Abstract
A family , , of the Friedrichs models with the perturbation of rank one, associated to a system of two particles, moving on the two-dimensional lattice is considered. The existence or absence of the unique eigenvalue of the operator lying below threshold depending on the values of and is proved. The analyticity of corresponding eigenfunction is shown.
1. Introduction
In celebrated work [1] of Simon and Klaus it has been considered a family of the SchrΓΆdinger operators and, a situation where as tends to some eigenvalue tends to , that is, as an eigenvalue is absorbed into continuous spectrum, and conversely, for any continuous spectrum gives birth to a new eigenvalue. This phenomenon in [1] is called coupling constant threshold.
In [2] the Hamiltonian of a system of two identical quantum mechanical particles (bosons) moving on the -dimensional lattice and interacting via zero-range repulsive pair potentials has been considered. For the associated two-particle SchrΓΆdinger operator the existence of coupling constant threshold has been proved: the operator has none eigenvalue for any , but for each it has a unique eigenvalue above the upper threshold of the spectrum.
Note that in [1] the existence of a coupling constant threshold has been assumed, at the same time in [2] the coupling constant threshold has been definitely found by the given data of the Hamiltonian.
We remark that for the Hamiltonians of a system of two arbitrary particles moving on or the coupling constant threshold vanishes, if and the coupling constant threshold is positive, if .
Notice also that for the Hamiltonians of a system of two identical particles moving on or the coupling constant threshold vanishes, if particles are bosons and the coupling constant threshold is positive, if particles are fermions.
In [3] for a wide class of the two-particle SchrΓΆdinger operators on the -dimensional lattice , , being the two-particle quasimomentum, it has been proved that if the following two assumptions (i) and (ii) are satisfied, then for all , the discrete spectrum of below its threshold is nonempty. The assumptions are (i) the two-particle SchrΓΆdinger operator , corresponding to the zero value of the quasimomentum , has a coupling constant threshold and (ii) the one-particle free Hamiltonians in the coordinate representation generate positivity preserving semigroups.
In [4] a family of the Friedrichs models with perturbation of rank one associated to a system of two particles on the three-dimensional lattice has been considered. In some special case of the multiplication operator and under the assumption that the operator has a coupling constant threshold , the existence of a unique eigenvalue, below the threshold of the spectrum of for all nontrivial values of , has been proved.
In the present paper, a family of the Friedrichs models , , where is a -neighborhood of the point with perturbation of rank one associated to a system of two particles on the two-dimensional lattice interacting via pair local potentials, is considered and the following results have been obtained.(i)If the parameters of the Friedrichs model satisfy some conditions (see Theorem 2.3), then there exists a coupling constant threshold for any the operator has none eigenvalue; at the same time for any it has a unique eigenvalue , lying below its threshold of the spectrum. Moreover an explicit expression for the corresponding eigenfunction is found and its analyticity is proven.(ii)If the parameters of the Friedrichs model do not satisfy conditions mentioned in (i), then the operator has none positive coupling constant threshold, that is, for any the operator has a unique eigenvalue , lying below its threshold of the spectrum.(iii)A criterion for being the threshold of the spectrum of a virtual level of the operator is proven.
Note that the generalized Friedrichs models appear in the problems of quantum mechanics [5], solid state physics [6], and quantum field theory [7, 8] and the existence of its eigenvalues and resonances have been studied in [9, 10].
In [11] a special family of generalized Friedrichs models has been considered and the existence of eigenvalues for some values of quasimomentum of the system, lying in a neighborhood of some , has been proved.
2. Notions and Assumptions: The Main Results
Let be the one-dimensional hypercubic lattice and be the two-dimensional torus, the dual group of (Brillion zone). Note that operations addition and multiplication by number of the elements of torus is defined as operations in by the module .
Let be the Hilbert space of square-integrable functions defined on the torus and be one-dimensional complex Hilbert space.
We consider the family of generalized Friedrichs model acting in as follows: Here where is inner product in and is the multiplication operator by function , that is, Note that for any and the equality holds.
