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Spectral properties of a two-particle hamiltonian on a d-dimensional lattice
M. I. Muminov1, A. M. Khurramov2 1Universiti Teknologi Malaysia (UTM), 81310 Skudai, Johor Bahru, Malaysia 2Department of Mechaniical and Mathematics, Samarkand State University, Uzbekistan mmuminov@mail.ru, xurramov@mail.ru
DOI 10.17586/2220-8054-2016-7-5-880-887
A system of two arbitrary quantum particles moving on d-dimensional lattice interacting via some attractive potential is considered. The number of eigenvalues of the family h(k) is studied depending on the interaction energy of particles and the total quasi-momentum k G Td (Td -d-dimensional torus). Depending on the interaction energy, the conditions for h(0) that has simple or multifold virtual level at 0 are found. Keywords: two-particle hamiltonian, virtual level, multiplicity of virtual level.
Received: 17 May 2016 Revised: 6 June 2016
1. Introduction
Lattice two-particle Hamiltonians have been investigated in [1-3]. In [1], the problem of the two-particle bound states for the transfer-matrix in a wide class of Gibbs fields on the lattices in the high temperature domains of (t > 1), as well in [2] the appearance of bound state levels standing in a definite distance from the essential spectrum has been shown for some quasi-momenta values. The spectral properties of the two-particle operator depending on total quasi-momentum have been studied in [3].
In [4], it was proven that if the operator h(0) has a virtual level at the lower edge of essential spectrum, then the discrete spectrum of h(k) lying below the essential spectrum is always nonempty for any k g Td \ {0}. In [5], assuming that dispersion relations e1(^) and e2(-) are linearly dependent, it was proven that the positivity of h(0) implies the positivity of h(k) for all k.
In recent work [6], conditions were obtained for the discrete two-particle Schrodinger operator with zero-range attractive potential to have an embedded eigenvalue in the essential spectrum depending on the dimension of the lattice. In [7], the discrete spectra of one-dimensional discrete Laplacian with short range attractive perturbation were studied.
In [8], a system of two arbitrary particles in a three-dimensional lattice with some dispersion relation was considered. Particles interact via an attractive potential only on the neighboring knots of lattice. The existence and absence of eigenvalues of the family h(k) depending on the energy of interaction and quasi-momentum k g T3 (T3 - three dimensional torus) have been investigated. Moreover, depending on the interaction energy, the conditions were found for h(0) to have a simple, two-fold, or three-fold virtual level at 0. In [9], the two-particle Schrodinger operator h(k), k g T3, associated with a system of two particles on the three-dimensional lattice, was considered. Here, some 6N-dimensional integral operator is taken as the potential and the dispersion relation is chosen depending on N. In this work, the existence or absence of eigenvalues has also been studied for the family h(k) depending on the interaction energy and total quasi-momentum k. Moreover, dependending on the interaction energy, conditions were found for the operator h(0) that has 3N-fold eigenvalue and a 3N-fold virtual level.
The current work is a generalization of [8]. In this work, we consider the system of two arbitrary quantum particles moving on the d-dimensional lattice and interacting via an attractive potential. For all values of k g Td (Td - d-dimensional torus) the dependence of the number of eigenvalues of the family h(k) on the interaction energy is studied. The conditions for that h(0) has simple or multifold virtual level (eigenvalue) at 0 are found for d =3, 4 (d > 5).
2. Statement of the Main Result
Let L2(Td) be the Hilbert space of square-integrable functions defined on d-dimensional lattice Td.
Consider the two-particle Schrodinger operator h(k), k g Td, associated with the direct integral expansion of Hamiltonian of the system of two arbitrary particles, interacting via short-range pair potential [8], acting in L2(Td) as
h(k) = h0(k) - v,
here h0 (k) - multiplication operator by a function:
Ek (p) = £l(p) + £2(k - p)
and v is an integral operator with kernel
d
v(P - «) = P0 + Pa cos(pa - Sa), pa > 0.
a=1
Assumption 1. Additionally, we assume that £;, l =1, 2 are real-valued, continuous, even and periodic functions with period n in every variable.
Please note that the Weyl theorem on the essential spectrum [10] implies that the essential spectrum aess(h(k)) of the operator h(k) coincides with the spectrum of the unperturbed operator h0(k):
&ess (h(k)) = a(ho(k)) = [m(k),M (k)],
where m(k) = min Ek(p), M(k) = maxEk(p).
p£Td p£Td
Since v > 0, one has:
sup(h(k)f, f) < sup(ho(k)f, f) = M(k)(f, f), f e L2(Td), and, thus, h(k) does not have eigenvalues lying above the essential spectrum:
a(h(k)) n (M(k), +to) = 0.
