On Absence of Threshold Resonances for Schrodinger and Dirac Operators

Using a unified approach employing a homogeneous Lippmann-Schwinger-type equation satisfied by resonance functions and basic facts on Riesz potentials, we discuss the absence of threshold resonances for Dirac and Schrodinger operators with sufficiently short-range interactions in general space dimensions. More specifically, assuming a sufficient power law decay of potentials, we derive the absence of zero-energy resonances for massless Dirac operators in space dimensions $n \geq 3$, the absence of resonances at $\pm m$ for massive Dirac operators (with mass $m>0$) in dimensions $n \geq 5$, and recall the well-known case of absence of zero-energy resonances for Schr\"odinger operators in dimension $n \geq 5$.


Introduction
Happy Birthday, Gisèle, we sincerely hope that our modest contribution to threshold resonances of Schrödinger and Dirac operators will create some joy.
The principal purpose of this paper is a systematic investigation of threshold resonances, more precisely, the absence of the latter, for massless and massive Dirac operators, and for Schrödinger operators in general space dimensions n ∈ N, n 2, given sufficiently fast decreasing interaction potentials at infinity (i.e., short-range interactions).
In the case that (−∞, E 0 ) and/or (E 1 , E 2 ), E j ∈ R, j = 0, 1, 2, are essential (resp., absolutely continuous) spectral gaps of a self-adjoint unbounded operator A in some complex, separable Hilbert space H, then typically, the numbers E 0 and/or the numbers E 1 , E 2 are called threshold energies of A. The intuition behind the term "threshold" being that if the coupling constant of an appropriate interaction potential is varied, then eigenvalues will eventually "emerge" out of the continuous spectrum of the operator A and enter the spectral gap (−∞, E 0 ) and/or (E 0 , E 1 ). The precise phenomenon behind this intuition is somewhat involved, depending, in particular, on the (possibly singular) behavior of the resolvent of A at the points E j , j = 0, 1, 2. In the concrete case of Dirac and Schrödinger operators with sufficient short-range interactions at hand, this behavior is well understood (see, e.g., [56], [57], [59] and the literature cited therein) and E 0 = 0 is then a threshold for Schrödinger operators h, with σ ess (h) = [0, ∞) (cf. (2.16)-(2.18)), whereas ±m are thresholds for massive Dirac operators H(m) corresponding to mass m > 0, with σ ess (H(m)) = (−∞, −m] ∪ [m, ∞) (cf. (2.9)-(2.11)). The case of massless Dirac operators H is more intricate as σ ess (H) = R (cf. (2.4), (2.6)), and hence H exhibits no spectral gap. However, since the potential coefficients tend to zero at infinity, 0 is still a threshold point of H that is known to possibly support an eigenvalue and/or a resonance.
Threshold resonances for Dirac and Schrödinger operators then are associated with distributional solutions ψ of hψ = 0, or Ψ ±m of H(m)Ψ ±m = ±mΨ ±m , respectively, distributional solutions Ψ of HΨ = 0, that do not belong to the domains of h, H(m), respectively H, but these functions are "close" to belonging to the respective operator domains in the sense that they (actually, their components) belong to L p (R n ) for p ∈ (p 0 , ∞) ∪ {∞} for appropriate p 0 > 2. More precisely, we then recall in our principal Section 3 that Schrödinger operators have no zeroenergy resonances in dimensions n 5 (a well-known result, cf. Remark 3.5 (iii)), that massless Dirac operators have no zero-energy resonances in dimension n 3 (the case n = 3 was well-known, cf. Remark 3.9 (iii)), and that massive Dirac operators have no resonances at ±m in dimensions n 5. We prove these results in a unified manner employing a homogeneous Lippmann-Schwinger-type equation satisfied by threshold resonance functions and the use of basic properties of Riesz potentials. While our power law decay assumptions of the potentials at infinity are not optimal, optimality was not the main motivation for writing this paper. Instead, we wanted to present a simple, yet unified, approach to the question of absence of threshold resonances with the added bonus that our results in the massless case for n 4 and in the massive case appear to be new. Since we explicitly permit matrix-valued potentials in the (massless and massive) Dirac case, we note that electric and magnetic potentials are included in our treatment (cf. Remark 2.2).
We also emphasize that threshold resonances and/or threshold eigenvalues of course profoundly influence the (singularity) behavior of resolvents at threshold energies, the threshold behavior of (fixed energy, or on-shell) scattering matrices, as well as dispersive (i.e., large time) estimates for wave functions. These issues have received an abundance of attention since the late 1970's in connection with Schrödinger operators, and very recently especially in the context of (massless and massive) Dirac operators (the case of massless Dirac operators having applications to graphene). While a complete bibliography in this connection is clearly beyond the scope of our paper, we refer to [2]- [11], [12,Ch. 4], [13], [15]- [21], [28]- [43], [45], [46], [50]- [54], [58]- [62], [65]- [73], [76]- [78], and the literature therein to demonstrate some of the interest generated by this circle of ideas. We also mention the relevance of threshold states in connection with the Witten index for non-Fredholm operators, see, for instance, [22]- [26].
In Section 2 we present the necessary background for Dirac and Schrödinger operators, including a description of Green's functions (resp., matrices) and their threshold asymptotics in the free case, that is, in the absence of interaction potentials in terms of appropriate Hankel functions. Appendix A discusses the failure of our technique to produce essential boundedness (of components) of threshold resonances and eigenfunctions for space dimensions sufficiently large.
We conclude this introduction with some comments on the notation employed in this paper: If H denotes a separable, complex Hilbert space, I H represents the identity operator in H, but we simplify this to I N in the finite-dimensional case C N , N ∈ N.
If T is a linear operator mapping (a subspace of) a Hilbert space into another, then dom(T ) and ker(T ) denote the domain and kernel (i.e., null space) of T . The spectrum, point spectrum (the set of eigenvalues), and the essential spectrum of a closed linear operator in H will be denoted by σ( · ), σ p ( · ), and σ ess ( · ), respectively. Similarly, the absolutely continuous and singularly continuous spectrum of a selfadjoint operator in H are denoted by σ ac ( · ) and σ sc ( · ).
The Banach space of bounded linear operators on a separable complex Hilbert space H is denoted by B(H).
If Lebesgue measure is understood, we simply write L p (M ), M ⊆ R n measurable, n ∈ N, instead of the more elaborate notation L p (M ; d n x). ⌊ · ⌋ denotes the floor function on R, that is, ⌊x⌋ characterizes the largest integer less than or equal to x ∈ R. Finally, if x = (x 1 , . . . , x n ) ∈ R n , n ∈ N, then we abbreviate x := (1 + |x| 2 ) 1/2 .

