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Monotonicity of the eigenvalues of the two-particle Schrodinger operatoron a lattice
J. I. Abdullaev12, A. M. Khalkhuzhaev12, L. S. Usmonov2
1 Institute of Mathematics of the Academy of Sciences of the Republic of Uzbekistan, Mirzo Ulugbek 81, Tashkent 100170, Uzbekistan
2 Samarkand State University, University Boulevard 15, Samarkand 140104, Uzbekistan [email protected], [email protected], [email protected]
PACS 02.30.Tb DOI 10.17586/2220-8054-2021-12-6-657-663
We consider the two-particle Schrodinger operator H(k), (k € T3 = (—n, n]3 is the total quasimomentum of a system of two particles) corresponding to the Hamiltonian of the two-particle system on the three-dimensional lattice Z3. It is 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 nondecreasing function in each k(i) € [0, n], i = 1, 2, 3. Under some additional conditions potential v, the monotonicity of each eigenvalue zn (k) = zn (k(1), k(2), k(3)) of the operator H(k) in k(i) € [0, n] with other coordinates k being fixed is proved.
Keywords: two-particle Schrodinger operator, Birman-Schwinger principle, total quasimomentum, monotonicity of the eigenvalues. Received: 22 October 2021 Revised: 20 November 2021
1. Introduction
Coherent optical fields provide a strong tool for manipulating ultracold atoms and a unique setting for quantum simulations of interacting many-body systems because of high-degree of controllable parameters such as optical lattice geometry and dimensionality, particle masses, tunneling, two-body potentials, temperature etc. [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 the short-range case. However, recent experimental and theoretical results show that integrating plasmonic systems with cold atoms, especially 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 realization of Hubbard models and engineer effective long-range interaction in many body dynamics [5-7].
Hamiltonians, corresponding to systems of particles on a lattice, were first considered in the 1990s by D. S. Mattis [8], A. I. Mogilner [9], and after that, research has rapidly developed. The kinematics of quantum quasiparticles on a lattice is rather peculiar, even in the two-particle case. For example, because the discrete analog of the Laplacian or of its generalization is not translation invariant, the Hamiltonian of the system cannot be divided into two parts with one part corresponding to the motion of the center and the other corresponding to the internal degrees of freedom. This is the so-called phenomenon of "excess mass" for lattice systems: the effective mass of the two-particle bound state exceeds the sum of the effective masses of the quasiparticles constituting the system (see, e.g., [8,9]). In contrast to the continuous case, where the center-of-mass motion can be separated, the two-particle problem on a lattice reduces to studying the one-particle problem using the Gelfand transformation. Namely, the Hilbert space ^2((Z3)2) can be decomposed into the direct (continuous) von Neumann integral associated with the representation of the Abelian (discrete) group Z3 formed by commutative operators on the lattice. Then the two-particle Hamiltonian can also be decomposed into the direct (continuous) von Neumann integral. In contrast to the continuous case, the corresponding fiber operators H(k), k g T3, associated with the decomposition of the direct integral depend parametrically on the quasimomentum k, which ranges the first Brillouin zone R3 \ (2nZ)3. Because the spherical symmetry of the problem is lost, the spectra of the family H(k), k g T3, become rather sensitive to variations in the quasimomentum k.
Spectral properties of the two-particle discrete Schrodinger operator H(k) = H0(k) - V, k g T3 are studied in the more works (see.i.e. [10-13]). In work [13] a two-particle discrete Schrodinger operator H(k), k g T3 with zero range potential v(ni - n2) = ^Snin2 was considered and the existence of a unique eigenvalue z(k) of the operator H(k) was established. In [13] it is proved that the eigenvalue z(k) = z k g T3 is symmetric and
even in each variable k(i) g [—n, n], i = 1, 2, 3 and strictly increases in each k(i) g [0, n], i = 1, 2,3. In particular, it was shown that the two-particle operator H(k), k = 0 has a positive eigenvalue below the essential spectrum, provided that H(0) has a virtual level at zero.