We set: _
±(, ) = ci(k; z) + si(k; z) ± y7(cj(k; z) - sj(k; z))2 + 4g2(k; z) Pi (; z)= 2[ci(k; z)Si(k; z) - £2(k; z)] ,
where
f cos2 sj ds f sin2 s j ds
ci(k;z) = —, si(k;z) = vr~\—,
J Ek(s) - z J Ek (s) - z
T d T d
f sin s j cos sj ds
fi(k; z) = —---, z < m(k).
J Ek(s) - z Td
Recall that cj(k; z)sj(k; z) - (k; z) > 0. There exist (finite or infinite) limits:
Km b(k;z), lim ci(k;z), Um si(k;z), lim f»2(k;z),
z^m(k)- 0 z^m(k)- 0 z^m(k)- 0 z^m(k)- 0
where
b(k; z)= ' ds
J Ek(s) - z'
Td
Lemma 1. For any k e T there exists finite limits:
P0(k) = lim ——, (2.1)
z^m(k)-0 b(k; z)
P±(k) = lim p±(k; z), i =1,...,d. (2.2)
z^m(k)-0
Moreover,
P-(k) < P+(k) for all k e Td, i = 1,..., d.
Let us define the functions:
'0 if P0 e (0; p0(k)], 1 if P0 e (p0(k); to),
/ M I 0 11 P0 e (0; P (k)b /0 ^
a(p;k) = \1 _ (2.3)
(0 if Pi e (0; p-(k)],
A(p; k) = ^ 1 if Pi e (p-(k); p+(k)], (2.4)
[2 if Pi e (p+(k); to)
for all i = 1 , . . . , d.
Theorem 1. Let m = (mo, • • • , Md) g r++1. Then, counting multiplicity, h(k) has exactly:
"(m; + k)
i=1
eigenvalues below the essential spectrum.
Assumption 2. Assume that m(0) = min E0(p) = 0 and
peTd
M = {p G Td : m(0) =0} = {p1, • • • ,pn}, n < to.
Moreover, assume that around points of M E0(p) is of order p > 0:
c|p - p;|p < Eo(p) < C1 |p - pi|p as p ^ p;, l =1,...,n.
Let C(Td) be a Banach space of continuous periodic functions on Td and G(k; z) denote the (BirmanSchwinger) integral operator in L2(Td) with the kernel:
G(k; z;p,q) = v(p - q)(ffc(q))-1, p, q G Td.
Definition 1. We say that the operator h(0) has a virtual level at 0 (lower edge of essential spectrum) if 1 is an eigenvalue of G(0;0) with some associated eigenfunction ^ G L2(Td) satisfying:
V>(0
¿o(0
G L1(Td) \ L2(Td).
The number of such linearly independent vectors ^ is called the multiplicity of virtual level of h(0). We set:
m?
1
1
ca(0; 0)' s?(0; 0) J ' We define the following sets depending on c?(0; 0) and s?(0; 0):
a = 1,..., d.
L?2 =
L?3 = Ma1 =
Ma2 = Ma3 =
0: 1
?: c?(0;0)
0: 1
?: c?(0;0)
0: 1
?: c?(0;0)
0: 1
?: s?(0; 0)
0: 1
?: s?(0; 0)
0: 1
?: s?(0; 0)
>m?
^a, p?
2 or p?
— for all i = 1, • • • , n >,
2 ' ' r '
M?, p? = 2 or p? = - 2 at least one i = 1,...,n
0
> M?
= M?, p? = 0 or p? = n for all i = 1,..., n
= M?, p? = 0 or p? = n at least one i = 1,..
where p? - a-th coordinate of minimum point p4 of E0( • ). Let us define the following functions:
^(Mo) = Y(a) = Y(a) =
n(a) = n(a) =
0 if Mo G (0;M0(0)),
1 if mo = M0(0),
0 if m? G (0; m?) orm? G L?1 U L?2,
1 if m? G L?3,
0 if m? G (0; m?) or m? G L?1 U L?3,
1 if m? G L?2,
0 if m? G (0; m?) or m? G M?1 U M?2,
1 if m? G M?3,
0 if m? G (0; m?) or m? G M?1 U M?3,
1 if m? G M?2.