Some Background Material
This preparatory section is primarily devoted to various results on Green's functions for the free Schrödinger operator (i.e., minus the Laplacian) and free (massive and massless) Dirac operators (i.e., in the absence of interaction potentials) in dimensions n ∈ N, n 2.
To rigorously define the case of free massless n-dimensional Dirac operators, n 2, we now introduce the following set of basic hypotheses assumed for the remainder of this manuscript.

free massless Dirac operator
Under these assumptions on V , the massless Dirac operator H in [L 2 (R n )] N is defined via We recall that Then H 0 and H are self-adjoint in [L 2 (R n )] N , with essential spectrum covering the entire real line, σ ess (H) = σ ess (H 0 ) = σ(H 0 ) = R, (2.6) a consequence of relative compactness of V with respect to H 0 . In addition, We also recall that the massive free Dirac operator in [L 2 (R n )] N associated with the mass parameter m > 0 is of the form (with β = α n+1 ) (2.8) and the corresponding interacting massive Dirac operator in [L 2 (R n )] N is given by In this case, (2.10) and (2.11) In the special one-dimensional case n = 1, one can choose for α 1 either a real constant or one of the three Pauli matrices. Similarly, in the massive case, β would typically be a second Pauli matrix (different from α 1 ). For simplicity we confine ourselves to n ∈ N, n 2, in the following.
In connection with the free massive Dirac operator H 0 (m) = H 0 + m β, m > 0, one computes, (2.25) Assuming m > 0, z ∈ C\(R\[−m, m]), Im z 2 − m 2 1/2 > 0, x, y ∈ R n , x = y, n ∈ N, n 2, (2.26) and exploiting (2.24), one thus obtains for the Green's function G 0 (m, z; · , · ) of H 0 (m), Here we employed the identity ([1, p. 361]), We also recall the asymptotic behavior (cf. [1, p. 360], [53, p. 723-724]) (2.29) |x − y| n , m > 0, x, y ∈ R n , x = y, n ∈ N, n 3, (Here the remainder term O z 2 − m 2 ln z 2 − m 2 depends on x, y ∈ R 2 , but this is of no concern at this point.) In particular, G 0 (m, z; · , · ) blows up logarithmically as z → ±m in dimensions n = 2, just as g 0 (z, · , · ) does as z → 0. By contrast, the massless case is quite different and assuming z ∈ C + , x, y ∈ R n , x = y, n ∈ N, n 2, (2.34) one computes in the case m = 0 for the Green's function G 0 (z; · , · ) of H 0 , The Green's function G 0 (z; · , · ) of H 0 continuously extends to z ∈ C + . In addition, in the massless case m = 0, the limit z → 0 exists 1 , lim z→0, z∈C+\{0} |x − y| n , x, y ∈ R n , x = y, n ∈ N, n 2, (2.36) and no blow up occurs for all n ∈ N, n 2. This observation is consistent with the sufficient condition for the Dirac operator H = H 0 + V (in dimensions n ∈ N, n 2), with V an appropriate self-adjoint N × N matrix-valued potential, having no eigenvalues, as derived in [54, The following remark is primarily of a heuristic nature; at this point it just serves as a motivation for our detailed discussion of absence of threshold resonances for Schrödinger and Dirac operators in Section 3.
Lemma 2.5. Let n ∈ N and x 1 , x 2 ∈ R n . If k, ℓ ∈ [0, n), ε, β ∈ (0, ∞), with k + ℓ + β n, and k + ℓ = n, then We conclude this section by recalling the following interesting results of McOwen [61] and Nirenberg-Walker [64], which provide necessary and sufficient conditions for the boundedness of certain classes of integral operators in L p (R n ): is bounded if and only if c < n/p and d < n/p ′ .
is bounded if and only if c < n/p and d < n/p ′ .

