The nature of bound states of two-particle cluster operators for small parameter values was first studied in detail by Minlos and Mamatov [1] and then in a more general setting by Minlos and Mogilner [2]. In [3], Howland showed that the Rellich theorem on perturbations of eigenvalues does not extend to the resonance theory. Studying bound states of a two-particle system Hamiltonian H on the d-dimensional lattice ℤ d reduces to studying [2] [4] [5] [6] [7] the eigenvalues of a family of Shrödinger operators H ( k ) , k ∈ T d , where k is the total quasi-momentum of a system. Moreover, eigenfunctions of H ( k ) are interpreted as bound states of the Hamiltonian H, and eigenvalues, as the bound state energies. The bound states of H of a system of two fermions on a one-dimensional lattice were studied in [4], a system of two bosons on a two-dimensional lattice was studied in [6], and perturbations of the eigenvalues of a two-particle Shrödinger operator on a one-dimensional lattice were studied in [8]. The finiteness of the number of eigenvalues of Shrödinger operator on a lattice was studied in the works [7] [9].

The discrete spectrum of the two-particle continuous Shrödinger operator

h λ = − Δ + λ V

was studied by many authors, with the conditions for the potential V formulated in its coordinate representation. The condition for the finiteness of the set of negative elements of the spectrum and the absence of positive eigenvalues of h λ can be found in [10]. If V ≤ 0 , then the number of negative eigenvalues N ( λ ) is a nondecreasing function of λ ∈ ( 0, ∞ ) , and each eigenvalue z n ( λ ) decreases on the half-axis ( 0, ∞ ) . It is known that when the coupling constant λ decreases, the bound state energies of h λ tend to the boundary of the continuous spectrum (see [10] ) and for some finite λ are on the boundary. Two questions then arise: Does a bound or virtual state correspond to such a threshold state (i.e., is the corresponding wave function square-integrable)? And where do the bound states “disappear to” as λ decreases further? The study of the first question was the subject in [11] [12]. Regarding the second question, it turns out that the bound state disappears by being absorbed into the continuous spectrum and becomes a resonance [5].

Here, we consider bound states of the Hamiltonian H ^ (see (1)) of a system of two fermions on the three-dimensional lattice ℤ 3 with the special potential v ^ (see (5)). In other words, we study the discrete spectrum of a family of the Shrödinger operators H ( k ) , k = ( k 1 , k 2 , k 3 ) ∈ T 3 , (see (3)) corresponding to H ^ in the invariant subspace L 123 − ( T 3 ) .

Restriction of the operator H ( k ) in the invariant subspace L 123 − ( T 3 ) is denoted by H 123 − ( k ) .

In the case k = π → : = ( π , π , π ) , the operator H ( π → ) has an infinite number of eigenvalues of the form 6 − v ^ ( n ) , n ∈ ℤ 3 and the essential spectrum consists of the single point 6. Here, the potential v ^ is defined by (5) and v ¯ : ℕ → ℝ is a decreasing function on ℕ and v ¯ ∈ l 2 ( ℕ ) . These eigenvalues z n ( π → ) = 6 − v ¯ ( n ) , n ∈ ℕ are arranged in ascending order, z 1 ( π → ) < ⋯ < z n ( π → ) < ⋯ , and the smallest eigenvalue z 1 ( π → ) = 6 − v ¯ ( 1 ) is threefold, z 2 ( π → ) = 6 − v ¯ ( 2 ) is sevenfold, and the other eigenvalues z n ( π → ) = 6 − v ¯ ( n ) , n ≥ 3 are ninefold. All ninefold eigenvalues z n ( π → ) = 6 − v ¯ ( n ) , n ≥ 3 of the operator H ( π → ) are simple eigenvalues for the operator H 123 − ( π → ) .

Further, we investigate eigenvalues and eigenfunctions of the restriction operator H 123 − ( k ) .

In the case k = ( k 1 , k 2 , π ) the corresponding operator H 123 − ( k 1 , k 2 , π ) has infinitely many invariant subspaces ℜ 123 − ( n ) : = L 2 − ( T ) ⊗ L 2 − ( T ) ⊗ L − ( n ) , n ∈ ℕ . It is proved that the restriction H 123 n − ( k 1 , k 2 , π ) of the operator H 123 − ( k 1 , k 2 , π ) in the invariant subspace ℜ 123 − ( n ) has no more than one eigenvalue. If exists, it can be calculated explicitly. For every ( k 1 , k 2 ) ∈ ( − π , π ) 2 the operator H 123 − ( k 1 , k 2 , π ) has only a finite number of eigenvalues.

For any perturbation β > 0 , the essential spectrum { 6 } of H ( π → ) becomes the essential spectrum σ e s s ( H ( π − 2 β , π , π ) ) = [ 6 − 2 sin β ,6 + 2 sin β ] . If the potential v ^ is of the form (5), the Shrödinger equation H 123 − ( π − 2 β , π , π ) f = z f ,   f ∈ ℜ 123 − ( n ) can be exactly solved (see Theorem 1).

The Shrödinger equations H ( π − 2 β , π , π ) f = z f and H ( π − 2 β , π − 2 β , π ) f = z f ,   f ∈ ℜ 123 − ( n ) with small β are solved by using methods invariant subspaces and operator theory.

The free Hamiltonian H ^ 0 of a system of two fermions on a three-dimensional lattice ℤ 3 usually corresponds to a bounded self-adjoint operator acting in the Hilbert space l 2 a s ( ℤ 3 × ℤ 3 ) : = { f ∈ l 2 ( ℤ 3 × ℤ 3 ) : f ( x , y ) = − f ( y , x ) } by the formula

H ^ 0 = − 1 2 m Δ 1 − 1 2 m Δ 2 .

