Eigenvalue bounds for non-selfadjoint Dirac operators

We prove that the eigenvalues of the n-dimensional massive Dirac operator D0+V\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {D}}_0 + V$$\end{document}, n≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\ge 2$$\end{document}, perturbed by a potential V, possibly non-Hermitian, are contained in the union of two disjoint disks of the complex plane, provided V is sufficiently small with respect to the mixed norms Lxj1Lx^j∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^1_{x_j} L^\infty _{{\widehat{x}}_j}$$\end{document}, for j∈{1,⋯,n}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j\in \{1,\dots ,n\}$$\end{document}. In the massless case, we prove instead that the discrete spectrum is empty under the same smallness assumption on V, and in particular the spectrum coincides with the spectrum of the unperturbed operator: σ(D0+V)=σ(D0)=R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma ({\mathscr {D}}_0+V)=\sigma ({\mathscr {D}}_0)={\mathbb {R}}$$\end{document}. The main tools used are an abstract version of the Birman–Schwinger principle, which allows in particular to control embedded eigenvalues, and suitable resolvent estimates for the Schrödinger operator.

or e.g. the more recent books [3,15] where also physical applications may be found.
In this paper, we deal with the free Dirac operator D 0 perturbed by a potential The importance of this family of operators in quantum physics is well known: in two dimensions, the operator D V is related to the quantum theory of graphene, while in the 3D case the Hamiltonian D V models the dynamic of a relativistic quantum particle of spin 1 2 subject to an external electric field. We consider the operator D V acting on the Hilbert space of spinors H = L 2 (R n ; C N ), where n ≥ 2, N := 2 n/2 and · is the ceiling function. The free Dirac operator D 0 with non negative mass m is defined as where c is the speed of light, is the reduced Planck constant and the matrices α k ∈ C N ×N , for k ∈ {0, . . . , n}, are elements of the Clifford algebra (see [37]) satisfying the anti-commutation relations α j α k + α k α j = 2δ j,k I C N , for j, k ∈ {0, . . . , n}, (1.1) where δ j,k is the Kronecker symbol. Without loss of generality we can take and core C ∞ 0 (R n ; C N ). The potential V : R n → C N ×N will be any complex matrix-valued function with |V | ∈ L 2 loc (R n ; R), where |V (x)| is the operator norm of the matrix V (x). With the usual slight abuse, the same symbol V will denote the multiplication operator by V on H, with initial domain dom(V ) = C ∞ 0 (R n ; C N ). In the case of the non-selfadjoint Schrödinger operator − + V , properties of the point spectrum σ p (− + V ) were investigated among others by Frank [22,23], Frank and Sabin [24], Frank and Simon [25]. In particular, the eigenvalues of − + V satisfy the bound |z| γ ≤ D γ,n R n |V (x)| γ +n/2 dx, 0 < γ = 1 2 , if n = 1, where the constant D γ,n > 0 is independent of z and V . This localization estimate is due to Abramov, Aslanyan and Davies [2] for n = 1, with sharp constant D 1/2,1 = 1/2, and for larger n to Frank [22]. The proof combines the Birman-Schwinger principle with the uniform Sobolev inequalities by Kenig et al. [33], where p = p/( p − 1) is the dual exponent of p. In [35,39], Laptev and Safronov conjectured that the range of γ for n ≥ 2 can be extended to 0 < γ < n/2, and Frank and Simon [25] proved the conjecture to be true for radial symmetric potentials. Consider now the structure of σ p (D V ). In the selfadjoint case there exists an extensive literature on the spectral properties of D V , see for example the monograph by Thaller [40]. In the non-selfadjoint case, the study of the spectrum of D V was initiated by Cuenin et al. [10] in the 1D case, followed by [8,11,18]. For the higher dimensional case, we refer to the works [9,16,20,38].
In the 1D paper [10], it is proved of two disjoint closed disks, with centers and radius respectively In particular, in the massless case (m = 0), the spectrum of D V is R. Moreover, this inclusion is sharp. The proof is essentially based on the combination of the Birman-Schwinger principle with the resolvent estimate for the free Dirac operator It should be remarked that, in higher dimensions n ≥ 2, L p (R n ) → L p (R n ) estimates for (D 0 − z) −1 do not exist, as observed in the Introduction of [8]. Indeed, Cuenin points out that, due to the Stein-Thomas restriction theorem and standard estimates for Bessel potentials, the resolvent and thus we are forced to choose n = 1. The situation for the Schrödinger operator is better since the right-hand side of the above range is 2/n, as per the Kenig-Ruiz-Sogge estimates.
In [9], Cuenin localized the eigenvalues of the perturbed Dirac operator in terms of the L p -norm of the potential V , but in an unbounded region of the complex plane. Indeed, Theorem 6.1.b of [9] states that, if n ≥ 2 and |V | ∈ L p , with p ≥ n, then any where C is a constant independent of z and V . Similar unbounded enclosing regions were obtained in [12], where Cuenin and Tretter study arbitrary non-symmetric perturbations of selfadjoint operators. In particular, for the massless Dirac operator in R 2 , if |V | ∈ L p with p > 2, they obtain that Considering instead the massive Dirac operator with Coulomb-like potential in R 3 , the authors in [12] A different result on the localization of eigenvalues in an unbounded regions was proved by Fanelli and Krejčiřík in [20]: in 3D, if |V | ∈ L 3 (R 3 ) and z ∈ σ p (D V ), then The advantage of the last result lies in the explicit condition which is easy to check in applications. However, also in this result the eigenvalues are localized in an unbounded region around the continuous spectrum σ (D 0 ) = (−∞, −m] ∪ [m, +∞) of the free Dirac operator D 0 . We finally mention the recent paper [7], where the authors obtain results on the absence of eigenvalues for the Schrödinger and Pauli operators with a constant magnetic field and non-Hermitian potentials, and for the purely magnetic Dirac operators. However, Dirac operators with electric perturbations can not be treated by the multiplicative techniques of [7] (recall that the square of a purely magnetic Dirac operator is a diagonal magnetic Laplacian, which allows one to use the multiplier method).
The main goal of the present paper is to generalize the results by Cuenin et al. [10] to higher dimensions, enclosing the eigenvalues of the massive (m > 0) Dirac operator D V in a compact region of the complex plane, provided the potential V satisfies a suitable smallness condition in a mixed Lebesgue norm. In the case of the massless (m = 0) Dirac operator, we obtain that the point spectrum of the perturbed operator D V is empty, and then σ ( Before we formalize our results in Theorems 1 and 2 below, we introduce a few notations used throughout the paper.