The following effect was discovered in [14] for a wide class of two-particle Schrodinger operators H(k), associated with the Hamiltonian of the system of two arbitrary particles. If the discrete Schrodinger operator H(0), 0 = (0,0,0) g T3, has a virtual level or an eigenvalue on the lower threshold of the essential spectrum, then the operator H(k), has an eigenvalue below the threshold of the essential spectrum for all nonzero values of the quasimo-mentum k g T3. Similar results was discussed in [15] for d- dimensional lattice case. In [16] was studied the discrete spectrum of the two-particle Schrodinger operator HMi>(k), k g T2, associated to the Bose-Hubbard Hamiltonian HMia of a system of two identical bosons interacting on site and nearest-neighbor sites in the two dimensional lattice Z2, with interaction magnitudes ^ g R and A g R, respectively, and completely described the spectrum of HMi>(0) and established the optimal lower bound for the number of eigenvalues of HMi>(k) outside its essential spectrum for all values of k g T2. Namely, the A) -plane was partitioned that in each connected component of the partition, the number of bound states of below or above its essential spectrum cannot be less than the corresponding number of bound states of HMjA(0) below or above its essential spectrum. In [17] a two-particle Schrodinger operator H on the d- dimensional diamond lattice was considered and a sufficiency condition of finiteness for discrete spectrum eigenvalues of H was found.
In this note, we consider the two-particle operator H(k) = H0(k) - V, k g T3 with general potential v. For the potential v(x) = (Fv)(x) we assume:
v(x) > 0, Vx g Z3, v g ¿i(Z3). (1.1)
Non-negativity v(x) > 0 will ensure the positivity interaction operator V. We denote by V1/2 its positive square root.When proving monotonicity the eigenvalue zn(k) of the operator H(k) with respect to k(i) g [0, n], we will use the monotonicity property of the operator-valued function:
G(k,z) = V1 (H0(k) - zl)-1V2
by z g (-to, Emin(k)) and k(i) g [0, n], where the number £min(k) is the left edge of the essential spectrum of the operator H(k). For any k g (-n, n)3 the operator G(k, z) converges uniformly to the limit operator G(k, Emin(k)) as z ^ Emin (k). Under the condition (1.1), it is proved that G(k, Emin(k)) belongs to the class E1 (see. proof of the Lemma 3.1.) Since G(k, Emin(k)) is monotonic in each k(i) g [0, n], i = 1,2, 3 it follows that the number N(k) of eigenvalues lying below the essential spectrum of the operator H(k) is nondecreasing function with respect in each k(i) g [0, n], i = 1,2,3 (Theorem 2.1).
We will prove the monotonicity G(k, z) by z g (-to, Emin(k)), that is G(k, z1) < G(k, z2) at z1 < z2. This implies that each eigenvalue An (k, z) of the operator G(k, z) is increasing function with respect to z g ( -to , Emin (k)) (Lemma 3.4). Given v(2s, n(2), n(3)) = 0, Vs g Z or v(2s + 1, n(2), n(3)) = 0, Vs g Z the operator-valued function G(k, z) decreases by k(1) g [0, n] (Lemma 3.5). It follows that each eigenvalue zn(k) of the two-particle operator H(k) increases in G [0, n] (Theorem 2.3).
2. Representation of Hamiltonian associated to a system of two particle on a lattice. Statement of the main result
Energy operator Hi of a system of two quantum particles on a three-dimensional lattice Z3 acts in the Hilbert space ^2((Z3)2) by:
# = H0 - V,
where the free energy operator H0 acts in ^2((Z3)2) as:
Ho =--—-.
0 2m1 2m2 x2
Here, m1, m2 > 0 are denoted the masses of particles, which in the future are considered equal to one, Axi = A (g> I and Ax2 = I ( A, lattice Laplacian A is a difference operator describing the transfer of a particle from a site to neighboring site:
3
(AVO(x) = ^(x + ei) + Vi(x - ei) - 2^(x)], G ^2(Z3),
i=1
where e1 = (1,0,0), e2 = (0,1,0), e3 = (0,0,1) are the unit vectors in Z3. The interaction of two particles is described by the operator V :
(vV0(xi, x2) = V(xi - x2)^(xi, x2), G ^2((Z3)2).