Here, m is the fermion mass, which we assume to be equal to unity in what follows, Δ 1 = Δ ⊗ I and Δ 2 = I ⊗ Δ , where I is the identity operator, and the lattice Laplacian Δ is a difference operator that describes a translation of a particle from a side to a neighboring side,

( Δ ψ ^ ) ( x ) = ∑ j = 1 3 [ ψ ^ ( x + e j ) + ψ ^ ( x − e j ) − 2 ψ ^ ( x ) ] ,     x ∈ ℤ 3 ,     ψ ^ ∈ l 2 ( ℤ 3 ) ,

where e 1 = ( 1,0,0 ) , e 2 = ( 0,1,0 ) , e 3 = ( 0,0,1 ) are unit vectors in ℤ 3 . The total Hamiltonian H ^ acts in the Hilbert space l 2 a s ( ℤ 3 × ℤ 3 ) and is the difference of the free Hamiltonian H ^ 0 and the interaction potential V ^ 2 of the two fermions (see [8] [13] ):

H ^ = H ^ 0 − V ^ 2 , (1)

where

( V ^ 2 ψ ^ ) ( x , y ) = v ^ ( x − y ) ψ ^ ( x , y ) ,   ψ ^ ∈ l 2 a s ( ( ℤ 3 ) 2 ) : = l 2 a s ( ℤ 3 × ℤ 3 ) .

Hereafter, we assume that

v ^ ∈ l 2 ( ℤ 3 )       and     v ^ ( x ) = v ^ ( − x ) ≥ 0     for   all     x ∈ ℤ 3 . (2)

Under this condition, the Hamiltonian H ^ is a bounded self-adjoint operator in l 2 a s ( ( ℤ 3 ) 2 ) .

We pass to momentum representation using the Fourier transform [2] [4] [7]

F : l 2 a s ( ℤ 3 × ℤ 3 ) → L 2 a s ( T 3 × T 3 ) .

The Hamiltonian H = H 0 − V = F H ^ F − 1 in the momentum representation commutes with the unitary operators U s , s ∈ ℤ 3 , given by

( U s f ) ( k 1 , k 2 ) = exp ( − i ( s , k 1 + k 2 ) ) f ( k 1 , k 2 ) ,     f ∈ L 2 a s ( T 3 × T 3 ) .

It follows that there exist decompositions of L 2 a s ( T 3 × T 3 ) and the operators U s and H into direct integrals (see [7] [9] and [10] )

L 2 a s ( T 3 × T 3 ) = ∫ T 3 ⊕ L 2 a s ( F k ) d k ,     U s = ∫ T 3 ⊕ U s ( k ) d k ,     H = ∫ T 3 ⊕ H ˜ ( k ) d k .

Here,

F k = { ( k 1 , k 2 ) ∈ T 3 × T 3 :   k 1 + k 2 = k } ,   k ∈ T 3 ,

and U s ( k ) is an operator of multiplication by the function exp ( − i ( s , k ) ) in L 2 a s ( F k ) . The fiber operator H ˜ ( k ) of H also acts in L 2 a s ( F k ) and is unitarly equivalent to H ( k ) : = H 0 ( k ) − V , which is called the Shrödinger operator. This operator acts in the Hilbert space L 2 o ( T 3 ) : = { f ∈ L 2 ( T 3 ) : f ( − q ) = − f ( q ) } by the formula

( H ( k ) f ) ( q ) = ε k ( q ) f ( q ) − ( 2 π ) − 3 2 ∫ T 3   v ( q − s ) f ( s ) d s . (3)

The unperturbed operator H 0 ( k ) is an operator of multiplication by the function

ε k ( q ) = ε ( k 2 + q ) + ε ( k 2 − q ) = 6 − 2 cos k 1 2 cos q 1 − 2 cos k 2 2 cos q 2 − 2 cos k 3 2 cos q 3 . (4)

From (3) and (4), it follows that

H ( k 1 , k 2 , k 3 ) = H ( − k 1 , k 2 , k 3 ) = H ( k 1 , − k 2 , k 3 ) = H ( k 1 , k 2 , − k 3 ) ,

so we can assume k 1 , k 2 , k 3 ∈ [ 0, π ] .

The perturbation operator V is an integral operator in L 2 o ( T 3 ) with the kernel

( 2 π ) − 3 2 v ( q − s ) = ( 2 π ) − 3 2 ( F v ^ ) ( q − s ) ,

and belongs to the class of Hilbert-Schmidt operators Σ 2 .

In this work, we consider the operator H ( k ) with the potential v ^ of the form

v ^ ( n ) = v ^ ( n 1 , n 2 , n 3 ) = ( v ¯ ( | n | ) , | n 1 | + | n 2 | ≤ 1 0, | n 1 | + | n 2 | ≥ 2 (5)

where | n | = | n 1 | + | n 2 | + | n 3 | . Supporter is in the cylinder:

D = { n = ( n 1 , n 2 , n 3 ) ∈ ℤ 3 : n 3 ∈ ℤ , | n 1 | + | n 2 | ≤ 1 } .

Since for every function ψ ^ ∈ l 2 a s ( ( ℤ 3 ) 2 ) the equality ψ ^ ( x , x ) = 0, x ∈ ℤ 3 holds, then the value of the potential v ^ at the origin can be set arbitrary, since it does not affect the result, for simplicity, we assume that v ^ ( 0 ) = 0 .

