CC BY
19
Finiteness of discrete spectrum of the two-particle Schrodinger operator
on diamond lattices
M. I. Muminov, C. Lokman
Department of Mathematic of Sciences, Faculty of Science, Universiti Teknologi Malaysia, 81310 UTM Johor Bahru, Johor, Malaysia
DOI 10.17586/2220-8054-2017-8-3-310-316
We consider a two-particle Schrodinger operator H on the d—dimensional diamond lattice. We find a sufficiency condition of finiteness for discrete spectrum eigenvalues of H.
Keywords: two-particle Hamiltonian on lattice, Birman-Schwinger principle, discrete spectrum.
Received: 21 March 2017 Revised: 5 May 2017
1. Introduction
The spectrum of the many particle Schrodinger operator is closely connected to the spectrum of two-particle Schrodinger operator. To obtain the two-particle Schrodinger operator (in the continuous case) from the total Hamiltonian, we can separate the energy of motion of the center of mass such that the one-particle "bound states" are eigenvectors of the energy operator with separated total momentum (in this case, such an operator is indeed independent of the total momentum values) [1]. On the lattice case, the "separation of the center of mass" of a system is associated with the realization of the Hamiltonian as a "laminated operator", i.e., as the direct integral of the family h(k), k g Td (where T is a one dimensional torus), of the energy operators of two particles, where k is the value of the total quasi-momentum [2].
Conditions for the finiteness of the negative spectrum and for the absence of positive eigenvalues of the two-particle continuous Schrodinger operator H were presented in [3]. The finiteness of the number of bound states for two-particle cluster operators at some values of the clustering parameter was established in [4]. The sufficient condition of finiteness of discrete spectrum of two-particle lattice Schrodinger operators was given in [5]. An example one-dimensional lattice Schrodinger operator having at the same time an infinite number of discrete and embedded eigenvalues was given in the paper [6]. The existence conditions for eigenvalues of the family h(k) depending on the energy of interaction and quasi-momentum k have been investigated in [7].
The models which can be obtained investigating differential operators on graphs have already been used by physicists, a good review of such publications can be found, for example, in [8,9]. Two particle scattering theory on graphs was studied in [9]. The obtained results are applied to the qualitative description of a simple three-electrode nanoelectronic device. In [10] and [11], the problem of quantum particle storage in a nanolayered structure was considered. The authors numerically solved an eigenvalue problem of the corresponding Hamiltonian.
K. Ando et al. [12] described the Schrodinger operators on square, triangular, hexagonal, Kagome, diamond, subdivision lattices and the spectral properties of these Schrodinger operators were studied with compactly supported potentials. Conditions for the finiteness of the discrete spectrum and the non-existence of embedded eigenvalues of these Schrodinger operators with compactly supported potentials were given. The inverse scattering for discrete Schrodinger operators with compactly supported potentials on Zd and on the hexagonal lattice were studied in [13,14] in part, in these papers, the discreteness of embedded eigenvalues of these operators was proved.
We consider a discrete Schrodinger operator H on the d-dimensional diamond lattice with any continuous potential Q, i.e. a perturbation of discrete Laplacian with compact operator. The aim of the present paper is to prove the finiteness of the discrete spectrum of H. To show this, we use the technique proposed in [15]. First, we, using the well-known Birman-Schwinger principle, we reduce the study number of discrete spectrum N-(z)(N+(z)) of H, lying to the left (right) from z to the study of the number of eigenvalues n(1,TT(z)) of the compact operator TT(z), lying to the right from 1, i.e. we prove the equalities NT(z) = n(1, TT(z)). Further, we show that the operator-valued function T±(•) is well defined at the limits of the essential spectrum and apply the Weyl inequality.
2. Statement of the Main Results
We first give descriptions of the d—dimensional diamond lattice and a discrete Schrodinger operator on the d—dimensional diamond lattice [12].
Discrete Laplacian on the graph. We denote by G = (V(G), E(G)) the graph that consists of a vertex set V(G), whose cardinality is at most countable, and an edge set E(G), each element of which connects a pair of vertices. We assume that the graph is simple, i.e. there are neither self-loop, which is an edge connecting a vertex to itself, nor multiple edges, which are two or more edges connecting the same vertices. Let v, u g V(G), and e g E(G). We denote by v ~ u, when v is adjacent to u by e; by Nv the set of vertices which are adjacent to v, i. e. Nv = {u g V(G) : u ~ v}. We denote by deg(v) = #Nv the degree of v. We assume that the graph G is connected, which implies that deg(v) > 0 for any v g V(G). The discrete Laplacian Ad on G is defined as (see [16])
(Ad/)(v) = degv) E f(u) - /(v^
6V ' ueNv
for the function / on V(G). It is well known that —Ad is bounded, self-adjoint on
¿2(G) = {/ : £ |f (v)|2 deg(v) < .