Under the conditions (1.1), the energy operator H is the bounded self-adjoint operator in the space ^2((Z3)2). Transition to momentum representation is performed by using the Fourier transform F : L2 (( T3)2) ^ (( Z3 )2 ). Operator energy H = F-1HF in the momentum representation commutes with the group of unitary operators Us, s g Z3 :
(Uf )(ki, k2) = exp ( - i(s, ki + k2))/(kl, k2), f G L2((T3)2).
From the last fact we obtain [18] that there are decompositions of the space L2((T3)2), operators Us and H into direct integrals:
L2((T3)2) = y eL2(Fk)dk, Us = y eUs(k)dk, H = J e#(k)dk.
T3 T3 T3
Here
Fk = {(ki,k2) G (T3)2 : ki + k2 = k}; Us(k), k g T3 is the multiplication operator by the function exp(-i(s, k)) in the space L2(Fk), and fiber operators H"(k), k g T3 in L2(Fk) are defined according to the formula
(H"(k)f )(q, k - q) = (E(q) + E(k - q))f (q, k - q) - (2n)-3 J v(q - s)f (s, k - s)ds
t3
and it is unitarily equivalent to the operator H (k) = H0(k) - V, the so-called the Schrodinger operator. Unitarity is carried out using the unitary transformation:
kk
uk : L2(Fk) ^ L2(T3), (ukg)(q) = g(^ - q, ^ + q). H0 (k) is the multiplication operator by the function:
kk
Ek(q) = E(q)+ E(k - q),
where:
3
E(q) = E(1 - cos q(j)) j=l
and V is the integral operator in L2(T3), generated by the kernel (2n)-3/2v(q - s). The kernel v of the integral operator V is the Fourier transform of the potential V. The potential V satisfies the conditions (1.1), therefore, the function v is continuous on T3.
We denote by N(k) the number of eigenvalues of the operator H(k), lying to the left Emin(k) = minqet3 Ek(q).
Theorem 2.1. N(k) = N(k(1), k(2), k(3)) is nondecreasing function in each k(i) G [0, n] with other coordinates of k G T3 being fixed.
Assumption 2.2. Let:
v(2s,n(2),n(3) ) = 0, Vs G Z (2.1)
or:
V(2s + 1,n(2),n(3))=0, Vs G Z. (2.2)
Theorem 2.3. Let assumption 2.2 be fulfilled. Then, each eigenvalue zn(k) = zn - of the operator
H(k) increases in G [0, n].
Remark 2.4. Let:
£(n(1), 2s,n(3)) = 0, Vs G Z
or:
V(n(1), 2s + 1,n(3)) = 0, Vs G Z
(respectively
i(n(1),n(2), 2s) = 0, Vs G Z
or:
V(n(1),n(2), 2s + 1) = 0, Vs G Z). Then, each eigenvalue zn(k) of the operator H(k) increases in
G [0, n] (respectively in
G [0, n]).
3. Eigenvalues of the two-particle operator
Let us investigate the essential and discrete spectra of families of two-particle discrete Schrodinger operator H(k), k g T3. Here, we will study the number N(k) of eigenvalues of the operator H(k), lying below the essential spectrum and dependence of the eigenvalues zn (k) on the total quasi-momentum k g T3.
We introduce the following notation: for a self-adjoint operator B acting in a Hilbert space H and not having any essential spectrum on the right from the point denote by HB C H, ^ g R subspaces such that nonzero elements f g HB satisfy the inequality (Bf, f) > ^(f, f) and put:
n(^, B) = sup dimHB(/«).