The function v ¯ : ℕ → ℝ in (5) is decreasing in ℕ i.e.,

v ¯ ( 1 ) > v ¯ ( 2 ) > ⋯ (6)

and belongs to l 2 ( ℕ ) . The kernel v , of the integral operator V, i.e., the Fourier transform v ( p ) = ( F v ^ ) ( p ) , of the potential v ^ , has the form

v ( p ) : = ( F v ^ ) ( p ) = 1 ( 2 π ) 3 / 2 ∑ n ∈ ℤ 3     v ^ ( n ) e i ( n , p ) = 1 ( 2 π ) 3 / 2 [ 2 v ¯ ( 1 ) ( cos p 1 + cos p 2 + cos p 3 )     + 2 v ¯ ( 2 ) ( cos 2 p 3 + 2 cos p 1 cos p 2 + 2 cos p 1 cos p 3 + 2 cos p 2 cos p 3 )     + 2 ∑ n = 1 ∞     v ¯ ( n + 2 ) ( cos ( n + 2 ) p 3 + 2 cos ( n + 1 ) p 3 ( cos p 1 + cos p 2 )     + 4 cos p 1 cos p 2 cos n p 3 ) ] . (7)

Eigenvalues of the operator H ( k ) . We note that the spectra of the operators H 0 ( k ) and V are known. The operator H 0 ( k ) does not have eigenvalues, its spectrum is continuous and coincides with the range of the function ε k :

σ ( H 0 ( k ) ) = [ m ( k ) , M ( k ) ] ,     where       m ( k ) = min q ∈ T 3 ε k ( q ) ,       M ( k ) = max q ∈ T 3 ε k ( q ) .

The spectrum of V consists of the set { 0, v ¯ ( n ) , n ∈ ℕ } . Under condition (2), the operator V is a Hilbert-Schmidt operator and is hence compact. By the Weyl theorem [10], the essential spectrum of H ( k ) coincides with the spectrum of H 0 ( k ) :

σ e s s ( H ( k ) ) = [ m ( k ) , M ( k ) ] .

If k = π → , then the spectrum of H ( π → ) = 6 I − V consists of eigenvalues of the form 6 − v ¯ ( n ) , n ∈ ℕ and the essential spectrum is { 6 } . If k j = π (for some j ∈ { 1,2,3 } ), then there exists a potential v ^ such that H ( k ) has an infinite number of eigenvalues outside the continuous spectrum (see [4] [14] ).

We recall some notations and known facts. For any self-adjoint operator B acting in a Hilbert space H without an essential spectrum to the right of μ ∈ ℝ , we let n ( μ , B ) denote the number of its eigenvalues to the right of μ . We let N ( k , z ) denote the number of eigenvalues of H ( k ) to the left of z ≤ m ( k ) , i.e., N ( k , z ) = n ( − z , − H ( k ) ) . The number N ( k , m ( k ) ) in fact coincides with the number of eigenvalues outside the continuous spectrum of H ( k ) . It follows from the self-adjointness of H ( k ) = H 0 ( k ) − V and positivity of V that

σ ( H ( k ) ) ∩ ( M ( k ) , ∞ ) = ∅ ,

and hence σ d i s c ( H ( k ) ) ⊂ ( − ∞ , m ( k ) ) . Therefore we seek only eigenvalues z less than m ( k ) .

For any k ∈ T 3 and z < m ( k ) , we define the integral operator

G ( k , z ) = V 1 2 r 0 ( k , z ) V 1 2 ,

where r 0 ( k , z ) is the resolvent of the unperturbed operator H 0 ( k ) . Under condition (2), the operator V is positive, and we let V 1 2 denote the positive square root of the positive operator V. A solution f of the Schrödinger equation

H ( k ) f = z f

and the fixed points φ of G ( k , z ) are connected by the relations

f = r 0 ( k , z ) V 1 2 φ       and       φ = V 1 2 f .

The following proposition (the Birman-Schwinger principle) holds [9].

Lemma 1. The number of eigenvalues of H ( k ) to the left of z < m ( k ) coincides with the number of eigenvalues of G ( k , z ) greater than unity, i.e., the equality

N ( k , z ) = n ( 1, G ( k , z ) )

holds.

Lemma 2. If for some k ∈ T 3 the limit operator lim z → m ( k ) − G ( k , z ) = G ( k , m ( k ) ) exists and is compact, then the equality

N ( k , m ( k ) ) = n ( 1, G ( k , m ( k ) ) ) (8)

holds.

Equality (8) states that the number of eigenvalues of H ( k ) , to the left of m ( k ) is equal to the number of eigenvalues of G ( k , m ( k ) ) greater than unity.

In this section, we study the invariant subspaces with respect to the operator H ( k ) .

Let L 2 − ( T ) = { f ∈ L 2 ( T ) : f ( − p ) = − f ( p ) } be a subspace of the space L 2 ( T ) , consisting of odd functions on T = [ − π , π ] , and L 2 + ( T ) = { f ∈ L 2 ( T ) : f ( − p ) = f ( p ) } be a subspace of L 2 ( T ) , consisting of even functions on T . In addition, we use the notation

L 123 − ( T 3 ) : = L 2 − ( T ) ⊗ L 2 − ( T ) ⊗ L 2 − ( T ) ,   L 123 + ( T 3 ) : = L 2 + ( T ) ⊗ L 2 + ( T ) ⊗ L 2 + ( T ) .

Note that L 123 − ( T 3 ) is a subspace of the space L 2 o ( T 3 ) . It is natural to expect the invariance of the subspace L 123 − ( T 3 ) with respect to the operator H ( k ) . It turns out that this subspace is invariant under the operator H ( k ) , i.e. the following statement holds.

Lemma 3. Let the potential v ^ have the form (5). Then the subspace L 123 − ( T 3 ) is invariant under the action of H ( k ) .

