The spectrum of non-local discrete Schrödinger operators with a δ -potential

The behaviour of the spectral edges (embedded eigenvalues and resonances) is discussed at the two ends of the continuous spectrum of non-local discrete Schrödinger operators with a δ -potential. These operators arise by replacing the discrete Laplacian by a strictly increasing C 1 -function of the discrete Laplacian. The dependence of the results on this function and the lattice dimension are explicitly derived. It is found that while in the case of the discrete Schrödinger operator these behaviours are the same no matter which end of the continuous spectrum is considered, an asymmetry occurs for the non-local cases. A classification with respect to the spectral edge behaviour is also offered.


Non-local discrete Schrödinger operators on lattice
The spectrum of discrete Schrödinger operators has been widely studied for both combinatorial Laplacians and quantum graphs; for some recent summaries see [3,4,7,9,11,16,19] and the references therein. Specifically, eigenvalue behaviours of discrete Schrödinger operators on l 2 Z d are discussed in e.g. [2,8,10,12,13,15]. However, for discrete non-local Schrödinger operators only few results are known. Typical examples include discrete fractional Schrödinger operators.
In this paper we define generalized discrete Schrödinger operators which include discrete fractional Schrödinger operators and others whose counterparts on L 2 R d are currently much studied [5,6,17,18]. In [14] we have introduced a class of generalized Schrödinger operators whose kinetic term is given by so called Bernstein functions of the Laplacian. These operators are non-local and via a Feynman-Kac representation generate subordinate Brownian motion killed at a rate given by the potential. Their discrete counterparts studied in this paper also have a probabilistic interpretation in that they generate continuous time random walks with jumps on Z d . In the present paper we consider a class of Schrödinger operators obtained as a strictly increasing C 1 -function of the discrete Laplacian and a δ-potential. This includes, in particular, Bernstein functions (see below) of the discrete Laplacian. In the presence of a δ-potential the above probabilistic picture then describes free motion with a "bump" which can be interpreted as an impurity in space. Our aim here is to investigate the spectrum of such operators, specifically, embedded eigenvalues and resonances at the edges of the continuous spectrum.
Let d ≥ 1 and L be the standard discrete Laplacian on l 2 Z d defined by We give a remark on the definition of Laplacians for the reader's convenience. In the previous paper [15] we defined the discrete Laplacian by L 0 ψ(x) = 1 2d |x−y|=1 ψ(y), and found that the spectrum of L 0 is the closed interval [−1, 1]. In this paper we define the negative Laplacian by ψ(x) → 1 2d |x−y|=1 (ψ(y) − ψ(x)), and flipping the signature, we define the positive Laplacian (1.1). Thus the spectrum of L is positive, i.e., σ (L) = [0, 2]. Hence we can consider the Bernstein functions of L. Also, let V (x) = vδ x,0 be δpotential with mass v concentrated on x = 0, i.e., V ψ(x) = 0 for x = 0 and V ψ(x) = vψ(x). Then the operator The scalar product on H is denoted by Then the discrete Laplacian L transforms as where i.e., the right hand side above is a multiplication operator on H . In this paper we use a non-local discrete Laplacian (L) defined for a suitable function by applying Fourier transform.

Definition 1. For a given
Also, we call non-local discrete Schrödinger operator with δ-potential.
An example of such a function is (u) = u α/2 , 0 < α < 2, which describes a discrete Laplacian of fractional order α/2. Other specific choices will be given in Example 1 below.
Under Fourier transform (1.7) is mapped into (2)]. In what follows we consider the spectrum of H v instead of h. Note that the map → v( , ) is a rank-one operator, and thus the continuous spectrum of the rank-one perturbation H v of L is [ (0), (2)], for every v ∈ R. See e.g. [1,20] for rank-one perturbations.

( * )-resonances and ( * )-modes
As it will be seen below, for a sufficiently large value is an eigenvalue of H v 0 , we call the eigenvector associated with (0) a (0)-mode.
Similarly, for a sufficiently large v > 0 it will be seen that there exists an eigenvalue E + (v) strictly larger than (2). (2) is an eigenvalue of H v 2 , we call the eigenvector associated with (2) a (2)-mode, and a (2)-resonance whenever (2) is not an eigenvalue of For the discrete Schrödinger operator L+V these modes and resonances were studied in e.g. [15], in particular, their dependence on the dimension d. For d = 1, 2, there is no 0-mode, 2-mode, 0-resonance or 2-resonance, for d = 3, 4 there are 0 and 2-resonances, and for d ≥ 5 there are 0 and 2-modes. This shows that the eigenvalue behaviour at both edges (0 and 2) is the same. See Table 1.
As it will be seen below, for the case of the fractional Laplacian we have the remarkable fact that the edge behaviours are in general different at the two sides. See Table 2 and note that σ √ L = 0, √ 2 .