Hb (m)
If some point of the essential spectrum B is greater then n(^, B) is equal to infinity, and if n(^, B) finite, then it is equal to the number of eigenvalues of the operator B, which is greater than ^ (see., for example, Glazman's lemma [19])
The number n(^, B) is the same as the number of eigenvalues of the operator B lying to the right of For any k g (-n, n)3 and z < Emin(k) we define the integral operators G(k, z) and Q(k, z), acting in the space L2 (T3) with the kernels:
G(k, z; p, q) = / v2 (P - t)(Ek(t) - z)-1v2 (t - q)dt
T3
and:
Q(k,z; p, q) = (2n)-3 v1 (p - q)((Ek(q) - z)1 )-1,
where:
2 (P) = (Fv 1 )(P) = (2n1)3/2 Y, T^expWn p)).
nez3
Note that for any z < Emin(k) the equalities
G(k, z) = V1 r0(k,z)V2, Q(k,z) = V1 r02 (k,z),
hold, where r0(k, z) is the resolvent of the unperturbed operator H0(k), and V1 is the positive square root of the positive operator V. In the limiting case z = Emin(k), we have the following assertion.
Lemma 3.1. For any k g (-n, n)3 the operator Q(k, Emin(k)) belongs the Hilbert-Schmidt class S2.
Proof. By virtue of conditions (1.1) for the potensial v() the function v1 () belongs to L2 (T3). The function Ek(p) -Emin(k) can be represented as:
3 k(i)
Ek(p) -Emin(k) = 2^cos —(1 - cosp(i)), k G (-n,n)3, (3.1)
i_1 2
and it has only nondegenerate minimum at the point p = 0, therefore:
(2-)3 / |v2(p)|2^p / F (q) (k)
(2n)3 J J Ek(q) -Emin(k)
T3 T3 T3 T3
It means that Q(k, Emin(k)) belongs to the Hilbert-Schmidt class S2. □
From the representation G(k, z) = Q(k, z)(Q(k, z))* it follows positivity and the operator G(k, z) belongs to the class E1 with all k g (-n, n)3 and z < Emin(k).
Lemma 3.2. The number z < Emin(k) is an eigenvalue of the operator H (k) if and only if A = 1 is an eigenvalue of the operator G(k, z).
Proof of Theorem 2.1. Using the view (3.1) we get that the function:
|(V1/2^)(p)|2dp_ [ |(V 1/2^)(p)|2dp
J J |Q(k, Emin(k); p, q)|2dpdq = J |v1 (p)|2dp J
<.
(G(k, Emin(k))^,^)= , ^ c , „_3 fcW
J Ek(p) -Emin(k) J 2 V"3_ cos -¡^ (1 - cosp(i))
T3 t3 2 '
is non-decreasing in each k(i) g [0, n] with fixed other coordinates. This means that the function N(k) also has this property. □
Let us denote by A1(k, z) > A2(k, z) > ••• > An(k,z) > ••• eigenvalues of the compact positive operator G(k, z). Each eigenvalue An(k, z) is the even function by k(i) G [-n, n]. Now we will prove the monotonicity of each eigenvalue An(k, z) by z G (-to, Emin(k)) and k(i) G [0, n].
v
The following lemma is a Birman-Schwinger principle for the operator H(k).
Lemma 3.3. For any k g (—n, n)3 and z < Emin(k) the equality:
n(—z, —H (k)) = n(1,G(k,z)), (3.2)
holds.
Proof. A proof of a similar lemma is given in the paper [15]. □
Lemma 3.4. For any k g (—n, n)3 each positive eigenvalue An(k,z) of the operator G(k, z) increases by z g
(—TO, Emin(k)).
Proof. For any ^ g L2(T3) and z1 < z2 g (—to, Emin(k)) the inequality holds
f |(V 1/2VQ(p)|2dp < r |(V1/2^)(p)|2dp J Ek(p) — z1 ~J Ek(p) — z2 '
T3 T3
Hence (G(k, z1)^, < (G(k, z2)^, so An(k, z1) < An(k, z2). Now, we show the strict inequality:
An(k,z1) <A„(k,z2). (3.3)
Let H[anjTO)(G(k, z1)) be subspace generated by the eigenfunctions of the operator G(k, z1), corresponding eigenvalues A1(k, z1) > A2(k, z1) > .„ > An(k, z1) > 0. For any non-zero ^ g H[anjTO)(G(k, z1)) we obtain:
i > i =(G<k.zD^) > A„(k,z,)W,,)'
T3 T3
Hence, strict inequality (3.3) holds. □
Lemma 3.5. Let assumption 2.2 be fulfilled. Then, for any z g (—to, Emin(k)), each positive eigenvalue An (k, z) of the operator G(k, z) decreases in kW G [0, n].