The following assumption will be needed throughout the paper.
Assumption 2.1. The following conditions are satisfied: (i)the function is nontrivial real-analytic on ;(ii)the function is real-analytic on and has a unique nondegenerated minimum at .
The perturbation is positive operator of rank one. Consequently, by well-known Weyl's theorem [12], the essential spectrum of fills the following segment on the real axis: where
By Assumption 2.1 there exist such -neighborhood of the point and analytic vector function that for any the point is a unique nondegenerated minimum of the function (see Lemma 3.2).
Moreover, in the case the following integral exists (see Lemma 3.4) and we introduce a parameter as If , then we define as .
Definition 2.2. The number is called a virtual level of the operator , if the equation has a nonzero solution , where is the Banach space of integrable functions on . The corresponding solution is called a virtual state of the operator .
In the following theorem we assert that for any there exists a unique eigenvalue , lying below the essential spectrum, of the operator , but for the operator has none eigenvalue outside the essential spectrum. It is proved that for fixed , the function is analytic in .
Moreover, this theorem provides a criterion, for being the bottom of the essential spectrum of , a virtual level of the operator .
Theorem 2.3. Let Assumption 2.1 holds and . Then the following assertions are true. (i)If , then the operator has a unique eigenvalue , lying below the essential spectrum of . The function is monotonously decreasing real-analytic function in the interval and the function is real-analytic in . The corresponding eigenfunction is analytic on , where is a normalizing constant. Moreover, the mappings are vector-valued analytic functions in and , respectively.(ii)If and , then the operator has none eigenvalue in .(iii)If ,β and , then the number is a virtual level of the operator and the corresponding virtual state is of the form: where is a normalizing constant.(iv)If ,β and , then the number is an eigenvalue of the operator and the corresponding eigenfunction is of the form where is a normalizing constant.
Remark 2.4. Notice that if , then . So, in this case the number is neither a virtual level nor an eigenvalue for the operator .
Remark 2.5. From the positivity of it follows that the operator has none eigenvalue lying above .
3. Proof of the Results
We postpone the proof of the theorem after several lemmas and remarks.
For any and we define in an analytic function (the Fredholm determinant , associated to the operator ) as where
Lemma 3.1. For any and the number is an eigenvalue of the operator if and only if, when The corresponding eigenfunction is analytic on , where is a normalizing constant [4].
Lemma 3.2. Let Assumption 2.1 holds. Then there exist such a -neighborhood of the point and analytic function that for any the point is a unique non degenerated minimum of .
Proof. By Assumption 2.1, the square matrix is positively defined and . Then by the implicit function theorem in analytic case there exist a -neighborhood of and a unique analytic vector function such that and Hence for any the point is a unique non degenerated minimum of the function .
Remark 3.3. We note that by the parametrical Morse lemma for any there exists a map of the sphere with radius and center at to a neighborhood of the point that in the function can be represented as Here the function (resp., ) is analytic in (resp., ) and . Moreover, the Jacobian of the mapping is analytic in and positive, that is for all and for all .
Lemma 3.4. Let Assumption 2.1 holds. Then the integral exists and defines an analytic function in .
Proof. We represent the function
as
where
and is a neighborhood of .
Observe that by Assumption 2.1 for any the function is analytic at the point .
According to Remark 3.3 in the integral for a change of variables implies
where is the Jacobian of the mapping .
Passing to spherical coordinates as , we obtain
where is a unit sphere in and its element. Inner integral can be represented as
where the Pizetti coefficients are analytic in [13].
Thus we have that
From (3.15) it follows that the following limit exists
and, consequently,
where . Observe that the functions in the right hand side of (3.17) are analytic in . So, the function is analytic in .
Proposition 3.5. For the following equalities hold: where is real for and is an analytic function in a neighborhood of the origin [14].
In the following lemma we establish an expansion of in a half neighborhood of the point .
Lemma 3.6. Assume Assumption 2.1. Then for any ,β and sufficiently small the function can be represented as the following convergent series: where are real numbers.