Proof. We prove that this subspace is invariant first with respect to H 0 ( k ) , and then with respect to V. It follows from representation (4) that the function ε k belongs to the subspace L 123 + ( T 3 ) , and it follows from the inclusion f ∈ L 123 − ( T 3 ) that ε k   f ∈ L 123 − ( T 3 ) . This proves that L 123 − ( T 3 ) is invariant with respect to H 0 ( k ) .

Simple calculations show that the function (see (7))

( V f ) ( p 1 , p 2 , p 3 ) = 1 ( 2 π ) 3 2 ∫ T 3   v ( p 1 − s 1 , p 2 − s 2 , p 3 − s 3 ) f ( s 1 , s 2 , s 3 ) d s 1 d s 2 d s 3

belongs to the subspace L 123 − ( T 3 ) for f ∈ L 123 − ( T 3 ) . Hence, we prove the invariance of L 123 − ( T 3 ) with respect to V, and it follows that L 123 − ( T 3 ) is invariant with respect to H ( k ) = H 0 ( k ) − V .

H 123 − ( k ) denotes the restriction of H ( k ) to the respective subspace L 123 − ( T 3 ) . The action of H 0 ( 123 ) − ( k ) : = H 0 ( k ) is unchanged, the unperturbed operator H 0 ( k ) is an operator of multiplication by the function ε k . We present the formula for V 123 − = V | L 123 − ( T 3 ) operator V acts on the element f ∈ L 123 − ( T 3 ) according to the formula

( V 123 − f ) ( p ) = 1 π 3 ∑ n = 1 ∞     v ¯ ( n + 2 ) ∫ T 3 sin p 1 sin p 2 sin n p 3 sin q 1 sin q 2 sin n q 3 f ( q ) d q .

Note that for k = π → , the spectrum of H ( π → ) = 6 I − V consists only of the eigenvalues 6,6 − v ¯ ( n ) , n ∈ ℕ and the essential spectrum { 6 } . Under condition (6) the number z 1 ( π → ) = 6 − v ¯ ( 1 ) is a threefold eigenvalue of H ( π → ) , with the corresponding eigenfunctions

sin p 1 , sin p 2 , sin p 3 ,

the number z 2 ( π → ) = 6 − v ¯ ( 2 ) is a sevenfold eigenvalue with the corresponding eigenfunctions

sin 2 p 3 , cos p 1 sin p 2 , sin p 1 cos p 2 , cos p 1 sin p 3 , sin p 1 cos p 3 , cos p 2 sin p 3 , sin p 2 cos p 3 ,

for each n ≥ 3 , the number z n ( π → ) = 6 − v ¯ ( n ) is a ninefold eigenvalue, and the corresponding eigenfunctions are

sin ( n + 2 ) p 3 , sin p 1 cos ( n + 1 ) p 3 , sin p 2 cos ( n + 1 ) p 3 , sin ( n + 1 ) p 3 cos p 1 , sin ( n + 1 ) p 3 cos p 2 , sin n p 3 cos p 1 cos p 2 , sin p 2 cos p 1 cos n p 3 , sin p 1 cos p 2 cos n p 3 , sin p 1 sin p 2 sin n p 3 .

The number z ∞ ( π → ) = 6 is an eigenvalue of an infinite multiplicity, and the corresponding eigenfunctions are

ψ ( n 1 , n 2 , n 3 ) − − − ( p ) = sin n 1 p 1 sin n 2 p 2 sin n 3 p 3 ,         n 3 ∈ ℕ ,   n 1 + n 2 ≥ 3.

All ninefold eigenvalues z n ( π → ) = 6 − v ¯ ( n ) , n ≥ 3 of the operator H ( π → ) are simple eigenvalues for the operator H 123 − ( π → ) , and the number z ∞ ( π → ) = 6 is an eigenvalue of an infinite multiplicity.

If the third coordinate k 3 of the total quasimomentum k is equal to π , then the operator H ( k 1 , k 2 , π ) has infinitely many invariant subspaces ℜ 123 − ( n ) , n ∈ ℕ .

Next, we give a description of the invariant subspace ℜ 123 − ( n ) , n ∈ ℕ .

The system of functions

{ ψ n − ( q ) = 1 π sin n q } n ∈ ℕ

is an orthonormal basis in the space L 2 − ( T ) . Let us denote by L − ( n ) , n ∈ ℕ the one-dimensional subspace spanned by the vector ψ n − . The space L 2 − ( T ) can be decomposed into the direct sum

L 2 − ( T ) = ∑ n = 1 ∞ ⊕ L − ( n ) .

This decomposition produces another decomposition

L 123 − ( T 3 ) = ∑ n = 1 ∞ ⊕ { L 2 − ( T ) ⊗ L 2 − ( T ) ⊗ L − ( n ) } = ∑ n = 1 ∞ ⊕ { L 12 − ( T 2 ) ⊗ L − ( n ) } = ∑ n = 1 ∞ ⊕ ℜ 123 − ( n ) ,

where

ℜ 123 − ( n ) : = L 12 − ( T 2 ) ⊗ L − ( n ) ,   L 12 − ( T 2 ) = L 2 − ( T ) ⊗ L 2 − ( T ) .

Lemma 4. Let the potential v ^ have the form (5). Then the subspace ℜ 123 − ( n ) is invariant under H 123 − ( k 1 , k 2 , π ) for any n ∈ ℕ .

