Eigenvalue bounds for non-selfadjoint Dirac operators
aa r X i v : . [ m a t h . SP ] J un Eigenvalue bounds for non-selfadjoint Dirac operators
Piero D’Ancona, Luca Fanelli, Nico Michele Schiavone
Keywords:
Dirac operator, non-selfadjoint perturbation,localization of eigenvalues, BirmanSchwinger principle
MSC2020: primary 35P15, 35J99, 47A10, 47F05, 81Q12
Abstract
In this work we prove that the eigenvalues of the n -dimensional massive Diracoperator D + V , n ≥
2, perturbed by a possibly non-Hermitian potential V , arelocalized in the union of two disjoint disks of the complex plane, provided that V is sufficiently small with respect to the mixed norms L x j L ∞ b x j , for j ∈ { , . . . , n } . Inthe massless case, we prove instead that the discrete spectrum is empty under thesame smallness assumption on V , and in particular the spectrum is the same of theunperturbed operator, namely σ ( D + V ) = σ ( D ) = R .The main tools we employ are an abstract version of the Birman-Schwinger prin-ciple, which include also the study of embedded eigenvalues, and suitable resolventestimates for the Schr¨odinger operator. In recent years, non-selfadjoint operators are attracting increasing attention, not onlyin view of applications in quantum mechanics and other branches of physics, but alsofor the interesting mathematical challenges they present. While the theory of selfadjointoperators is consolidated in a vast literature, references on the study of non-selfadjointoperators are more sparse, so much that, quoting E.B. Davies [15], “it can hardly be calleda theory” . We refer to the books [3, 39] for milestones on the history of the argument,and to [14] for some physical applications.In this paper, we deal with the free Dirac operator D perturbed by a potential,formally defined by D V = D + V. The relevance of these kind of operators in quantum physics is common knowledge, sincein the 2-dimensional case the operator D V is related to the quantum theory of graphene,while in the 3-dimensional case the Hamiltonian D V determines the dynamic of a rela-tivistic quantum particle of spin , subject to an external electric field described by thepotential V . Department of Mathematics “Guido Castelnuovo”, University of Rome “La Sapienza”, Piazzale AldoMoro 5, 00185 Rome, Italy.e-mails: [email protected], [email protected], [email protected]
1e consider the operator D V acting on the Hilbert space of spinors H = L ( R n ; C N ),where n ≥ N := 2 ⌈ n/ ⌉ and ⌈·⌉ is the ceiling function. The free Dirac operator D withnon-negative mass m is defined as D = − ic ~ α · ∇ + mc α = − ic ~ n X k =1 α k ∂∂x k + mc α , where c is the speed of light, ~ is the reduced Planck constant and the matrices α k ∈ C N × N ,for k ∈ { , . . . , n } , are elements of the Clifford algebra (see [34]) satisfying the anti-commutation relations α j α k + α k α j = 2 δ j,k I C N , for j, k ∈ { , . . . , n } , (1.1)where δ j,k is the Kronecker symbol. Without loss of generality we can take α = (cid:18) I C N/ × N/ − I C N/ × N/ (cid:19) and renormalize the unit measures such that c = ~ = 1. The free Dirac operator hasdomain dom( D ) = { ψ ∈ H : ∇ ψ ∈ H n } and it is selfadjoint with core C ∞ ( R n ; C N ).The potential V : R n → C N × N is allowed to be any non-Hermitian matrix-valuedfunction with | V | ∈ L ( R n ; R ), where | V | is the operator norm of V . Thus the resultingoperator may be non-selfadjoint. With an abuse of notation, we use the same symbol V toindicate the multiplication operator generated by the matrix V in H with initial domaindom( V ) = C ∞ ( R n ; C N ).In this work, we are interested in location of eigenvalues for D V in the complex plane.For the non-selfadjoint Schr¨odinger operator − ∆ + V , we refer to the works by Frank[22, 23], Frank and Sabin [24], Frank and Simon [25]. In particular, we have that theeigenvalues of − ∆ + V satisfy the bound | z | γ ≤ D γ,n Z R n | V ( x ) | γ + n/ dx, < γ ( = , if n = 1, ≤ , if n ≥ D γ,n > z and V . This localization estimatewas proven by Abramov, Aslanyan and Davies [2] for n = 1 with the sharp constant D / , = 1 /