L?1
n
M
n
Theorem 2. (i) Let p = 2, p0 e (0; p0(0)], pa e (0, p0a], a = 1,..., d. Then 1) if d = 3, 4, then 0 is
d
^(P0) + ¿[y (a) + n(a)]
a=1
d
-fold virtual level of h(0). In addition, if [J La2 n Ma2 = 0, then 0 is simultaneously
a=1
d
J3[7(a) + n(a)]
-fold eigenvalue of h(0). 2) if d > 5, then 0 is
^(P0) + ¿[7 (a) + n(a)]
-fold eigenvalue of h(0).
(ii) Let p e (-, d), d > 3, p0 e (0; p0(0)], pa e (0, p°a\, a = 1,..., d. Then 0 is at least
d
^(P0) + ¿[y (a) + n(a)]
a=1
-fold virtual level of h(0).
Remark 1. 1) By definition of sets La2 and Ma2 for each a = 1,..., d one has La3 U Ma3 = 0. Moreover, in
ia pa,
this case, the multiplicity of the virtual level of h(0) is always not less than d if pa = p°a, a = 1,..., d.
2) For p = 2 the function
d
Eo(0 = Eo(p) = £ 1 (p) + £2(p), £1(p) = £2(p) = cos2 p1 + ^(1 +cos2pj)
1
satisfies the assumptions of Theorem 2 with U La2 n Ma2 = 0. In addition, L12 = 0.
a=1
5) For p e ( —, dj the function:
/ d \p/2
Eo(p) = £1(p) + £2(p), £1(p) = £2(p) = ($3(1 - cos 2pj) \
satisfies the assumptions of Theorem 2.
3. Eigenvalues of h(k)
Proof of Lemma 1. Note that proof of (2.1) is obvious.
By definition p-(k; z) < p+(k; z) for any z < m(k) and k e Td. Notice that:
2 f cos2 sads f sin2 tadt f sin sa cos sads f sin ta cos tadt
ca(k; z)sa(k;z) - (k;z) = [ c°sf s"ds [
J Ek (s) - z j
Ek (s) - zj Ek(t) - z J Ek(s) - z J Ek (t) - z
Td Td Td Td
12 2 12 2
2 cos sa sin tadsdt f I" sin sa cos sa sin ta cos tadsdt f f 2 cos ta sin sadsdt
(Ek (s) - z)(Ek (t) - zHi (Ek (s) - z)(Ek (t) - z) ././(Ek (s) - z)(Ek (t) - z)
Td Td Td Td Td Td
= 1 r r sin2(sa - ta)dsdt (31)
= 2 J J (Ek(s) - z)(Ek(t) - z). ( . )
Td Td
Hence, ca(k; z)sa (k; z) - £2(k; z) > 0 for all z < m(k) and k e Td.
The function z) we estimate as follows:
+ ) _ ca(k; z) + sa(k; z) + v/(ca(fc; z) - sa(k; z))2 + 4gg(k; z) Ma ( ; z)_ 2[ca(k; z)sa(k; z) - £(k; z)]
ca(k; z) + sa(k; z) + ^(ca(k; z) + sa(k; z))2 - 4[ca(k; z)sa(k; z) - ^^(k; z)]
2[ca(k; z)sa(k; z) - ^(k; z)]
< ca(k;z) + ga(k;z) (3 2)
ca(k; z)sa(k; z) - ^^(k; z)' '
sin2 (s — t )
Since — ?-— > 0 for any z < m(k) and for a.e. k, s,t G Td, there exists S > 0 such that:
Ek (t) - z
From here and from (3.1) we get:
/sin (sa - ta)dsdt -- ^
Ek (t) - z
Td
ca(k;z)sa(k;z) -(k;z) >2/ dS
2 7 Ek(s) - z'
Td
Since
ca(k; z) + sa(k; z) _ J ds from (3.2) we get uniform upper estimate:
Ek(s) - z
Td
M+(k;z) < 2-j.
From here we get (2.2). Lemma is proved.
Lemma 2. z < m(k) is an eigenvalue of h(k) if and onlj if A(k; z) _ 0, where
d
A(k; z) _ (1 - Müb(k; z)) ^ ([1 - Maca(k; z)][l - Masa(k; z)] - mO^(k; z)) . (3.3)
a=1
Proof. Let z < m(k) be an eigenvalue of h(k) with associated eigenfunction f _ 0. Then h(k)f _ zf and
so:
f _ ro(z)vf, (3.4)
where r0(z) is a resolvent of h0(k). Introduce the following notations:
¥>o _ J f (s)ds, (3.5)
_J cos saf (s)ds, (3.6)
_ j sin saf (s)ds, a _ 1, 2, 3, ...d.