Proof. Let the condition (2.1) be satisfied. Then for the function v1 (p), the following equality
v1 (p(1) + n,p(2),p(3)) = —v2 (p(1),p(2),pC3)) holds. Similarly, if satisfing the condition (2.2), then
v1 (p(1) + n,p(2),p(3)) = v2 (pC1),pC2),pC3)).
Therefore, in both cases |(V2^)(p)| = |<^(p)| is a n - periodic function by argumentp(1). For any ^ g L2(T3) we have
(G(k,z)^) = f = fi f__W (3.4)
( ( , ^^ J Ek(p) — z J \ J B('k,' p; z) — 2cos kr cospC1) J p
T3 T2 -n 2
Here, 'k = (k(2),k(3)), 'p = (pC2),p(3)) g T2,
kC2) kC3)
B('k,' p; z) = 6 — 2 cos cosp(2) — 2 cos cosp(3) — z > 0, z < Emin(k).
The inner integral of the right-hand side of the equality (3.4) is represented as the sum of two integrals over the intervals [—n, 0] and [0, n]. In the first integral, making the replacement variable p(1) = n + q(1) and using the identity
cos(n + x) = — cos x and property |^(pC1) + n,' p) | = |y(p) | we have:
(G(k,z)^)=2 / B('k,' p; z)(/-|y(p)|2dp -— W (3.5)
v v ' ' J K '{J B2('k,' p; z) — 4 cos2 k^ cos2 pC1) * '
t2 0 2
Since B('k,' p; z) > 0 with all 'p G T2, z < Emin(k), the inner integral in (3.5) strictly decreases with increasing
kC1)
G [0, n]. The monotonicity of the integral implies that:
(G(k, z)V>, V) > (G(k', z)^, (3.6)
if ^ = V1/2^ is a nonzero element in L2(T3) and k = (kC1), k(2), k(3)), k' = (k(1), k(2), k(3)) at 0 < k(1) < k(1) < n. Note that from the inclusion ^ G H[An,TO)(G(k', z)) \ {0} it follows that V1/2^ = 0. Therefore, from
the inequality (3.6), the assertion (G(k, z)^, > An(k', z)(^, holds for all ^ G H[an,TO)(G(k', z)) \ {0}. This proves that A„(k, z) > A„(k', z) by 0 < k(1) < k(1) < n. □
Proof of Theorem 2.3. Let k = (k(1), k(2), k(3)) and ki = (k(1), k(2), k(3)) be two arbitrary points such that 0 < k(1) < k(1) < n. Let the operator H(k) has N = N(k) eigenvalues z1(k) < z2(k) < • • • < zN(k), lying below Emin(k). Existence is not less than N(k) eigenvalues of the operator H(k1) follows from Theorem 2.1. From here, it follows that the operator G(k, z) has N(k) eigenvalues:
A1(k, z) > A2(k, z) > • • • > An(k, z) > 1
by z G (zN(k), Emin(k)]. The continuity of G(k, z) with respect to the totality of variables k G T3 and z < Emin(k) implies the continuity of An(k, z), 1 < n < N, with respect such arguments k and z. It is easy to show that
lim ||G(k, z)|| =0. (3.7)
From the inequality An(k, z) < ||G(k, z)|| it follows that for any n G {1,2,..., N} the equation An(k, z) = 1 has a unique solution z = zn(k) G (-to, Emin(k)). Uniqueness follows from the monotonicity of An(k, •) in (-to, Emin(k)). By virtue of Lemma 3.2, the number zn(k) is the eigenvalue of the operator H(k). Using the definition of zn(k), the inequality An(k, z) > An+1(k, z) and monotonicity of the function An(k, •) we obtain that zn(k) < zn+1(k),n = 1,N - 1. Now, let's show the monotonicity of zn(k) in each k(1) G [0, n]. By virtue of Lemma 3.5 an eigenvalue An(k, z) is the decreasing function with respect to k(1) g [0, n], and hence:
1 = A„(k, z„(k)) > A„(ki, z„(k)).