2, and for larger n by Frank [22], combining the Birman-Schwinger principlewith the uniform Sobolev inequalities by Kenig, Ruiz and Sogge [30], i.e. (cid:13)(cid:13) ( − ∆ − z ) − (cid:13)(cid:13) L p → L p ′ ≤ C | z | − n/ n/p − , n + 1 ≤ p − p ′ ≤ n , where p ′ = p/ ( p −
1) is the dual exponent of p . In [32, 36], Laptev and Safronov conjecturedthat the range of γ for n ≥ < γ < n/
2, and Frank and Simon [25]proved the conjecture to be true for radial symmetric potentials.Let us return to the Dirac operator D V . If we suppose V : R n → C N × N is an Hermitianmatrix-valued function, such that the operator D V is selfadjoint, we have an extensive2iterature on its spectral properties, see for example the monograph by Thaller [38]. In thenon-selfadjoint case, the study of the spectrum of D V was initiated by Cuenin, Laptevand Tretter in [10] in the 1-dimensional case, followed by [8, 11, 18]. For the higherdimensional case, we refer to the works [9, 16, 20, 35].In [10], the authors proved that, for n = 1, if V = ( V ij ) i,j ∈{ , } with V ij ∈ L ( R ) and k V k L ( R ) = Z R | V ( x ) | dx ≤ , then every non-embedded eigenvalue z ∈ ρ ( D ) of D V lies in the disjoint union z ∈ B R ( x − ) ∪ B R ( x +0 )of the two closed disks in the complex plane with centers and radius respectively x ± = ± s k V k − k V k + 24(1 − k V k ) + 12 , R = s k V k − k V k + 24(1 − k V k ) − . In particular, in the massless case ( m = 0), the spectrum of D V is R . They also showedthat this inclusion is sharp. Their proof is essentially based on the combination of theBirman-Schwinger principle with the resolvent estimate for the free Dirac operator (cid:13)(cid:13) ( D − z ) − (cid:13)(cid:13) L ∞ ( R ) → L ( R ) ≤ s
12 + 14 (cid:12)(cid:12)(cid:12)(cid:12) z + mz − m (cid:12)(cid:12)(cid:12)(cid:12) + 14 (cid:12)(cid:12)(cid:12)(cid:12) z − mz + m (cid:12)(cid:12)(cid:12)(cid:12) , z ∈ ρ ( D ) . It should be remarked that, in higher dimensions n ≥ L p ( R n ) → L p ′ ( R n ) estimates for( D − z ) − do not exist, as observed in the Introduction of [8]. Indeed, Cuenin pointsout that, due to the Stein-Thomas restriction theorem and standard estimates for Besselpotentials, the resolvent ( D − z ) − : L p ( R n ) → L p ′ ( R n ) is bounded uniformly in | z | > n + 1 ≤ p + 1 p ′ ≤ n , and thus we are forced to choose n = 1. The situation for the Schr¨odinger operator isbetter since the right-hand side of the above range is 2 /n , as stated in 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 ≥ | V | ∈ L p , with p ≥ n , then any eigenvalue z ∈ ρ ( D ) of D V satisfies |ℑ z/ ℜ z | ( n − /p |ℑ z | − n/p ≤ C k V k L p ( R n ) , where C is a constant independent on z and V . A similar result was proved by Fanelliand Krejˇciˇr´ık in [20], where they show that, for dimension n = 3, | V | ∈ L ( R ) and z ∈ ρ ( D ) ∩ σ p ( R n ), then (cid:18) ℜ z ) ( ℜ√ m − z ) (cid:19) − / < ( π/ / √ e − + 2 e − k V k L ( R ) . σ ( D ) = ( −∞ , − m ] ∪ [ m, + ∞ ) of the free Dirac operator D .Here, our main results try to generalize the one by Cuenin, Laptev and Tretter [10]in higher dimensions, recovering the enclosure of the eigenvalues of the massive ( m > D V in a compact region of the complex plane, imposing the smallnessof the potential V with respect to suitable mixed Lebesgue norms. In the case of themassless ( m = 0) Dirac operator, we obtain that the point spectrum of the perturbedoperator D V is empty, and then σ ( D V ) = σ ( D ) = R . We also mention the recent paper[7], in which similar results are obtained by multiplication techniques.Before to formalize our results in Theorems 1 & 2, we need to introduce the followingnotations used throughout the paper. Notations.
