Estimates on Complex Eigenvalues for Dirac Operators on the Half-Line

We derive bounds on the location of non-embedded eigenvalues of Dirac operators on the half-line with non-Hermitian L1-potentials. The results are sharp in the non-relativistic or weak-coupling limit. In the massless case, the absence of discrete spectrum is proved under a smallness assumption.


Introduction
The aim of this paper is to obtain estimates for eigenvalues of the Dirac operator on L 2 (R + , C 2 ) subject to separated boundary conditions at zero, ψ 1 (0) cos(α) − ψ 2 (0) sin(α) = 0, α ∈ [0, π/2], 1 (1.2) and perturbed by a matrix-valued (not necessarily Hermitian) potential V ∈ L 1 (R + , Mat(2, C)), The author gratefully acknowledges the support of Schweizerischer Nationalfonds, SNF, through the postdoc stipend PBBEP2 136596. He would also like to thank the Institut Mittag-Leffler for the kind hospitality within the RIP (Research in Peace) programme 2013, during which part of this manuscript was written. Special thanks go to Ari Laptev for useful discussions. Finally, the author thanks an anonymous referee for helpful comments. IEOT where the norm in the integral is the operator norm in C 2 (with respect to the Euclidean norm on C 2 ). Here, we are only concerned with eigenvalues that are not embedded in the spectrum of D 0 , σ(D 0 ) = (−∞, −mc 2 ] ∪ [mc 2 , ∞).
For the purpose of investigating the non-relativistic limit, we have made the dependence on c (the speed of light) explicit, whereas the reduced Planck constant is set to unity. This work is a continuation of [2] where corresponding eigenvalue estimates for Dirac operators on the whole line were established. More precisely, it was shown there that if v := V 1 /c < 1, then any eigenvalue z ∈ C\σ(D 0 ) of D 0 + V is contained in the union of two closed disks in the left and right half plane with centres ±mc 2 x 0 and radii mc 2 r 0 , where x 0 and r 0 depend non-linearly on v and diverge as v → ∞ in such a way that the disks cover the entire complex plane minus the imaginary axis. In the non-relativistic limit (c → ∞), the Dirac operator D 0 + V − mc 2 converges to the Schrödinger operator − 1 2m d 2 dx 2 + V (say, for V a multiple of the identity matrix) in the norm-resolvent sense, and the spectral estimate reduces to the bound in [1]: Similar estimates for Schrödinger operators on the half-line were established in [4]: Any eigenvalue λ ∈ C\[0, ∞) of − d 2 / dx 2 +V , with boundary condition ψ (0) = σψ(0), σ ≥ 0, satisfies (1.3) if the constant 1/2 is replaced by 1; in the case of Dirichlet boundary conditions ψ(0) = 0, the sharp estimate The aim of this note is to obtain corresponding results for the Dirac operator on the half-line. As in [2], an interesting distinction between the massive (m = 0) and the massless (m = 0) Dirac operator occurs: The former behaves like the Schrödinger operator in the non-relativistic limit c → ∞, while the latter (m = 0 may be regarded as the "ultra-relativistic" limit) has no complex eigenvalues for sufficiently small L 1 -norm of the potential (see [2] for the case of the whole line and Theorem 2.1 for the half-line case). This fact may be expressed by saying that the whole spectrum (which is R in this case) is non-resonant. This is quite remarkable, considering that the point zero is resonant for the (scalar) relativistic operator |p| on the real line, Vol. 79 (2014) Estimates on Complex Eigenvalues for Dirac operators on the half-line 379 i.e. there are eigenvalues for arbitrarily small perturbations, even for realvalued potentials. In that case, the absence of eigenvalues for small L 1 -norm of the potential would be equivalent (by the variational characterization of eigenvalues and Hölder's inequality) 2 to the boundedness of the resolvent of |p| from L 1 to L ∞ , which in turn would be equivalent to the boundedness of the Fourier transform of its symbol; however, the Fourier transform of p.v. 1 |p| diverges logarithmically. In contrast, the symbol of the resolvent of the Dirac operator on the line behaves like p.v. 1 p (the Hilbert transform), which has a bounded Fourier transform due to cancellations.