v£v£V(G)
Higher-dimensional diamond lattice. Let Zd, d > 2, be a d-dimensional integer lattice, (Zd)2 = Zd x Zd be the Cartesian power of Zd, and l2((Zd)2) be the Hilbert space of square-integrable functions defined on (Zd)2. Let Ad be a subset of Zd+1 defined as follows
d+1
Ad = {x G Zd+1 : £xi = 0}
i=i
and e1 = (1,0,..., 0), e2 = (0,1,0,..., 0), ..., ed+1 = (0,..., 0,1), vi = ed+1 - ei, i = 1,..., d. Then, Ad is a lattice (see [12]) of rank d in Rd with basis vi, i = 1, • • • , d, i.e.
d
Ad = {v(n) : v(n) = £ njVj, n = (n1, • • • ,nd) G Z^.
j=1
The lattice Ad is called d-dimensional diamond lattice. We put
V = Ad U (p + Ad), p = d+ï(v1 + ••• + vd).
The set V is vertex set of d-dimensional lattice Ad.
The set of adjacent points of v(n) G Ad and p + v(n') g P + Ad are defined by
Nv(n) = {p + v(n') : n - n' = (0,..., 0), (1,0,..., 0),... (0,..., 0,0)},
Np+v(n') = {v(n) : n - n' = (0,..., 0), (1, 0,..., 0),... (0,..., 0, 0)}. Hence deg(v) = d +1 for v G Nv(n) or v G Np+v(n).
Discrete Laplacian on higher-dimensional diamond lattice. Using the definition of Discrete Laplacian on Graph from the definition of adjacent sets on V Discrete Laplacian Ad on V is defined by
((d +1)(Ad + 1)f)(v) = (g 1(n),g2(n)),
where
g 1(n) = /2(n) + f2(n - e1) +-----+ /2(n - ed),
h(n) = /1(n) + /1(n + e1)+-----+ /1(n + ed).
Any function / on V is written as /(n) = (/1(n),/1(n)), n G Zd, where /1(n) := /1(v(n)), /2(n) := /2(p + v(n)). Hence ^2(V) is the Hilbert space equipped with the inner product
(/,g k(V) := £ Â(v)g1(v)deg(v) + £ /2(v)g2(v)deg(v).
veAd ve(P+Ad)
The discrete Schrodinger operator. Let Td = Rd/(2n)d. We denote by ^Td) the Hilbert space with inner product
(f,g)Lv) = (fi,gi) + (f2 ,д2), (fj ,gj) = J fj (x)gj (x)dx.
We then define a unitary operator U : l2(V) ^ L22)(Td)
(Uf)j = (2n)-d/2Vd+lYJ fj(n)enx.
nezd
Passing to the Fourier series, we rewrite — (Ad + 1) into the following form:
(U( — (Ad + 1))U-lf)(x) = (Hof)(x), f g L2s)(Td),
where Ho is a matrices operator for a 2 x 2 matrix Ho (x)
Ho(x) =
0
E(x)
E (x) 0
E(x)
d + 1
(1 + eixi + ...eixd) .
Note that
Hence
IE(x)l2 =
(d +1)2
d +1 + 2J2 cosxi + 2^^cos(xi - Xj)
j=1 i<j
j = 1, ••• ,d.
min |E(p)| = 0, max IE(p)l = 1, p p
and the point 0 = (0,..., 0) g Td is a unit degenerated maximum point of function |E(• ) |2.
Let Q be the potential on l2(V) defined as multiplication operator by real valued, diagonal 2 x 2 matrices
(Qf)(n) =
Qi(n) 0
fi(n)
0 Q 2(n) \ f 2 (n) / '
where
Qi(n):= Qi(v(n)), Q2(n):= Q2(p + v(n)), n G Zd.
Throughout the paper, we shall assume that
^ |Qj(n)| < j = 1, 2.
nezd
The discrete Schrodinger operator is denoted by
H = —(d + 1)(Ad + 1) + Q.
Passing to the Fourier series, we rewrite H into the following form
H = Ho + Q,
where
' (Qifi)(x) (Q2 f2)(x)
(Qf )(x)
j, f G L22)(Td), (Qj fj)(x) = J Qj(x - t)fj(t)dt, j = 1, 2,
(2.1)
Qj(x) = (UQj)(x), j = 1, 2.
1
1
The Main Results. Note that from (2.1), it follows that the function Q(•) is continuous on Td. Hence the operator Q is a compact operator. By the Weyl theorem, the essential spectrum aess(H) of the operator H coincides with the spectrum of the unperturbed operator H0. Lemma 2.1. The spectrum <t(H0) of H0 coincides with the set
{A : |E(x)|2 = A2 for some x g Td},
i.e.,
a(Ho) = [—1,1].