We use the symbols σ ( H ), σ p ( H ), σ e ( H ) and ρ ( H ) respectively for the spec-trum, the point spectrum, the essential spectrum and the resolvent of an operator H .Since for a non-selfadjoint closed operator there are various definitions for essential spec-trum, we define σ e ( H ) = { z ∈ C : H − z is not a Fredholm operator } , whereas the discrete spectrum is defined as σ d ( H ) = { z ∈ C : z is an isolated eigenvalue of H of finite multiplicity } . For z ∈ ρ ( H ), we denote with R H ( z ) := ( H − z ) − the resolvent operator of H . We recallalso that σ ( − ∆) = σ e ( − ∆) = [0 , + ∞ ) ,σ ( D ) = σ e ( D ) = ( −∞ , − m ] ∪ [ m, + ∞ ) . We use the symbol ( · , · ) H for the inner product on the Hilbert space H = L ( R n ; C N ),that is ( φ, ψ ) H = Z R n φ † · ψ dx where · is the scalar product.Fixed j ∈ { , . . . , n } and x = ( x , . . . , x n ) ∈ R n , we denote b x j := ( x , . . . , x j − , x j +1 , . . . , x n ) ∈ R n − , (ø x, b x j ) := ( x , . . . , x j − , ø x, x j +1 , . . . , x n ) ∈ R n . Define the mixed Lebesgue spaces L px j L q b x j ( R n ) as the spaces of the functions with finitemixed-norm k f k L pxj L q b xj := Z R (cid:18)Z R n − | f ( x j , b x j ) | q d b x j (cid:19) p/q dx j ! /p , where the obvious modifications occur for p = ∞ or q = ∞ .For any matrix–valued function M : R n → C N × N , we set k M k L pxj L q b xj := k| M |k L pxj L q b xj | M | : R n → R is obtained considering the operator norm | M ( x ) | of M ( x ) for almost every fixed x ∈ R n .We also indicate with[ f ∗ x j g ]( x ) := Z R f ( y j , b x j ) g ( x j − y j , b x j ) dy j , [ F x j f ]( ξ j , b x j ) := 1 √ π Z R e − ix j ξ j f ( x j , b x j ) dx j , [ F − ξ j f ]( x j , b x j ) := 1 √ π Z R e ix j ξ j f ( ξ j , b x j ) dξ j , respectively the partial convolution respect to x j , the partial Fourier transform respect to x j and its inverse. In a similar way one can define the partial (inverse) Fourier transformrespect to b x j and the complete (inverse) Fourier transform respect to x .Finally, let us consider the function spaces X ≡ X ( R n ) := n \ j =1 L x j L b x j ( R n ) , Y ≡ Y ( R n ) := n \ j =1 L x j L ∞ b x j ( R n ) , with norms defined as k f k X = max j ∈{ ,...,n } k f k L xj L b xj , k f k Y = max j ∈{ ,...,n } k f k L xj L ∞ b xj . The dual space of X is given (see e.g. [4]) by X ∗ ≡ X ∗ ( R n ) := n X j =1 L ∞ x j L b x j ( R n ) , with the norm k f k X ∗ := inf ( n X j =1 k f j k L ∞ xj L b xj : f = n X j =1 f j ) . (1.2)We can announce now our results. Theorem 1.
Let m > . There exists a constant C > , independent on V , such that if k V k Y < C , then every eigenvalues z ∈ σ p ( D V ) of D V lies in the union z ∈ B R ( x − ) ∪ B R ( x +0 ) of the two closed disks in C with centers in x − , x +0 and radius R , with x ± := ± m V + 1 V − , R := m VV − , V ≡ V ( V ) := (cid:20) ( n + 1) C k V k Y − n (cid:21) > . Theorem 2.