The second crucial point is the behaviour of the resolvent (D 0 − z) −1 when the spectral parameter z is close to the real axis. For z = λ + i , λ > 0, its symbol picks up singularities on the sphere of radius λ 1/2 when → 0. In fact, from the well-known formula it follows that the (scalar part of) the symbol of (D 0 − z) −1 for m = 0 has a bounded Fourier transform. We emphasize that in higher dimensions n ≥ 2 there can be no L p → L q estimate (p and q being dual exponents, i.e. q = p/(p − 1)) for the resolvent of the Dirac operator that is uniform in the spectral parameter. The reason is that the analogue of (1.6) in higher dimensions implies that The bound on the left is imposed by the Stein-Tomas restriction theorem, see [10], while the bound on the right is dictated by standard estimates for Bessel potentials of order one, see e.g. [6, Cor. 6.16]. Both conditions are known to be sharp. Unfortunately, this forces n = 1. For the Laplacian, the situation is better since the right hand side of (1.7) is then replaced by 2/n, see [8,Theorem 2.3]. Based on the latter, eigenvalue estimates for multi-dimensional Schrödinger operators with L p -potentials were established in [3].

Main Results
Let D 0 be the Dirac operator in (1.1), with domain consisting of all square integrable functions ψ ∈ L 2 (R + , C 2 ) that are absolutely continuous on R + , satisfy (1.2), and such that D 0 ψ ∈ L 2 (R + , C 2 ). In the following, we tacitly assume that the potential V is smooth and has compact support. This assumption allows us to define the sum D 0 +V in an unambiguous way (as an operator sum). However, it is in no way essential, as the attentive reader will gather, and can easily be disposed of. In fact, the assumptions imposed on V in Theorems 2.1-2.3 are sufficient to define a closed extension of D ⊃ D 0 + V 380 J.-C. Cuenin IEOT via the resolvent formula (3.4), see [2] and the references therein for details. In particular, the Birman-Schwinger principle remains valid for this extension, by construction. Incidentally, for (3.4) to be well-defined, the existence of a point z 0 ∈ C \ σ(D 0 ) for which I + Q(z 0 ) has a bounded inverse, has to be assumed, and once such a point is shown to exist, it will automatically belong to C \ σ(D). Hence, from this point of view, Theorems 2.1-2.3 (with V satisfying only the regularity assumptions stated in the respective theorem) yield the existence of a closed extension and the spectral estimates as a byproduct.
is contained in the disjoint union of two closed disks with centres ±mc 2 x 0 and radii mc 2 r 0 , where (2.1) In particular, the spectrum of the massless Dirac operator (m = 0) with non-Hermitian potential V is R.
Proof. The proof is based on the Birman-Schwinger principle: Let z ∈ C \ σ(D 0 ) and define where the branch of the square root is chosen such that Im κ(z) > 0. Let us assume that α ∈ (0, π/2]. It can then be checked that is a solution to the differential equation (D 0 − z)ψ l (x; z) = 0 satisfying the boundary condition (1.2). In the case α = 0, formally corresponding to cot(α) = ∞, the solution is On the other hand, Vol. 79 (2014) Estimates on Complex Eigenvalues for Dirac operators on the half-line 381 is a solution that lies in L 2 (R + ). The resolvent R 0 (z) = (D 0 − z) −1 is then given by (see e.g. [11,Satz 15.17]) is the Wronskian and (·, ·) denotes the Hermitian scalar product on C 2 (which we define to be linear in the second variable). The values of the resolvent kernel R 0 (x, y; z) are linear maps in C 2 , given by where θ denotes the characteristic function of (0, ∞). Note that W = 0 since D 0 has no eigenvalue [11, p.137]; alternatively, this may be seen as follows: By assumption, σ := cot(α) ≥ 0, and thus the solution ζ = i σ of W = 0 lies in the (open) upper half plane. However, the function ζ defined in (2.2) takes values in the (open) lower half plane. Indeed, ζ(z) is a (holomorphic) branch of the square root of (z + mc 2 )/(z − mc 2 ) for z ∈ C \ σ(D 0 ). The range of the latter is the cut plane C \ R + , thus any branch of the square root must have values either in the upper or in the lower half plane. One easily checks that Imζ(0) < 0.