On the other side:
1 = An(ki
, zn (ki)) > A„(ki , zn (k)). Since An(k, •) is an increasing function in (-to, Emin(k)), we get zn(ki) > zn(k). □
Notice that the assumption (2.1) is essential. The following example shows that if the assumption (2.1) is not satisfied, then there is a potential v and the segment [n - J, n], such that the eigenvalue E0(k) of the operator H(k) strictly decreases in
k(1) G [n - J, n].
Example 3.6. Let £(0) = 2w(e1) = 2v(-e1) = 2, v(n) = 0 at n = 0, n = ±e1. Then the operator H(n, n, n) has simple eigenvalue E0 = 4. Using perturbation theory, we obtain that the operator H(n - n, n) has a unique simple eigenvalue E0(n - n, n) in the neighborhood of E0 for small ft, and for E0(n - n, n) the following asymptotic formula holds [20]:
E0(n - ft,n,n) = E0 - £(0n\- 3!!(e1) 1 ft2 + O(ft4) at ft ^ 0. v(0) - £(e1) 4
This implies the existence of the segment [n - J, n], where E0(k(1), n, n) strictly decreases. 4. Conclusion
We study the two-particle Schrodinger operator H(k), (k g T3 = (-n,n]3 is the total quasimomentum of a system of two particles) corresponding to the Hamiltonian of the two-particle system on the three-dimensional lattice Z3. We prove that the number N (k) = N (k(1), k(2), k(3)) of eigenvalues below the essential spectrum of the operator H(k) is nondecreasing function in each k(i) g [0, n], i = 1, 2,3. We show the monotonicity property of each eigenvalue zn(k) = zn(k(1), k(2), k(3)) of the operator H(k) in k(i) G [0, n] with other coordinates k being fixed. In [21], for the case d =1 and card{n G Z : v(n) > 0} = to, the limit result:
lim N(k) = +to,
k^n —
for the number N(k) of the eigenvalues of the operator H(k) was proved. We remark that in our case if card{n G Z3 : v(n) > 0} = to, then one can prove the above limit result.
In the following, we give some generalizations of the statement of Theorem 2.3. If the potential v satisfies one of the conditions:
£(2s(1) + 1, 2s(2) + 1, 2s(3) + 1) = 0, Vs = (s(1), s(2), s(3)) G Z3, £(2s(1), 2s(2), 2s(3)) =0, Vs = (s(1), s(2), s(3)) G Z3,
then the eigenvalue zn(k(1), k(2), k(3)) of the operator H(k) increases with respect to each argument k(1), k(2) and k(3) in [0, n].
One can show that the statement of Theorem 2.3 is preserved, if the dimension d of the lattice Zd is greater than three. It is clear that for d = 1,2, is impossible to define the Birman-Schwinger operator G(k, z) in the whole space at the point z = Emin(k). Let us denote by:
Le(Td) = {f e L2(Td): f (-p)= f (p)} and L2(Td) = {f e ¿2(Td) : f (-p) = -f(p)}.
For the even potential the subspaces L2(Td) and L2(Td) are invariant under the operator H(k). The operator Go(k, z), corresponding to the operator Ho(k) = H(k)|L°(Td), can be defined as a compact operator on the boundary z = Emin(k) of the essential spectrum. In this case, one can prove a similar result concerning to the monotonicity of the eigenvalues of the operator Ho(k) with respect to k(i) e [0, п].
Acknowledgement
The authors expresses gratitude to the referee for valuable remarks. References
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