Notations
We use the symbols σ (H ), σ p (H ), σ e (H ) and ρ(H ) respectively for the spectrum, the point spectrum, the essential spectrum and the resolvent of an operator H . More explicitly, we define whereas the discrete spectrum is defined as Recall that for non-selfadjoint operators, the essential spectrum defined above is not the complement of the discrete spectrum, see e.g. [17]. For z ∈ ρ(H ), we denote with R H (z) := (H − z) −1 the resolvent operator of H . We recall also that For j ∈ {1, . . . , n} and x = (x 1 , . . . , x n ) ∈ R n , we write The mixed Lebesgue spaces L p x j L q x j (R n ) are the spaces of measurable functions on R n with Obvious modifications occur for p = ∞ or q = ∞ (see e.g. [4] for general properties of such spaces). For any matrix-valued function M : R n → C N ×N , we set where |M(x)| denotes the operator norm of the matrix M(x). Further, we write to denote the partial convolution respect to x j , the partial Fourier transform with respect to x j , and its inverse, respectively. The partial (inverse) Fourier transform with respect to x j and the complete (inverse) Fourier transform with respect to x are defined in a similar way. Finally, we shall need the function spaces with norms defined as follows The dual space of X is given by (see e.g. [5]), endowed with the norm We can now state our results.
of the two closed disks in C with centers in x − 0 , x + 0 and radius R 0 , with then D V has no eigenvalues. In this case, we have σ (

Remark 1.1
The crucial tool in our proof is a sharp uniform resolvent estimate for the free Dirac operator. This approach is inspired by [22], where the result by Kenig et al. [33] was used for the same purpose. In our case, we prove in Sect. 2 the following estimates, of independent interest: and These can be regarded as precised resolvent estimates of Agmon-Hörmander type. Note also that similar uniform estimates, but in non sharp norms, were proved earlier by the first and second Author in [13,14,19]. In Sect. 3, we combine our uniform estimates with the Birman-Schwinger principle, enabling us in Sect. 4 to complete the proof of Theorems 1 and 2.