Proof. Let ( f ψ n − ) ( p 1 , p 2 , p 3 ) : = f ( p 1 , p 2 ) ψ n − ( p 3 ) , where f ∈ L 12 − ( T 2 ) , ψ n − ∈ L − ( n ) is an arbitrary element of ℜ 123 − ( n ) . We consider the action of H 123 − ( k 1 , k 2 , π ) = H 0 ( k 1 , k 2 , π ) − V 123 − on f   ψ n − :

( H 0 ( k 1 , k 2 , π ) f ψ n − ) ( p ) = [ ( 6 − 2 cos k 1 2 cos p 1 − 2 cos k 2 2 cos p 2 ) f ( p 1 , p 2 ) ] ψ n − ( p 3 ) , (9)

( V 123 − f ψ n − ) ( p ) = [ v ¯ ( n + 2 ) π 2 ∫ T 2 sin p 1 sin q 1 sin p 2 sin q 2 f ( q 1 , q 2 ) d q 1 d q 2 ] ψ n − ( p 3 ) . (10)

To obtain the last formula (10), we use the orthogonality of the system of functions { ψ n − } n ∈ ℕ in L 2 − ( T ) . Relations (9) and (10) imply the equality

( H 123 − ( k 1 , k 2 , π ) f ψ n − ) ( p 1 , p 2 , p 3 ) = ( H 0 ( k 1 , k 2 , π ) f ψ n − ) ( p 1 , p 2 , p 3 ) − ( V 123 − f ψ n − ) ( p 1 , p 2 , p 3 ) = [ ( 6 − 2 cos k 1 2 cos p 1 − 2 cos k 2 2 cos p 2 ) f ( p 1 , p 2 ) ] ψ n − ( p 3 )       − [ v ¯ ( n + 2 ) π 2 ∫ T 2 sin p 1 sin q 1 sin p 2 sin q 2 f ( q 1 , q 2 ) d q 1 d q 2 ] ψ n − ( p 3 ) (11)

which completes the proof of the lemma.

We denote by H 123 n − ( k 1 , k 2 , π ) restriction of the operator H 123 − ( k 1 , k 2 , π ) in the invariant subspace ℜ 123 − ( n ) . Formula (11) shows that the restriction H 123 n − ( k 1 , k 2 , π ) to the subspace ℜ 123 − ( n ) = L 12 − ( T 2 ) ⊗ L − ( n ) has the form

H 123 n − ( k 1 , k 2 , π ) = [ 2 I + H 0 ( k 1 , k 2 ) − v ¯ ( n + 2 ) V 11 ] ⊗ I , (12)

where I is the identity operator and H 123 ( n ) ( k ) : = 2 I + H 0 ( k ) − v ¯ ( n + 2 ) V 11 , k = ( k 1 , k 2 ) , is a two-dimensional two-particle operator acting in L 12 − ( T 2 ) by the formula

( H 123 ( n ) ( k ) f ) ( p ) = ( 2 + ε k ( p ) ) f ( p ) − v ¯ ( n + 2 ) π 2 ∫ T 2 sin p 1 sin p 2 sin q 1 sin q 2 f ( q ) d q ,

where ε k ( p ) = 4 − 2 cos k 1 2 cos p 1 − 2 cos k 2 2 cos p 2 , and V 11 is a one-dimensional integral operator in L 12 − ( T 2 ) with the kernel

v ( p , q ) = 1 π 2 sin p 1 sin p 2 sin q 1 sin q 2 .

Studying the eigenvalues of H 123 n − ( k 1 , k 2 , π ) by representations (12) reduces to studying the eigenvalues of

H 123 ( n ) ( k ) = 2 I + H 0 ( k ) − v ¯ ( n + 2 ) V 11 , k = ( k 1 , k 2 )

i.e. the three-dimensional problem reduces to the two-dimensional problem.

Our main goal in this section is to study the behavior of the nondegenerate eigenvalue z n + 2 ( π → ) = 6 − v ¯ ( n + 2 ) , n ∈ ℕ of H 123 − ( π → ) at small perturbations β ( k 1 = π − 2 β or k 2 = π − 2 β ), i.e. the eigenvalues of H 123 − ( π − 2 β , π , π ) (or H 123 − ( π , π − 2 β , π ) ) at small perturbations β . The studying of the eigenvalues of H 123 − ( π − 2 β , π , π ) is reduced to study the eigenvalues of the operator H 123 n − ( π − 2 β , π , π ) for each fixed n ∈ ℕ . In turn, the problem of studying the eigenvalues of the operator H 123 n − ( π − 2 β , π , π ) by virtue of (12) is reduced to study of the discrete spectrum of the operator

H 123 ( n ) ( π − 2 β , π ) = 2 I + H 0 ( π − 2 β , π ) − v ¯ ( n + 2 ) V 11 .

Studying the eigenvalues of H 123 ( n ) ( π − 2 β , π ) and H 123 ( n ) ( π , π − 2 β ) reduces to studying the eigenvalues of H λ ( k ) acting in L 2 − ( T ) by the formula

( H λ ( k ) f ) ( p ) = ε k ( p ) f ( p ) − λ π ∫ T sin p sin q   f ( q ) d q , ε k ( p ) = 2 − 2 cos k 2 cos p . (13)

It is known that the essential spectrum of H λ ( π − 2 β ) = H 0 ( π − 2 β ) − λ V 1 , β ∈ ( 0, π 2 ] consists of a segment [ m ( β ) , M ( β ) ] , where m ( β ) = 2 − 2 sin β , M ( β ) = 2 + 2 sin β .

Further we give some information about the eigenvalues and eigenfunctions of the operator H λ ( k ) . Combining Theorem 6.3 in [6], Theorem 5.10 in [15] and Lemmas 1 and 2 we obtain the following statement about eigenvalues of the operator H λ ( k ) .

