) J Am,7 (s,z)'
T3
Denote by Q the subspace projection operator L2(T3) ©
Let BMi7(z) be the operator restriction A^^z) to the subspace L2(T3) © th<
(Q^)(p) = ^(p) - (p), ^ G L2(T3).
Now, using some calculations, we have
-0(s)ds (s) rv y
□
(BM,7(z)^)(p) = (qa^q^xp) = 2, E /^(pvi(s)^= . . ,
a2(Y,zVAM,7(p,z) j=1 < VAM,7(s,z)
T3
where
^j(p) = c2(z)bj(z) — cos Pi (5.4)
and
f cos sjds
bi(z):= m^ny A;;^, i = 1,2,3.
T3
Let
^ W ^ i ^ (s)^j(s)ds ■ ■ 1 O Q «n
bij(z) := bij (M,Y,z)^^-(s z) , i,j = 1, 2, 3 (5.5)
t3 M'7
where by functional invariance of Ami7 (p, z) regarding the permutation of variables pj and pj it follows that
bn(z) = b22 (z) = &33(z), b12(z) = &21(z) = b23(z) = &32(z) = &13(z) = &31(z).
Lemma 5.3. Let
d(z) := d(u, Y, z) = »MY , .
a2 (Y, z)
Then for sufficiently large and positive m the number
A1 (z) = d(z)(6n(z) + 2b12(z)) (5.6)
is simple and
A2,3 (z) = d(z)(b11(z) — Mz)) (5.7)
is an eigenvalue with the multiplicity two of the operator BMi7 (z).
Proof. Suppose the equation
(B;,Y (z)V0 (p) = A^(p) has a nonzero solution ^ € L2 (T3) . From here,
V(p) = /Ad(z| . E Cj^j(p), (5.8)
V(p,z)
where
Cj = / ^'(s)^(s)ds, i = 1,2,3. (5.9)
J VA;,7 (s, z)
T3
Substituting the right hand side of the equality (5.8) in (5.9), we obtain a system of equations for C1, C2 and C3:
(d(z)6n(z) — A) C1 + d(z)b12(z)C2 + d(z)b12(z)C3 = 0
d(z)b12(z)C1 + (d(z)6n(z) — A) C2 + d(z)b12(z)C3 = 0
^ d(z)b12(z)C1 + d(z)b12(z)C2 + (d(z)6n(z) — A) C3 = 0 Determinant D(A) of this system is a third degree polynomial with respect to A.
Solving the equation D(A) = 0, it makes sure that A1(z) and A2(z), defined by formulas (5.6) and (5.7), are simple and double zeros, respectively. After elementary calculations, we verify that
, , , (^1(p) + ^2(p) + C ^1(p) =-/a f ^-,
V/A;j7 (p,z)
is an eigenfunction corresponding to the eigenvalue A1 (z). General view of an element from the subspace of its own functions, corresponding to the double eigenvalue A2;3(z), looks like
^2(p) = (^1(p) — ^3(p)) C1 + (^2 (p) — ^3(p)) C2. VAMi7(p,z) VA;,7(p, z)
□
Lemma 5.4. Assume that m > 6(1 + y) and z > rmax,7(m, 0). Then the inequalities
(z;,Y(p) — 6 — 6Y)(z — 12 — 6Y^ 1 z • z;,Y(p)
< 1—7—T < -/ / u (5.10)
M(z — zMi7(p) — e(p)) AMi7(p, z) m(z — zMi7(p) — e(p))
hold, where zMi7 (p) is an eigenvalue of the two-particle operator hMi7 (p). Moreover, we obtain the following asymptotics
1 M ^1 + o( 1 ) ) (5.11)
A; ,Y(p, Tmax,Y ^ 0)) e(pR Vm as m ^
Proof. For all p € T3, by Lemma (3.1), we establish
/ds f ds _ 1
zM,7(p) — £(s) — Y£(p — s) w zMi7(p) — e(s) — y£(p + s) _ .
T3 T3
Observe that
A;,7 (p,z) = 1 — M j
ds
z — Eoi7 (p, s)
T3
/ds f ds
zMl7(p) — e(s) — Y£(p + s) W z — e(p) — e(s) — Ye(p + s)
T3 T3
ds
M(z — z;,y(p) — £(p)W [z—(p
./ [z;,7(p
[zMl7(p) — e(s) — Y£(p + s)] [z — Eo,7(p, s)]'
T3
(5.12)
Then, using the assertion 0 < e(s) < 6, we get
1 < „ _?_, < , * „ , (5.13)
z;,7 (p) zMi7 (p) — e(s) — y£(p + s) zMi7 (p) — 6 — 6y'
1
- <
1
<
z z - Eo,y(p, s) z - 12 - 67
(5.10) follows directly from relations (5.13), (5.14) and (5.12). Now, (5.11) can be obtained as in
1
1
z;,Y(p) - £(s) - y£(p + s) z;,Y(p) - 3 - 3y V z;,Y(p) - e(s) - 7e(p + s)
1-
£(s)+ y£(p + s)
527 (5.14)
_ g(s) + yc(p+s)+e(p)
z - e(s) - y£(p + s) - e(p) z - 6 - 3y V z - e(s) - Ye(p + s) - e(p)
1-
If we take into account the inequalities
9(1 + y )2
M + 3(1 + y) < z;,Y(n) < z;,Y(k) < z;,Y(0) < M + 3(1 + Y) + --— < z
M
for sufficiently large m > 0, from (5.12), we have
A;,Y(p, Tmax,Y (M, 0))
M (z - z;,y(p) - e(p))
[z;,Y(p) - 3 - 3Y] [z - 6 - 3Y]
1 + O -M
, m ^ to.