Proof. The function can be written as a sum of the following functions:
where
and is a neighborhood of the point .
Since for any , the function is analytic at . According to Lemma 3.4 the function is analytic in .
A change of variables in the integral yields
Passing to spherical coordinates by we obtain
and hence
where is unit sphere in . Since
where ,β are the Pizetti coefficients, we get
where . Using Proposition 3.5 we have
where and . Using relations (3.27) and (3.21) and putting (3.26) in (3.20) we get required relation (3.19).
Lemma 3.7. Let Assumption 2.1 hold. Then for any consider (i)if , then (ii)if , then .
Proof. We consider the following integral:
where is a continuous function on and . By Lemma 3.2 for any the function has a unique non degenerated minimum at . Then there exist a neighborhood of the point and positive number that
We represent the function as a sum of two functions:
where
From (3.30) we get that . In the integral for making a change of variables one obtains
where is the Jacobian of the mapping .
(i) Let ,β. Then from (3.33) we get
Passing to spherical coordinates by we get
Expanding the function to the Taylor series at we obtain
where is continuous in and . By condition of part (i) of this lemma and from equality (3.36) it follows that (3.35) has the following form:
Since the function is continuous in , we have . Taking into account , from (3.31) we get that .
Now we show that if the conditions of part (i) of Lemma 3.7 are not satisfied, that is, or , then the function defined by (3.28) does not belong to .
Let and . We will show that
Assume the converse, let
where and . Since the function are linearly independent, the last equality holds if and only if, when
Observe that . Consequently, the equalities (3.40) hold if and only if, when . This contradicts the fact that . Thus, . Hence the equality (3.35) has the form
Since
by the theorem on comparison of improper integrals, we get that and therefore.
In case of the relation can be proven analogously.
(ii) Let . Then from (3.33) we get
Passing to spherical coordinates by we obtain
By the condition of part (ii) of Lemma 3.7 and from (3.36) we get
Since the function under the integral sign is continuous in , it follows that . Thus . Taking into account we obtain . Consequently, from part (i) of Lemma 3.7 it follows that .
Lemma 3.8. Let the point a unique non degenerated minimum of the function , and . Then for any the equation has a nonzero solution if and only if In this case the nonzero function is a solution of (3.46), where a normalizing constant.
This lemma can be proved as Lemma 3.1 taking into account part (ii) of Lemma 3.7.
Now we prove the main results.
Proof of Theorem 2.3. (i) Observe that is continuous and monotonously decreasing in . Moreover,
By definition, if , then . So, for , Lemma 3.6 gives that
Analogously, if , then for the inequality
holds.
The continuity of function in yields that the equation has a unique solution and hence Lemma 3.1 yields that the operator ,β has a unique eigenvalue .
Since for any and the number is a solution of the equation and the function (resp., ) is real analytic in (resp., ), the implicit function theorem implies that (resp., ) is real analytic in (resp., ).
Note that the function monotonously decreases in and hence for any the eigenvalues and satisfy the relations:
Using the monotonicity of the function in we obtain that , that is, is monotonously decreases in .
Lemma 3.1 implies that if for any and the number is an eigenvalue of the operator ,β, then the function
is a solution of the equation
where is a a normalizing constant.
Analyticity of the function follows from the analyticity of the functions and in and the representation (3.53). The functions and are analytic in and for any the inequality holds, therefore representation (3.53) yields that the mapping is analytic in . Analogously the analyticity of the function ) implies that the mapping is analytic in .
(ii) Let and . Since
we have and Lemma 3.1 yields that the operator ,β does not have any eigenvalue in .
The statements (iii) and (iv) of Theorem 2.3 follows from Lemmas 3.7 and 3.8.
Acknowledgments
The first author wishes to thank the Mathematics Department of Faculty of Computer Science and Mathematics, MARA University of Technology (Malaysia), where the paper has been finished, for the invitation and hospitality. Authors are grateful to the referee(s) for useful remarks.