Lemma 5. Let   β ∈ ( 0, π 2 ] .

a) If λ < sin β   , then the operator H λ ( π − 2 β ) has no eigenvalues lying outside of the essential spectrum.

b) If λ = sin β   , then the left edge m ( β ) of essential spectrum of the operator H λ ( π − 2 β ) is a resonance.

c) If λ > sin β , then the operator H λ ( π − 2 β ) has a unique nondegenerate eigenvalue

z λ ( β ) = 2 − λ − 1 λ sin 2 β

which lying in the left of the essential spectrum with corresponding normalized eigenfunction

f λ − ( p ) = C λ   sin p 2 − 2 sin β cos p − z λ ( β ) ∈ L 2 − ( T ) . (14)

Here C λ is the normalizing multiplicity.

d) The operator H λ ( π − 2 β ) has no embedded eigenvalues in the interval ( m ( β ) , M ( β ) ) .

Hilbert space L 12 − ( T 2 ) = L 2 − ( T ) ⊗ L 2 − ( T ) can be written as a direct sum:

L 2 − ( T ) ⊗ L 2 − ( T ) = L 2 − ( T ) ⊗ L − ( 1 ) ⊕ ( L 2 − ( T ) ⊗ L − ( 1 ) ) ⊥ .

The following lemma establishes a connection between the operators H 123 ( n ) ( π − 2 β , π ) and H λ ( k ) .

Lemma 6. Let the potential v ^ have the form (5). Then:

a) the subspace L 2 − ( T ) ⊗ L − ( 1 ) and its orthogonal complement ( L 2 − ( T ) ⊗ L − ( 1 ) ) ⊥ are invariant under H 123 ( n ) ( π − 2 β , π ) .

b) restriction of the operator H 123 ( n ) ( π − 2 β , π ) to the invariant subspace ( L 2 − ( T ) ⊗ L − ( 1 ) ) ⊥ coinsides with the unperturbed operator H 0 ( π − 2 β , π ) .

c) restriction of the operator H 123 ( n ) ( π − 2 β , π ) to the invariant subspace L 2 − ( T ) ⊗ L − ( 1 ) can be represented as a tensor product:

H 123 ( n ) ( π − 2 β , π ) = [ 4 I + H 0 ( π − 2 β ) − v ¯ ( n + 2 ) V 1 ] ⊗ I . (15)

Here, I is the identity operator, and H λ ( n ) ( π − 2 β ) : = H 0 ( π − 2 β ) − λ ( n ) V 1 , λ ( n ) = v ¯ ( n + 2 ) is a one-dimensional two-particle operator acting in L 2 − ( T ) by the formula (13).

This lemma is proved in the same way as the Lemma 4. In particular, part b) of the lemma implies that the operator H 123 ( n ) ( π − 2 β , π ) has no eigenfunctions in ( L 2 − ( T ) ⊗ L − ( 1 ) ) ⊥ . Thus, studying the eigenvalues of the operator H 123 ( n ) ( π − 2 β , π ) is reduced to studying eigenvalues of the operator H λ ( n ) ( π − 2 β ) = H 0 ( π − 2 β ) − λ ( n ) V 1 .

From Lemmas 5 - 6 and tensor product (15) implies the following statement regarding operator H 123 ( n ) ( π − 2 β , π ) .

Theorem 1. Let   β ∈ ( 0, π 2 ] and n ∈ ℕ .

a) If v ¯ ( n + 2 ) < sin β , then the operator H 123 ( n ) ( π − 2 β , π ) has no eigenvalues lying outside of the essential spectrum.

b) If v ¯ ( n + 2 ) = sin β , then the left edge m ( β ) of essential spectrum of the operator H 123 ( n ) ( π − 2 β , π ) is a resonance.

c) If v ¯ ( n + 2 ) > sin β , then the operator H 123 ( n ) ( π − 2 β , π ) has a unique nondegenerate eigenvalue

z 123 ( n ) ( π − 2 β , π ) = 4 + z λ ( n ) ( β ) = 6 − v ¯ ( n + 2 ) − 1 v ¯ ( n + 2 ) sin 2 β , (16)

which lies in the left of the essential spectrum and with the corresponding normalized eigenfunction

f λ ( n ) − − ( p 1 , p 2 ) = f λ ( n ) − ( p 1 ) sin p 2 π = f λ ( n ) − ( p 1 ) ψ 1 − ( p 2 ) ∈ L 2 − ( T ) ⊗ L − ( 1 ) ,

where f λ ( n ) − is the normalized eigenfunction of the operator H λ ( n ) ( π − 2 β ) corresponding to the eigenvalue z λ ( n ) ( β ) , the operator H λ ( n ) ( k ) is defined by the formula (13).

d) The operator H 123 ( n ) ( π − 2 β , π ) has no embedded eigenvalues in the interval ( m ( β ) , M ( β ) ) .

Similar statement is true for the operator H 123 ( n ) ( π , π − 2 β ) . The eigenvalues of the operators H 123 ( n ) ( π , π − 2 β ) and H 123 ( n ) ( π − 2 β , π ) are same, but eigenfunctions differ with variable replacement p 1 and p 2 . In other words, the operators H 123 ( n ) ( k 1 , k 2 ) and H 123 ( n ) ( k 2 , k 1 ) are unitary equivalent. Therefore, the operators H 123 n − ( k 1 , k 2 , π ) and H 123 n − ( k 2 , k 1 , π ) are unitary equivalent too.