We now estimate the norm of cR 0 (x, y; z) as an operator on C 2 . Let us assume that α ∈ (0, π/2], so that ψ l is given by (2.3); the case α = 0 may always be recovered by letting cot(α) → ∞. We then have (suppressing the z-dependence of κ and ζ)
Using Hölder's inequality, we arrive at By the Birman-Schwinger principle, the left hand side of (2.9) is equal to 1 if z is an eigenvalue. If m = 0, then ζ(z) = ±1, depending on whether z is in the upper or lower half plane, and hence the right hand side of inequality (2.9) is equal to √ 2v. It follows that z cannot be an eigenvalue if v < 1/ √ 2. If m = 0, then for z in the left half plane the maximum equals 1 + |ζ(z)| 2 , while in the right half plane it equals 1 + |ζ(z)| −2 . Hence, for every eigenvalue z, if z is in the left half plane and |ζ(z)| ≤ ρ −1 if z is in the right half plane.
From (2.9) one sees that the eigenvalue estimate is equivalent to the inequality This should be compared to the result of [2] for the whole-line operator, which may also be written as It is instructive to note that if we replace V by λV , then in the weak coupling limit λ → 0, the inequalities (2.10) and (2.11) take the form with A = 1 in the case of (2.10) and A = 1/2 in case of (2.11), and ∓ indicating whether z tends to mc 2 or −mc 2 as λ → 0. Note that (2.12) has the semiclassical behaviour of a non-relativistic operator, the reason being that the weak-coupling limit is equivalent to the non-relativistic limit: If we Vol. 79 (2014) Estimates on Complex Eigenvalues for Dirac operators on the half-line 383 subtract (or add, respectively) the rest energy mc 2 (i.e. replace z ∓ mc 2 by z), we may consider c −1 as a small coupling constant (we now fix λ = 1, whereas before, we considered c fixed). In the limit c → ∞, the Dirac operator converges to the Schrödinger operator with Dirichlet or Neumann boundary conditions, see Sect. 3. On the other hand, for the massless operator (or for large eigenvalues of the massive operator), the inequalities (2.10) and (2.11) reduce to with B = 1/2 in the case of (2.10) and B = 1 in case of (2.11). Inequality (2.13) has the correct semiclassical behaviour of a relativistic operator. It is an open and interesting question whether there exists a bound on the number of complex eigenvalues of the massless Dirac operator in terms of the right hand side of (2.13). From the inequality it follows that the whole line estimate (2.11) continues to hold for the halfline operators if the constant 1/2 on the right hand side is replaced by 1. For "Dirichlet boundary conditions" ψ 1 (0) = 0 or ψ 2 (0) = 0 this may also be seen from the following argument: suppose ψ = (ψ 1 , ψ 2 ) t is an eigenfunction of the half-line operator with potential V to an eigenvalue z. Since the parity operator commutes with D 0 , it follows that z is an eigenvalue of the whole-line operator with potential Then every eigenvalue z = mc 2 (ζ 2 +1)/(ζ 2 −1) of the massive (m = 0) Dirac operator D 0 +V subject to the boundary conditions ψ 1 (0) cos(α)−ψ 2 (0) sin(α) = 0 satisfies with "−" if α = 0 and "+" if α = π/2, and with ζ = |ζ|e it , π < t < 2π.
It follows from Theorem 2.2 that the eigenvalues of D 0 + V may only emerge from ±mc 2 as the potential is "turned on". However, if the first moment of the potential is sufficiently small, then the eigenvalues can emerge only from one of those points.