A criterion for determining the eigenvalues
Consider the eigenvalue equation We introduce the functions: 3) The following result gives an integral test to spot the eigenvalues of H v .

Lemma 1. E is an eigenvalue of H v for a given v if and
(2.4) Table 1 Modes and resonances of L + V 2-mode 2-res. 0-mode 0-res.  Proof. To show the necessity part, suppose that E is an eigenvalue and an associated eigenvector. Assuming (L(θ)) has no point spectrum, this is a contradiction. This gives ( , For the sufficiency part, suppose now that with a chosen c. It is straightforward to see that satisfies holds. By J(E) = 0 it follows that there exists v such that (2.5) is satisfied, hence E is an eigenvalue of H v .
In order to investigate ( * )-resonances and ( * )modes we use Lemma 1 and estimate the two integrals I(E) and J(E) at the two ends E = ( * ) of the interval [ (0), (2)]. (2)]. Then there exists v = 0 such that E is an eigenvalue of H v .

The location of eigenvalues
Proof. In this case it is easily seen that I(E) < ∞ and J(E) = 0. Then E is an eigenvalue and v is given by (2.4).
Proof. Due to monotonicity of , there is a unique x ∈ (0, 2) such that (E) = (x). Thus It is directly seen that the right hand side diverges, and thus the lemma follows.
Next consider the cases E = (2) and E = (0). For a systematic discussion of the eigenvalue behaviour of H v we introduce the following concept.

Definition 2.
We say that is of (a, b)-type whenever Lemma 4. Let be of (a, b)-type. Then we have the following behaviour. Proof. Since is strictly increasing, the first statement follows directly.
Let be of (a, b)-type. Then we have at θ ≈ (0, . . . , 0), Hence and similarly Thus the lemma follows for E = (2). For the case of E = (0) the proof is similar.
From these lemmas we can derive the spectral edge behaviour of H v . The next theorem is the main result in this paper. http://www.pacific-mathforindustry.com/content/6/1/7 Theorem 1. Assume that is of (a, b)-type. Let v 2 = (2π) d /J( (2)) > 0 (2.8) The spectral edge behaviour of H v is as follows.
(1) Suppose that v > 0. Then the following cases occur: (2) Suppose that v < 0. Then the following cases occur: Proof. Consider the case v > 0 and let d < 1 + 2b. Then for all E > (2) we have I(E) < ∞ and J(E) = 0. Thus there exists v such that E is an eigenvalue of H v .
Let 1 + 2b ≤ d < 1 + 4b. Then for all E > (2) we have that I(E) < ∞ and J(E) = 0. Thus E is an eigenvalue of is also an eigenvalue. The cases for v < 0 can be dealt with similarly.

Remark 1. Note that in general
Remark 2. From the above it is seen that the spectral edge behaviour of H v depends on the dimension d as well as on the parameters a and b, and the result is different according to which edge is considered. For a summary see Table 3.
It is worthwhile to see the implications more closely for some specific choices of function .

A classification of spectral edge behaviour
The functions of the discrete Laplacian can be classified according to the behaviour of the eigenvalues at the two ends of the interval [ (0), (2)].

Normal type
Let be of normal type. In this case the spectral edge behaviour is the same as that of the discrete Schrödinger operator L + V . The following result has been obtained in [15].

Proposition 1.
Let be normal type. We have the following cases. Thus the spectral edge behaviour for positive and negative v is qualitatively the same, and the details only depend on the dimension d.

Fractional type
In the fractional type case we have the following spectral edge behaviour.

Theorem 2. Let
be of fractional type. The following cases occur.
(1) If v > 0, then the spectral edge behaviour is the same as for normal type with v > 0. from each other, in contrast with the normal type case.

The case of α = 1
For α = 1 the spectral edge behaviour of H v = √ L + V is displayed for dimensions d = 1, . . . , 4 and d ≥ 5 in Table 2. We have displayed the specific situations in Figures 1, 2, 3 and 4 below, where ⊕ denotes a resonance, • an eigenvalue, and × denotes a value which is not an eigenvalue. For dimension d = 1, 5 the edge behaviours at 0 and √ 2 are symmetric. See Figures 1 and 4. On the other hand for dimensions d = 2, 3, 4 the edge behaviours at 0 and √ 2 are again different. See Figures 2 and 3.