Similar statement can relatively be formulated for the operator H 123 ( n ) ( π − 2 β , π − 2 β ) . For this purpose, we introduce the following notation. Through

Δ n ( β , z ) = 1 − v ¯ ( n + 2 ) π 2 ∫ T 2 sin 2 p 1 sin 2 p 2 d p 1 d p 2 2 + 2 ( 2 − sin β cos p 1 − sin β cos p 2 ) − z

we denote the Fredholm determinant of the operator I − v ¯ ( n + 2 ) V 11 r 0 ( β , z ) , where   r 0 ( β , z ) is the resolvent of the operator 2 I + H 0 ( π − 2 β , π − 2 β ) , and V 11 is an integral operator with the kernel

v ( p , q ) = 1 π 2 sin p 1 sin p 2 sin q 1 sin q 2 .

Through C 11 − − denote the value of the following integral:

C 11 − − = 1 π 2 ∫ T 2 sin 2 p 1 sin 2 p 2 d p 1 d p 2 2 ( 2 − cos p 1 − cos p 2 ) = ∫ T 2 | ψ 1 − ( p 1 ) | 2 | ψ 1 − ( p 2 ) | 2 d p 1 d p 2 2 ε ( p ) .

Simple calculations reveal the following approximate value C 11 − − ≈ 0.302347 .

Theorem 2. Let β ∈ ( 0, π 2 ] , n ∈ ℕ .

a) If   v ¯ ( n + 2 ) < sin β C 11 − − , then the operator H 123 ( n ) ( π − 2 β , π − 2 β ) has no eigenvalues lying outside of the essential spectrum.

b) If   v ¯ ( n + 2 ) = sin β C 11 − − , then the left edge m ( β ) = 6 − 4 sin β of the spectrum of the operator H 123 ( n ) ( π − 2 β , π − 2 β ) is an eigenvalue.

c) If   v ¯ ( n + 2 ) > sin β C 11 − − , then the operator H 123 ( n ) ( π − 2 β , π − 2 β ) has a unique nondegenerate eigenvalue z 123 ( n ) ( π − 2 β , π − 2 β ) below the essential spectrum.

d) The operator H 123 ( n ) ( π − 2 β , π − 2 β ) has no embedded eigenvalues in the interval ( m ( β ) , M ( β ) ) .

This theorem is proved in similar way as Lemma 5. There are some differences:

1) In the Theorem 2, the eigenvalue z 123 ( n ) ( π − 2 β , π − 2 β ) was calculated with the accuracy of β 2 :

z 123 ( n ) ( π − 2 β , π − 2 β ) = 6 − v ¯ ( n + 2 ) − 2 v ¯ ( n + 2 ) sin 2 β + O ( β 4 )

and corresponding normalized eigenfunction has the form

f 123 ( n ) ( p 1 , p 2 ) = C n ( β ) sin p 1 sin p 2 6 − 2 sin β cos p 1 − 2 sin β cos p 2 − z 123 ( n ) ( π − 2 β , π − 2 β ) ∈ L 12 − ( T 2 ) , (17)

where C n ( β ) is the normalizing multiplicity.

2) Left edge m ( β ) = 6 − 2 sin β of the essential spectrum is a resonance for the operator H 123 ( n ) ( π − 2 β , π ) , but for the operator H 123 ( n ) ( π − 2 β , π − 2 β ) the left edge m ( β ) = 6 − 4 sin β of the essential spectrum is the eigenvalue, i.e. the equation H 123 ( n ) ( π − 2 β , π − 2 β ) f = m ( β ) f has a non-trivial solution

f ( p 1 , p 2 ) = C sin p 1   sin p 2 2 − cos p 1 − cos p 2

and it belongs to L 12 − ( T 2 ) .

1) We have shown that the operator H 123 − ( k 1 , k 2 , π ) has infinitely many invariant subspaces ℜ 123 − ( n ) , n ∈ ℕ . It has been proved that if condition v ¯ ( n + 2 ) > sin β holds then the operator H 123 n − ( π − 2 β , π , π ) has a unique simple eigenvalue z 123 ( n ) ( π − 2 β , π ) of the form (16), otherwise, the operator has no eigenvalues outside of the essential spectrum. A similar statement holds for the operator H 123 n − ( π − 2 β , π − 2 β , π ) .

2) Without loss of generality it can be assumed that v ¯ ( 3 ) ≤ 1 . Since, if v ¯ ( 3 ) > 1 then it follows from l i m n → ∞ v ¯ ( n ) = 0 that there exists a number m ∈ ℕ such that v ¯ ( m + 2 ) ≤ 1 and monotonicity of v ¯ implies that v ¯ ( n ) > 1 for n = 3 ,   4 , ⋯ , m + 1 , and in this case, the eigenvalues z 123 ( n ) ( π − 2 β , π ) , n = 1,2, ⋯ , m − 1 of H 123 − ( π − 2 β , π , π ) exist for all β ∈ [ 0, π / 2 ] .

For a fixed β ∈ ( 0, π / 2 ] there exists m ∈ ℕ such that sin β ∈ ( v ¯ ( m + 3 ) , v ¯ ( m + 2 ) ) and the operator H 123 − ( π − 2 β , π , π ) has m nondegenerate eigenvalues outside of the essential spectrum (see Theorem 1):

z 123 ( 1 ) ( π − 2 β , π , π ) : = z 123 ( 1 ) ( π − 2 β , π ) = 6 − v ¯ ( 3 ) − 1 v ¯ ( 3 ) sin 2 β ,

z 123 ( 2 ) ( π − 2 β , π , π ) : = z 123 ( 2 ) ( π − 2 β , π ) = 6 − v ¯ ( 4 ) − 1 v ¯ ( 4 ) sin 2 β ,

⋮

z 123 ( m ) ( π − 2 β , π , π ) : = z 123 ( m ) ( π − 2 β , π ) = 6 − v ¯ ( m + 2 ) − 1 v ¯ ( m + 2 ) sin 2 β .