Theorem 2.1. Let Q satisfy (2.1). Then, the number of eigenvalues of H lying in (—to, —1) U (1, to) is finite, i.e. the discrete spectrum of H is a finite set.
The proof of Theorem 2.1 implies the following theorem.
Theorem 2.2. Let vij-, i, j = 1,2 be continuous functions on (Td )2 and V = (Vj ) , where Vj is an integral
V /¿,¿=1
operator with kernel vij- (x, y), x, y g Td, d > 2. Then the discrete spectrum of H = H0 + V is finite set.
3. Proof of the Main results
Resolvent of H0. Proof of the Lemma 2.1. The operator H0 — AI has a matrix form
f —AI E \ ^ E —AI J ,
where I is an identity operator and E is operator multiplication by function E(x). Therefore, the inverse of this matrix has the form
f —AI E \-1_/ °-1(A) 0 \ i —AI — E \ ^ E —AI J = ^ 0 °-1(A) J ^ —E —AI J ,
where °(A) is operator multiplication by function °(A, x), °(A, x) = A2 — |E(x)|2. Let us denote by La and Aa 2 x 2 matrix operators
L = ( 0i,A> °L,A) ) ■ Aa = ( —EI —ei ).
Then the resolvent R0(A) = (H0 — AI)-1 of H0 has the form
Ro(A) = L-1Aa.
It follows from this that the operator R0(A) exists if and only if °(A, x) = 0 for all x g Td, i.e iff
A G {y = |E(x)| : x g Td} = [—1,1]. Hence, we have a(Ho) = [—1,1]. The lemma is thus proven.
Remember that °(A, x) > 0 as |A| > 1 for all x G Td. Therefore °(A) is a positive operator for all real A with |A| > 1. Hence La is a positive operator for all real A with |A| > 1. A positive root L_1/2 of L-1 has the form
L
-1/2 _ ( °-1/2(A) 0
a
0 °-1/2(A)
where ° 1/2(A) is an operator multiplication by function 1^V/°(A, •), |A| > 1. Let
AaO= ( —E(x) ).
For any fixing x G Td the eigenvalues of the matrices Aa(x) are ^_(A, x) = —A — |E(x)| and £+(A, x) = —A + |E(x)|. The numbers —A± |E(x)| are positive as A < —1 and negative as A > 1. Then, Aa > 0 as A < —1 and —Aa > 0 as A > 1. Since the operator L-1 is commutative with Aa, the operator L_1Aa is self-adjoint and L-1Aa > 0 as A < —1, —L-1Aa > 0 as A > 1.
The positive roots [Ro(A)]1/2, A < —1 and [—Ro(A)]1/2, A > 1 of the operators Ro(A), A > 1 and —Ro(A), A < 1 have the following forms, respectively:
Ro(A)1/2 = L_1/2A_1/2 as A > 1 (3.1)
and
[-Ro(A)]1/2 = L-1/2[-Ax]-1/2 as A < -1.
(3.2)
Lemma 3.1. The positive roots [A-x(x)]-i/2 and [—A-x(x)]-l/2 of the matrix Ax(x), X < —1 and —Ax (x), X > 1 are given, respectively, by
A-
1/2
1 ( Vt-(A, x) + VM.A,x)
V£-(A,x) - \Jt+(A,x)
2\ (/-M - Exk /-M +
E(x) \E(x)\
and
r A U/^ 1 l
HAJ1'2(x) = - I , _ ,_N
2 ^ (V-e-(A,x) - /-£+(A,x)) iex\
E(x)
V-£-(A,x) - V-£+(A,x)
y/-Ç-(A,x) + y/-£+(A,x)
E(x) \E(x)\
Proof. The eigenvectors of the matrix A\(x) corresponding to the eigenvalues £-(A,x) and £+(A,x) are p- = 1/V2 (1, E(x)/IE(x)l) and p+ = 1/V2 (' E(x)/|E(x)l, -1j, respectively, with ||p±|| = 1. Therefore the matrix Ax(x) in a sense operator can be represented as
Ax(x) = Î-(A,x)(^,^-)C2 P- + £+(A,x)(^,^+)C2 P+,
where (•, •)C2 is a usual scalar product of C2. Therefore the positive roots [AA(x)]1/2, A < -1 and [-AA]1/2(x), A> 1 of the matrices Ax(x), A < -1 and -Ax(x), A> 1 have the forms
[Ax(x)]1/2 = /£-(A,x)(^, <p-)C2p- + /£+(A, x)(•, p+)C2p+, A < 1,
[-Ax(x)]1/2 = /-£-(A,x)(;p-)&p- + /-£+(A,x)(;p+)c*p+, A > 1. These equalities prove the desired results of the lemma.
Note that the Lemma 3.1 shows that the matrix valued function [Aa(^)]1/2, A < -1 ([-Aa(-)]1/2, A 1 is
bounded for all x g Td and A <-1 (A > 1).