Remark 1.2 The space Y satisfies the embedding
where L p,q (R n ) denotes Lorentz spaces. Moreover, we have where W m, p (R n ) is a Sobolev space. In particular, in dimension 2 we obtain We refer to Fournier [21], Blei and Fournier [6] and Milman [36] for these inclusions.

Remark 1.3 According to the previous remark we have
. Thus in the massive 3-dimensional case the assumption V Y < C 0 implies both our result, Theorem (1), and the one by Fanelli and Krejčiřík [20], i.e. the eigenvalue bound (1.2). Although our result improves the latter one for large eigenvalues, bounding them in two compact regions, it may happen that, in a neighbourhood of z = −m and z = m, the bound in (1.2) improves the one stated in Theorem 1. Indeed, it is not hard to check that, supposing V Y sufficiently small, our disks are enclosed in the region found by Fanelli and Krejčiřík if This condition may not always be satisfied and depends on the norms of the potential V in the spaces L 3 (R 3 ) and Y (R 3 ). If this happens, the result in Theorem 1 and the one in [20] should be jointly taken in consideration for the eigenvalues bound. This situation is illustrated in Fig. 1.

Uniform resolvent estimates
Fix constants r , R, δ > 0 such that and consider the open cover ..,n} be a smooth partition of unity subordinate to S, that is to say a family of smooth positive functions such that From these, define the smooth partition of unity χ = {χ j } j∈{1,...,n} , with and correspondingly, for j ∈ {1, . . . , n}, the Fourier multipliers Note in particular that The following estimates hold true.
x j and j, k ∈ {1, . . . , n}, we have that In particular, it follows that x j and j ∈ {1, . . . , n} we have that where {χ j } j∈{1,...,n} are defined in (2.1) and C > 0 is the same as in Lemma 1. In particular, it follows that

Remark 2.1
Before we proceed further, we give a heuristic explanation for the choice of the localization in the frequency domain via the Fourier multiplier χ j (|z| −1/2 D).

Proof of Lemma 1
The last two estimates follow trivially from the first two estimates, (2.2) and the definitions of the norms on X and X * .
For simplicity, from now on C > 0 will stand for a generic positive constant independent of z and which may change from line to line. Clearly, by scaling, it is sufficient to consider |z| = 1, z = 1. Thus we are reduced to prove that n}. This is equivalent to where we have written z = λ + iε, λ 2 + ε 2 = 1 and z = 1. We proceed by splitting χ j in the functions which appear in its definition (2.1), localizing ourselves in the regions of the frequency domain near the unit sphere, i.e. S ± j , and far from it, i.e. S ∞ . Estimate on S ± j . We want to prove Let us define the family of operators Roughly speaking, the operator T ± j flattens the upper half unit sphere in the frequency domain {ξ ∈ R n : |ξ | = 1, ±ξ j > 0}. Writing more explicitly these operators, we have where we used the substitution ξ j → ξ j − ϕ( ξ j ) in the fourth step. Applying the Plancherel Theorem twice, we obtain that T ± j are isometries on L p x j L 2 x j : for p ∈ [1, +∞] we have We can then write where the last inequality follows from Young's inequality and Note that we dropped the absolute value in the definition of ϕ, i.e. |1 − | ξ j | 2 | = 1 − | ξ j | 2 , because supp{χ ± j • } ⊂ {ξ ∈ R n : | ξ j | ≤ 1}, thanks to the definition of S ± j and the assumption δ ≥ √ R 2 − 1. It is simple to see that a λ,ε (D)ψ ∈ S , where S is the space of Schwartz functions, since a λ,ε (D)ψ is the inverse Fourier transform of a smooth compactly supported function. Moreover, we can consider a λ,ε (D)ψ as a pseudodifferential operator with symbol a λ,ε applied to the Schwartz function ψ; letting λ + iε → 1 we have the pointwise convergence is uniformly bounded respect to z ∈ C with |z| = 1. We have proved By Plancherel's Theorem and Young's inequality, and by equality (2.5), we have where θ ≡ θ(x j ) is the Heaviside function. Inserting this in (2.6) we get (2.4).
Estimate on S ∞ . We shall now prove that We consider three cases, depending on whether we are localized in the regions defined by We set otherwise, and observe that χ ∞ = χ 1 ∞ + χ 2 ∞ + χ 3 ∞ , since χ ∞ ≡ 1 for |ξ | > R, from the assumptions on the cover S and the partition χ .
By Plancherel's Theorem and Hölder's, Young's and Minkowski's integral inequalities, for h ∈ {1, 2, 3} we infer Here and below, we always consider the principal branch of the complex square root function. Clearly, if we prove that C h for h ∈ {1, 2, 3} is bounded uniformly with respect to λ and ε, we recover (2.7). Estimate on C 1 R, j . Observing that χ 1 ∞ (ξ ) ≡ χ 1 ∞ ( ξ j ) and noting that we can explicitly compute the Fourier transforms: • if k = j, then • if s = 1, k = j, then Estimate on C 2 R, j . By the definition of the inverse Fourier transform in (2.8) and from the fact that χ 2 ∞ (ξ ) = 0 when |ξ | < r , we see that ∞ is compactly supported due to its definition. Estimate on C 3 R, j . By the inverse Fourier transform in (2.8), recalling the definition of χ 3 ∞ and exploiting the substitution ξ j → sgn{x j }ξ j , we have where, for fixed ξ j , x j , the complex function ψ(x j , ξ j , ·) : C → C is defined by Define, for a radius A > 0, the half circle γ A := {Ae iθ : θ ∈ [0, π]} in the upper half complex plane. Fixing ρ > R, by the Residue Theorem, we get Observing that we can consider x j = 0, letting ρ → +∞ we can apply Jordan's lemma to the integral on the curve γ ρ , finally getting where we used the relation (2.9).
Summing up, we can finally recover the desired estimate (2.3), where the positive constant C does not depend on λ and ε, but only on R and the partition χ .
Lemma 2 is now a straightforward corollary to Lemma 1.