The corresponding normalized eigenfunctions are of the forms:

f 123 λ ( 1 ) − − − ( p 1 , p 2 , p 3 ) = f λ ( 1 ) − ( p 1 ) ψ 1 − ( p 2 ) ψ 1 − ( p 3 ) ∈ L 2 − ( T ) ⊗ L − ( 1 ) ⊗ L − ( 1 ) ,

f 123 λ ( 2 ) − − − ( p 1 , p 2 , p 3 ) = f λ ( 2 ) − ( p 1 ) ψ 1 − ( p 2 ) ψ 2 − ( p 3 ) ∈ L 2 − ( T ) ⊗ L − ( 1 ) ⊗ L − ( 2 ) ,

⋮

  f 123 λ ( m ) − − − ( p 1 , p 2 , p 3 ) = f λ ( m ) − ( p 1 ) ψ 1 − ( p 2 ) ψ m − ( p 3 ) ∈ L 2 − ( T ) ⊗ L − ( 1 ) ⊗ L − ( m ) ,

where, f λ ( m ) − is the normalized eigenfunction of the operator H λ ( m ) ( π − 2 β ) corresponding to the eigenvalue z λ ( m ) ( β ) and the operator H λ ( m ) ( k ) is defined by the formula (13), λ ( m ) = v ¯ ( m + 2 ) .

The eigenvalues of the operators H 123 − ( π − 2 β , π , π ) and H 123 − ( π , π − 2 β , π ) are same but eigenfunctions differ with variable replacement p 1 and p 2 . In other words, the operators H 123 − ( π − 2 β , π , π ) and H 123 − ( π , π − 2 β , π ) are unitary equivalent.

In the case sin β = v ¯ ( m + 2 ) , the left edge m ( β ) = 6 − 2 sin β of the essential spectrum is a resonance of the operator H 123 − ( π − 2 β , π , π ) (see Theorem 1).

3) Let for some m ∈ ℕ the relation sin β ∈ ( v ¯ ( m + 3 ) C 11 − − , v ¯ ( m + 2 ) C 11 − − ) hold then the operator H 123 − ( π − 2 β , π − 2 β , π ) has m nondegenerate eigenvalues outside the essential spectrum (see Theorem 2) and for small β :

z 123 ( 1 ) ( π − 2 β , π − 2 β , π ) : = z 123 ( 1 ) ( π − 2 β , π − 2 β ) = 6 − v ¯ ( 3 ) − 2 v ¯ ( 3 ) sin 2 β + O ( β 4 ) ,

z 123 ( 2 ) ( π − 2 β , π − 2 β , π ) : = z 123 ( 2 ) ( π − 2 β , π − 2 β ) = 6 − v ¯ ( 4 ) − 2 v ¯ ( 4 ) sin 2 β + O ( β 4 ) ,

⋮

z 123 ( m ) ( π − 2 β , π − 2 β , π ) : = z 123 ( m ) ( π − 2 β , π − 2 β ) = 6 − v ¯ ( m + 2 ) − 2 v ¯ ( m + 2 ) sin 2 β + O ( β 4 ) .

The corresponding normalized eigenfunctions are of the forms:

f 123 ( 1 ) − ( p 1 , p 2 , p 3 ) = f 123 ( 1 ) ( p 1 , p 2 ) ψ 1 − ( p 3 ) ∈ L 12 − ( T 2 ) ⊗ L − ( 1 ) ,

f 123 ( 2 ) − ( p 1 , p 2 , p 3 ) = f 123 ( 2 ) ( p 1 , p 2 ) ψ 2 − ( p 3 ) ∈ L 12 − ( T 2 ) ⊗ L − ( 2 ) ,

⋮

f 123 ( m ) − ( p 1 , p 2 , p 3 ) = f 123 ( m ) ( p 1 , p 2 ) ψ m − ( p 3 ) ∈ L 12 − ( T 2 ) ⊗ L − ( m ) ,

where, f 123 ( m ) is the normalized eigenfunction of the operator H 123 ( m ) ( π − 2 β , π − 2 β ) corresponding to the eigenvalue z 123 ( m ) ( π − 2 β , π − 2 β ) defined by the formula (17).

In the case sin β = v ¯ ( m + 2 ) C 11 − − , the left edge m ( β ) = 6 − 4 sin β of the essential spectrum is the eigenvalue of H 123 − ( π − 2 β , π − 2 β , π ) (see Theorem 2) with the corresponding eigenfunction

f ( p ) = C sin p 1 sin p 2 2 − cos p 1 − cos p 2 ⋅ sin m p 3 ∈ L 12 − ( T 2 ) ⊗ L − ( m ) .

Remark 1. If the potential v ^ is even in all arguments p 1 , p 2 , p 3 and the condition v ^ ∈ l 2 ( ℤ 3 ) holds, then the statements of Lemmas 3 - 4 remain valid.

Remark 2. If k 3 ≠ π , then the subspaces ℜ 123 − ( n ) , n ∈ ℕ are not invariant under the operator H 123 − ( k 1 , k 2 , k 3 ) .

This work was supported by the Grant OT-F4-66 of Fundamental Science Foundation of Uzbekistan.

The authors declare no conflicts of interest regarding the publication of this paper.

Abdullaev, J.I. and Toshturdiev, A.M. (2021) Bound States of a System of Two Fermions on Invariant Subspace. Journal of Modern Physics, 12, 35-49. https://doi.org/10.4236/jmp.2021.121004