Td
(3.7)
Then, (3.4) is rewritten as:
1 d
f (P) _ e- /ü \ 0--+ ^T^- XI Ma [COS + sin (3.8)
Ek(p) - z Ek(p) - z a=i
From the n-periodicity of ( • ) in each argument, it follows that:
/cos s?ds i cos s? cos s^ ds i cos s? sin s^ ds i sin s? sin s^ ds
Efc(s) - z J Efc(s) - z J Efc(s) - z J Efc(s) - z
Td Td Td Td
0, a _ ß. (3.9)
Putting (3.8) in the relations (3.5)-(3.7) and using (3.9), we get that y>0, ..., y>d, ..., satisfy the system
of (2d + 1)-linear equations:
^0 = p0&(k; z)^0,
^a = paca(k; z)^a + pa£a(k; z)^a, a =1,...,d (3.10)
^a = Pa£a(k; z)^a + Pasa(k; z)^a, a = 1,..., d.
This system of equations has a nonzero solution (<^0,..., <^d, ... ^d) if and only if its determinant is zero, i.e. det D(k; z) = 0. It is easy to see that det D(k; z) = A(k; z).
Conversely, let A(k; z) = 0, z < m(k). Then, at least one of the equalities 1 - p0b(k; z) = 0, [1 -paca(k; z)][1 - pasa(k; z)] - pa£2(k; z) = 0, a e {1,..., d} holds. Thus, the vector c = (c0, • • • , c2d) where c0 = 1, ca = cd+a = is a solution of (3.10). Consequently, one of the functions:
1 1 , •
"Ma [<£>a COSpa + SinPa]
Ek(p) - z' Ek(p) - z
is an eigenfunction of h(k) associated with eigenvelue z < m(k).
Observe that A(k; •) is the Fredholm determinant of the operator I - r0(z)v, i.e. A(k; z) = det(1 - r0(z)v). Moreover, it is well-known [11] that geometric multiplicity of eigenvalue 1 of r0(z)v coincides with the multiplicity of zero z of A(k; •). Since the multiplicities of eigenvalues 1 and z of operators respectively r0(z)v and h(k) are the same, we get that multiplicity of zeros of A(k; •) is equal to the multiplicity of eigenvalues of h(k). The lemma is thus proved.
Proof of Theorem 1. Notice that the function:
Aa(k; z) = [1 - Paca(k; z)][1 - Pasa(k; z)] - pl£l(k; z), is a Fredholm determinant associated with the operator I - r0(z)va, where va - is an integral operator with kernel
Va (p - s) = Pa cos(pa - sa).
Since va is a two-dimensional operator, number of zeros ,0a(p; k) with multiplicities of the function Aa(k; •), lying below m(k), is not more than 2. Function Aa(k; •) can be represented as:
Aa(k; z) = [ca(k; z)sa(k; z) - ^(k; z)] ^a - M— (k; z)j (^Ma - M+(k; z)j . (3.11)
Since: one has:
llm nM±(k; z) = M±(k) < TO
J ^ 0 if Ma € (0,M±(k)], < 0 if Ma € (m±(k), to).
Pa - p± (k; m(k)) =
Consequently, from (3.11) and (3.1) it can be deduced that:
( 0 if Pa e (0,P-(k)], ^a(P; k) = < 1 if Pa e (P-(k),P+ (k)], [ 2 if Pa e (p+(k), to).
Observe that the function 1 - p0b(k; •) is monotonously decreasing in (TO,m(k)). Thus for the number of zeros a(P; k) of the function Aa(k; •) below m(k) it holds:
'0 if P0 e (0; P0(k)], 1 if P0 e (P0(k); to).
a(M;k) = ï 1 ^ .. ^ t .0/
If P0(k) = 0, then lim b(k; z) = +to. Obviously, in this case a(P; k) = 1 for any P0 > 0.
z^m(k)-0
The aforementioned facts imply that if: p = (p0, p1, ..., Pd) e R++1, then the function A(k; •) has exactly:
d
a(p; k) + ^ ^j(p; k) j=1
zeros (counting multiplicities) below m(k).
Then, from Lemma 1, we get that for p = (p0, p1,..., pd) e R++1 the operator h(k) exactly:
d
a(p; k) + ^ ^j(p; k) j=1
zeros (counting multiplicities) below m(k). This finishes the proof.