Proof of Lemma 2
Again, the last estimate in the statement follows from the first one, (2.2) and the definition of the X and X * norms.
From the anticommutation relations (1.1) we infer, for every z ∈ C, Thus as usual, for z ∈ ρ(D 0 ) we can write Write f j = χ j (|z 2 − m 2 | −1/2 D) f . By Lemma 1, it is easy to get x j as claimed.

The Birman-Schwinger principle
In this section, following the approach of Kato [31] and Konno and Kuroda [34], we define in a rigorous way the closed extension of a perturbed operator with a factorizable potential, formally defined as H 0 + B * A, in order to give an abstract version of the Birman-Schwinger principle. We mention also the recent work [30] by Hansmann and Krejčiřík, where, among others, a nice and innovative approach is developed to deal with the embedded eigenvalues.
Let H, H be Hilbert spaces and consider the densely defined, closed linear operators For simplicity, we assume also that σ ( The idea of the principle is easy to explain in the case of bounded operators A and B. Then H = H 0 + B * A is well defined as a sum of operators, and if z ∈ ρ(H 0 ), the Birman-Schwinger operator is also a bounded operator. One checks immediately that z ∈ σ p (H ) ∩ ρ(H 0 ) implies −1 ∈ σ p (Q(z)), and so Q(z) H →H ≥ 1. Hence, a bound for the norm of Q(z) gives information on the localization of the non-embedded eigenvalues of H .
We now return to the general case of an unbounded B * A. As in [34], we assume the following set of assumptions.
Assumption 1 For some, and hence for all, z ∈ ρ(H 0 ), the operator AR H 0 (z)B * , densely defined on dom(B * ), has a closed extension Q(z) in H , which we call the Birman-Schwinger operator, with norm bounded by Assumption 2 There exists z 0 ∈ ρ(H 0 ) such that −1 ∈ ρ(Q(z 0 )).
Let us collect some useful facts in the next lemma.

Lemma 3 Suppose Assumptions A and B and let z, z
. Then the following holds true: Proof See Lemma 2.2 in [29].
We can construct now the extension of the perturbed operator H 0 + B * A.