The Birman-Schwinger principle. We define the self-adjoint compact operators TT(z), acting in the Hilbert space L22) (Td) determined by
T-(z) = R0/2(z)QR0/2(z) for z < -1
and
T+(z) = - [-Ro(z)]1/2 Q [-Ro(z)]1/2 for 1 < z.
By N-(z) and N+(z), we denote the number of eigenvalues of the operator H lying to the left from z < -1 and lying to the right from z > 1, respectively.
Let A be a self-adjoint operator acting in a Hilbert space H, and let Ha(A), A > supaess(A), be the subspace consisting of the vectors f g H satisfying the condition (Af, f ) > A(f, f ). We set
n(A,A)= sup dimHa(A).
Ha (A)
By definition, we have N-(z) = n(-z, -H), -z > 1 and N+(z) = n(z, H), z > 1.
The following lemma is a modification of the well-known Birman-Schwinger principle for the operator H (see [17,18]).
Lemma 3.2. For the numbers N-(z) and N+(z) of eigenvalues (counted with multiplicities) of the operator H we have the equalities, respectively,
N-(z) = n(1, T-(z)), z < -1, (3.3)
and
N+ (z)= n(1, T+(z)), z > 1. (3.4)
Proof. We suppose that u g Hh(—z), i.e.,
(Hu, u) < z(u, u)
or
((H0 — zI)u, u) < —(Qu, u).
Then we have
(y,y) < — (R0/2(z)QR^/2(z)y,^ , y = (Ro — zI)1/2u.
Therefore, N_(z) < n (1, R^/2(z)QRj/2(z^.
By analogous arguments, we obtain the converse statement:
N_(z) > n (1, rJ/2(z)QR1/2(z)) .
Hence inequality (3.3) follows. The equality (3.4) can be proven similarly.
Proof of Theorem 2.1. To prove the theorem, we use the technique proposed in [15], i.e., we show that the operator-valued function T±(-) is well defined at the limits of the essential spectrum and we also use Lemma 3.2 and apply the Weyl inequality [19]
n(a + b, A + B) < n(a, A) + n(b, B),
which holds for compact operators A and B.
Let us first show that the operator-valued functions T_(-) and !+(•) are continuous in the norm, (—to, 0] and [1, to) respectively.
Note that since 0 is a unite maximum point of |E(-)|2, we have
C1x2 < |E(x)|2 < C2x2 for all x G Td.
From this, we get the estimation
1 1 C
m-T = . 2 ,,2 < n, V|z| > 1. (3.5)
\J °(z, x) Vz2 — |E(x)|2 |x|
Using (3.1) and (3.2) we rewrite T_(z) and T+ (z) as
T_(z) = L_1/2A_1/2QA_1/2L_1/2, z < —1
and
T+(z) = L_1/2[—Az]_1/2Q[—Az]_1/2L_1/2, z > 1. We denote by q±(z) the entries of the matrix operator
[±Aa]_1/2Q[±Aa]_1/2.
Then q±(z) are integral operators. Since Q(-, •) and £±(z, •) are continuous functions, respectively on (Td)2 and Td as |z| > 1, the kernel q±(z; •, •) of the integral operator q±(z) are bounded functions on (Td)2.
It follows from (3.5) that the kernel t±(z; x, y) = , 1 q±(z; x, y) , 1 of T±(z) estimated by
j vL(z,x) j vL(z,y)
C
|t±(z; x,y)| < "j - r as x, y G Td,
j |x||y|
where the constant C does not depend of z, |z| > 1. This implies that the functions (z; •, ^),z < —1 and t+ (z; •, •), z > 1 are square-integrable on (Td)2 and ti(z; •, •) converges almost everywhere to ti(^1; •, •) as z ^ ^0. By the Lebesgue theorem, the operator TT(z) then converges in the norm to TT(^1) as z ^ ^0. Further, using the Weyl inequality, from (3.3) and (3.4) we obtain
N_(z) < n (2, T_(z) — T_( —1)) + n ( 1, T_( —1)) , z < —1,
N+(z) < n (2, T+(z) — T+(1)) + n (2, T+(1^ , z > 1. Since the operator T(±1) is compact,
J1, T(±1)) < to.
For small |z + 1| and |z — 1| we have the equalities, respectively,
n
, T_(z) - T_(-1))
0
and
n
(2, T+(z) - T+(1))
0.
Hence, by Lemma 3.2, the number of eigenvalues of H in (-to, -1) U (1, to) must be finite. Acknowledgements
We thank the referee for valuable comments and gratefully acknowledge the support of the Malaysian Ministry of Education (MOE) through the Research Management Center (RMC), Universiti Teknologi Malaysia (Vote: QJ130000.2726.01K82)
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