Proof of Theorem 2. We shall study the equation:
G(0;0)^> _
Notice that the function A(k; z), defined as (3.3) is the Fredholm determinant of I - G(k; z). From Hypothesis 2, the function A(k; z) is defined for k _ 0, m(0) _ 0. Since Eo(• ) is even, the function
¡' sin s, cos s, ds
Ci(0; z) _ n - . - _0, z < 0. J Eo(s) - z
Consequently, the function A(0; z) can be represented as:
d
A(0; z) _ (1 - Mo6(0; z)) fl ([1 - M«ca(0; z)][1 - M«sa(0; z)]) .
a=1
The following lemma can be proved analogously to Lemma 2. Lemma 3. The number A _ 1 is an eigenvlue of G(0;0) if and only if A(m) _ A(0;0) = 0. In this case if 1 - M0b(0;0) _ 0 - Maca(0;0) _ 0 or 1 - ^asa(0;0) _ 0^, then the function _ 1 ^a(p) _ cospa or
^a(p) _ sinpaj is an eigenfucntion of the operator G(0;0), associated with 1.
Obviously, A(m) > 0 if m0 € (0; M(0)), Ma € (0; M^), a _ 1,..., d. By Lemma 3 A _ 1 is not eigenvalue of G(0; 0). Hence 0 is not an eigenvalue of h(0) for ^o € (0; m°(0)), Ma € (0; ^), a _ 1,..., d. Further, consider the equation G(0; 0)^> _ ^ for m0 _ Mü(0), Ma _ ML a _ 1,..., d. (i) a) Let p _ 2, mo _ M(0).
According to Lemma 3, A _ 1 is an eigenvalue of G(0;0), with associated eigenfunction yo(p) _ 1. It is easy to check that if d _ 3,4, then:
Fo(• ) € Li(Td) \ L2(Td),
and if d > 5, then:
Vo( •
where
Fo( • ) € L2(Td),
Fo(p) _ 1
Eo (p)'
It means that z = 0 is virtual level of h(0) for d = 3, 4, and eigenvalue for d > 5.
b) Let m? = M?, a = 1,..., d. Then m? belongs one and only one of the sets L?1, L?2, L?3 M?1, M?2,
M?3.
If m? G L?^m? G , then 1 - M?c?(0;0) > 0^1 - m?s?(0;0) > 0^. If m? G L?^m? G M?2j, then cos= 0 ^sin= 0^ for all i = 1,..., d. In this case
F?( • ) G L2(Td), ($?( • ) G ¿2(Td^, d > 3,
where
COs Pa ^ f , sin Pa ,
Fa(p)_ , $a(p)_ E0(P) , a _1,...,d,
and, so, z = 0 is not virtual level of h(0) for d >3, but is an eigenvalue of this operator.
If m? G L?3 ^m? G M?3), then cos= 0 ^sin= 0^ at least one of i = {1,..., d}. Consequently,
F?( • ) G L1(Td) \ L2(Td), ($?(• ) G L1(Td) \ L2(Td^ for d = 3,4,
F?( • ) G L2(Td), ($?(• ) G ¿2(Td)) for d> 4,
i.e. z = 0 is a virtual level (eigenvalue) of the operator h( 0) for d = 3, 4 (d > 4). From a) and b) we deduce the following:
if mo = M°(0), then z = 0 is virtual level (eigenvalue) of h(0) for d =3,4 (d > 4); if m? G L?1 U L?2, then z = 0 is not virtual level of h(0) for d > 3;
if m? G L?3, then z = 0 is virtual level (eigenvalue) of the operator h(0) for d =3,4 (d > 4);
if pa G La2, then z = 0 is eigenvalue of the operator h(0) for d > 3;
if pa G Mai U Ma2, then z = 0 is not virtual level of h(0) for d > 3;
if pa G Ma3, then z = 0 is a virtual level (eigenvalue) of h(0) for d = 3,4 (d > 4);
if pa G Ma2, then z = 0 is eigenvalue of h(0) for d > 3.
Part (i) of Theorem 2 is proved.
Part (ii) of Theorem 2 is proved analogously.
Acknowledgements
This work was supported in part by the Malaysian Ministry of Education through the Research Management Centre (RMC), Universiti Teknologi Malaysia (PAS, Ref. No. PY/2014/04068, Vote: QJ130000.2726.01K82).
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