Lemma 4 (Extension of operators with factorizable potential) Suppose Assumptions
A and B. Let z 0 ∈ ρ(H 0 ) such that −1 ∈ ρ(Q(z 0 )). Then the operator defines a densely defined, closed, linear operator H in H which has R H (z 0 ) as resolvent and which extends H 0 + B * A.
Proof We refer to Theorem 2.3 in [29]. See also the work by Kato [31].
We can finally formulate the abstract Birman-Schwinger principle.
for every ϕ ∈ H , where (·, ·) H is the scalar product on H . In particular . In order to treat the embedded eigenvalues, we will adapt the argument of Lemma 1 in [34] together with the limiting argument from Theorem 8 in [30].
Note that H ψ = zψ is equivalent to and hence we obtain from (3.2) that , which contradicts the assumption on z. Thus, we proved ψ = 0. Moreover, we can show the identity from which in particular φ = 0. Indeed, by (3.2) and (iii) of Lemma 3, it follows that which combined with (3.5) gives us (3.7). Multiplying by (1 + Q(z 0 )) −1 AR H 0 (z + iε) both sides of (3.6), we obtain and so, by (v) of Lemma 3 and by the resolvent identity, we have from which, using identity (3.7), we finally arrive at If z ∈ ρ(H 0 ), then ε = 0 and we completely proved case (i), the "in particular" part being straightforward. In the following, we suppose z ∈ σ (H 0 )\σ p (H 0 ) and H 0 self-adjoint. Fixed ϕ ∈ H , we get from (3.8) that Exploiting the Spectral Theorem and denoting the spectral measure of H 0 as E 0 , we have From the fact that and E 0 ({z}) = 0 since z / ∈ σ p (H 0 ), we infer that f ε → 0 as ε → 0 ± almost everywhere with respect to the spectral measure. Moreover hence by dominated converge theorem we conclude that I ε → 0 as ε → 0 ± , proving (3.3). Finally, since by (3.3) we have we get the first inequality in (3.4), while the second one is obvious by Assumption A.

Proof of the Theorems
We can now specialize to our problem the abstract theory developed in the last section. We choose H = H = L 2 (R n ; C N ) and H 0 the free Dirac operator D 0 . The factorization of V is given using the polar decomposition V = U W where W = (V * V ) 1/2 and the unitary matrix U is a partial isometry: then we may set A = W 1/2 and B = W 1/2 U * . It is easy to see that Assumption A holds thanks to Lemma 2 with Indeed, for ϕ ∈ C ∞ 0 (R n ; C N ), and hence by density (3.1). We used above the relation We show now that also Assumption B' holds. To find z 0 ∈ ρ(D 0 ) such that (z 0 ) < 1, let us define C 0 = [n(n + 1)C] −1 , V = [(n + 1)C 0 / V Y − n] 2 .
Since from the hypothesis of Theorems 1, 2 we have V Y < C 0 and so V > 1, the condition 1 ≤ (z) is equivalent to V ≤ |z/z| if m = 0, and to if m > 0. Then, if m = 0 it is sufficient to choose z 0 ∈ C\R, while if m > 0 we take z 0 ∈ ρ(D 0 ) outside the disks in the statement of Theorem 1.
Thus, we can apply Lemma 5, which combined with relation (4.1) and with relation V ≤ |z/z| proves Theorem 1 and Theorem 2 respectively.
For the final claim in Theorem 2, we will follow the argument in [10] to prove that the potential V ∈ Y = n j=1 L 1 x j L ∞ x j (R n ) leaves the essential spectrum invariant and that the residual spectrum of D V is absent. To get the invariance of the essential spectrum, it is sufficient to prove that, fixed z ∈ ρ(D 0 ) such that −1 ∈ ρ(Q(z)), the operator AR D 0 (z) is a Hilbert-Schmidt operator, hence compact. Thus identity (3.2) gives from which it follows that R D V (z) − R D 0 (z) is compact and so, by Theorem 9.2.4 in [17], To see that AR D 0 (z) is a Hilbert-Schmidt operator, we need to prove that its kernel A(x)K (z, x − y) is in L 2 (R n × R n ; C N ), where K (z, x − y) is the kernel of the resolvent (D 0 − z) −1 . By the Young inequality where 1/ p + 1/q = 2. Hence we need to find in which Lebesgue space L 2q (R n ; C N ) the kernel K (z, x) lies. For z ∈ ρ(− ) = C\[0, ∞), it is well-known (see e.g. [26]) that the kernel K 0 (z, x − y) of the resolvent operator (− − z) −1 is given by where K ν (w) is the modified Bessel function of second kind and we consider the principal branch of the complex square root. Fixed now z ∈ ρ(D 0 ) = C\{ζ ∈ material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.