Hence, again using the relations (5.15), one can make sure it's true (5.11).
(5.15)
□
Lemma 5.5. Assume that 7 > 0. Then there exists "Y > 0 such that for any " > "Y satisfying
a;2y (z)
C
< —,
M
which is carried out uniformly z > Tmax,Y (", 0), C is positive real number depending only on 7. Proof. Let 0 e L2(T3) and ||0|| = 1. Using the inequalities £(p) < 3 and Eo,y(p, s) > 0, we get
<
M
(e(p) + e(s) + yC(p+s))2 i0(s)ii0(p)|dsdp
(z - 6 - 3Y)2 J J (z - Eo,y (p, s)VA;,y (s,zVA;,y (p, z)
T3 T3
<
<
m(6 + 3y)2
I0(s)||0(p)|dsdp
(z - 6 - 3Y)3 J J v^y (s,z^A;,y (p,z)
T3 T3
m(6 + 3y )2
|0(s)|ds
(z - 6 - 3y)3 VA;,;(s,z)
Since z > z;,Y(p) +6 > Tmax,Y(m, 0), considering (5.10), if m > 6(1 + y), we get
,y (s)z
M(z - z;,Y (s) - e(s))
|0(s)|ds I <
Since
z - 6 - 3y
from (5.16) and (5.17) at m > 6(1 + y), we have
< z- / |0(s)|2ds / .
M 7 J e(s)
T3 T3
< 2, z > m + 3(1 + y),
< 4(6 + 3y)2w < g;
" m(1 -^ m
where CY = 8 (6 + 3y)2 W.
(5.16)
(5.17)
□
2
2
z
6. Proofs of the main results
The following Lemma plays an important role in the proof of the main results.
Lemma 6.1. Assume that 7 > 0. Then we obtain the following asymptotics:
Al(Tmax,7 (M, 0)) = 7 + 0( , (6.1)
71 VM /
A2,3(Tmax,7(M, 0))= ^ + o(-1) , (6.2)
72 Vm/
where 71 and y2 are defined by formula (2.2).
Proof. Let us prove equality (6.2). Taking into account equalities (5.4), (5.5), (5.11) and (5.15), we have
( M /"(cos2 si - cos si cos S2 )ds \ /1 A2,3(Tmax,7(M, 0)) = 7 7-77^-7--Ä-/-77vT\- +0 "
V(zM,7(0) - 6 - 37)J (s,zM,7 (0)) / VM
T3
= , (cos2 si - cos si cos s2)d^ 1 + / 1^ = zM,7(0) - £(s) - zM,7(s) V \m)J \Mj
f (cos2 si - cos si cos S2)ds + / A = _7 + /1 £(s) Vm/ 72 Vm
T3
□
Proof of Theorem 2.1 1. i) Assume that y € (0,y1). Then applying Lemma 5.5 and using (5.2), we obtain that there exists my > 0 such that for any m > M7, the operators A;,7 (z) and A;O7 (z) have the same number of eigenvalues greater than 1. From Lemma 5.1 and Lemma 5.2, we obtain
n [1,a;1Y (z)] = n [1,a;1YO)(z^ + n [m^z)] = n [1,a;oY6)(z)
From the statement of Lemma 5.3, one can conclude that the operator A;^ (rmax,7(m, 0)) have three eigenvalues A1(z), A2 3(z) taking into account the multiplicity. Since 0 < y < Y1, the inequalities A1(rmax,7(m, 0)) < 1, A2,3(rmax,7(m, 0)) < 1 are valid for sufficiently large m > 0. By the Birman-Schwinger principle (see Lemma 4.3) the operator H;,7 (0) has no eigenvalues z > rmax,7 (m, 0). The statements ii) and iii) can be proven similarly.
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Submitted 3 July 2023; revised 13 September 2023; accepted 14 September 2023
Information about the authors:
Ahmad M. Khalkhuzhaev - V. I. Romanovsky Institute of Mathematics of the Academy of Sciences of the Republic of Uzbekistan, Tashkent, Mirzo Ulugbek 81, 100170; ORCID 0009-0003-0569-6780; ahmadjx@mail.ru
Islom A. Khujamiyorov - Samarkand State University, University Boulevard 15, Samarkand 140104, Uzbekistan; ORCID 0000-0001-5439-8729; xujamiyorov1990@mail.ru
Conflict of interest: the authors declare no conflict of interest.