Nonexistence of Threshold Resonances
In this section we prove our principal results on nonexistence of threshold resonances in three cases: First, in the case of Schrödinger operators in dimension n 3; second, in the case of massless Dirac operators in dimensions n 2; and third, in the case of massive Dirac operators in dimensions n 3.
Schrödinger Operators in R n , n 3. We begin with the case of Schrödinger operators h in dimension n 3 as defined in (2.15), and start by making the following assumptions on the potential v.
Here we abbreviated We continue with the threshold behavior, that is, the z = 0 behavior, of h: (iii) 0 is called a regular point for h if it is neither a zero-energy eigenvalue nor a zero-energy resonance of h.
Additional properties of ψ are isolated in Theorem 3.3. While the point 0 being regular for h is the generic situation, zero-energy eigenvalues and/or resonances are exceptional cases.
Next, we introduce the following convenient abbreviation (for x, y ∈ R n , x = y): in the sense of distributions.
In the remainder of this section we will frequently apply [44, Prop. 6.10], that is, the fact that 0 < p < q < r ∞ implies L p ∩ L r ⊂ L q .

58) and
59) The inequality in (3.57) implies ψ ∈ L ∞ (R n ) for 5 n 7. In Remark A.1 we will illustrate why the same line of reasoning fails for n 8.
(ii) In physical notation (see, e.g., [63, footnote 3 on p. 300] for details), the zeroenergy resonances in Cases (II) and (IV ) for n = 3, 4, are s-wave resonances (i.e., corresponding to angular momentum zero) in the case where V is spherically symmetric (see also the discussion in [59]).
(iii) As mentioned in Remark 2.3, the absence of zero-energy resonances is wellknown in dimensions n 5, see [50].

68)
and if there exists Additional properties of Ψ are isolated in Theorem 3.8. While the point 0 being regular for H is the generic situation, zero-energy eigenvalues and/or resonances are exceptional cases.
Remark 3.9. (i) For basics on the Birman-Schwinger principle in the concrete case of massless Dirac operators see [32], [33].
(ii) In physical notation, the zero-energy resonances in Cases (II) and (IV ) for n = 2 correspond to eigenvalues ±1/2 of the spin-orbit operator (cf. the operator S in [54], [55]) when V is spherically symmetric, see the discussion in [33].
(iii) As mentioned in Remark 2.3, the absence of zero-energy resonances is wellknown in the three-dimensional case n = 3, see [7], [12,Sect. 4.4], [13], [16], [67], [68], [78]. In fact, for n = 3 the absence of zero-energy resonances has been shown under the weaker decay |V j,k | C x −1−ε , x ∈ R 3 , in [7]. The absence of zeroenergy resonances for massless Dirac operators in dimensions n 4 as contained in Theorem 3.8 (ii) appears to have gone unnoticed in the literature. ⋄ Massive Dirac Operators in R n , n 3. Finally, we turn to the case of massive Dirac operators H(m), m > 0, in dimension n 3 as defined in (2.8), and start by making the following assumptions on the matrix-valued potential V .
Hypothesis 3.10. Let n ∈ N, n 3. Assume the a.e. self-adjoint matrix-valued potential V = {V j,k } 1 j,k N satisfies for some C ∈ (0, ∞), In addition, alluding to the polar decomposition of V ( · ) (i.e., V ( · ) = U V ( · )|V ( · )|) in the following symmetrized form (cf. [49]), we suppose that and if there exists such that Ψ ±m defined by is called a regular point for H(m) if it is neither a threshold eigenvalue nor a threshold resonance of H(m).
Additional properties of Ψ ±m are isolated in Theorem 3.8. While the points ±m being regular for H(m) is the generic situation, threshold eigenvalues and/or resonances at energies ±m are exceptional cases.
then also and hence V * Hence, In addition, Ψ ±m ∈ [L 2 (R n )] N , and the same arguments as those in (3.126)-(3.127) and (3.138)-(3.158) (which now extend to include n ∈ {3, 4}, due to the assumption It follows that otherwise, ±m is an eigenvalue of H 0 (m) and Ψ ±m is a corresponding eigenfunction. However, this contradicts the fact that the spectrum of H 0 (m) is purely absolutely continuous. Hence, Thus, Φ ±m = 0, and that is, This concludes the proof.
(iii) The absence of threshold resonances for massive Dirac operators in dimensions n 5 as contained in Theorem 3.12 (ii) appears to have gone unnoticed in the literature. ⋄ Appendix A. Some Remarks on L ∞ (R n )-Properties of Threshold Eigenfunctions In this appendix we collect some negative results on L ∞ (R n )-properties of threshold eigenfunctions. We hope to return to this issue at a later date.