Let m = 0 . There exists a constant C > , independent on V , such that if k V k Y < C , then D V has no eigenvalues. In particular, we have σ ( D V ) = σ e ( D V ) = R . emark 1.1. The crucial tool in our proof is a uniform resolvent estimate for the re-solvent of the free Dirac operator. This approach is inspired by [22], where the result byKenig, Ruiz and Sogge [30] was used for the same purpose. In our case, we prove inSection 2 the following estimates, of independent interest: k R − ∆ ( z ) k X → X ∗ ≤ C | z | − / , k ∂ k R − ∆ ( z ) k X → X ∗ ≤ C. and k R D ( z ) k X → X ∗ ≤ C " n + (cid:12)(cid:12)(cid:12)(cid:12) z + mz − m (cid:12)(cid:12)(cid:12)(cid:12) sgn( ℜ z ) / . These can be regarded as precised resolvent estimates of Agmon–H¨ormander type. Notealso that similar uniform estimates, in less sharp norms, were proved earlier by the firstand second Authors in [12, 13, 19].In Section 3, we combine our uniform estimates with the Birman-Schwinger principle,enabling in Section 4 to complete the proof of Theorems 1 & 2.
Remark 1.2.
The following embedding for the space Y under consideration hold: Y ֒ → L n, ( R n ) ֒ → L n ( R n ) , (1.3) where L p,q ( R n ) is the Lorentz space. Moreover, we have W , ( R n ) ֒ → n \ j =1 L b x j L ∞ x j ( R n ) ֒ → L n/ ( n − , ( R n ) , where W m,p ( R n ) is the Sobolev space, and so, for the -dimensional case, we get in par-ticular W , ( R ) ֒ → Y = L x L ∞ x ( R ) ∩ L x L ∞ x ( R ) ֒ → L , ( R ) ֒ → L ( R ) . We refer to the papers by Fournier [21], Blei and Fournier [5] and Milman [33] for theseinclusions.
Let us start fixing constants r, R, δ > < r < R, √ R − < δ < , and consider the open cover S = {S + j , S − j , S ∞ } j ∈{ ,...,n } of the space R n defined by S ± j = { ξ ∈ R n : ± ξ j > δ | b ξ j | , | ξ | < R } , S ∞ = { ξ ∈ R n : | ξ | > r } . We can find a smooth partition of unity { χ + j , χ − j , χ ∞ } j ∈{ ,...,n } subordinate to S , i.e. afamily of smooth positive functions such thatsupp χ ± j ⊂ S ± j , supp χ ∞ ⊂ S ∞ , χ ∞ + n X j =1 [ χ + j + χ − j ] ≡ . χ = { χ j } j ∈{ ,...,n } , with χ j := χ + j + χ − j + 1 n χ ∞ , (2.1)and hence, for j ∈ { , . . . , n } , the Fourier multipliers χ j ( | z | − / D ) f = F − ξ [ χ j ( | z | − / ξ ) F x f ] . Note in particular that n X j =1 χ j ( | z | − / D ) f = f. (2.2)Therefore, the following estimates hold. Lemma 1.
For every z ∈ ρ ( − ∆) = C \ [0 , + ∞ ) , f ∈ L x j L b x j and j, k ∈ { , . . . , n } , wehave that (cid:13)(cid:13) χ j (cid:0) | z | − / D (cid:1) R − ∆ ( z ) f (cid:13)(cid:13) L ∞ xj L b xj ≤ C | z | − / k f k L xj L b xj , (cid:13)(cid:13) χ j (cid:0) | z | − / D (cid:1) ∂ k R − ∆ ( z ) f (cid:13)(cid:13) L ∞ xj L b xj ≤ C k f k L xj L b xj , where { χ j } j ∈{ ,...,n } are defined in (2.1) and C > does not depend on z . In particular, itfollows that k R − ∆ ( z ) k X → X ∗ ≤ C | z | − / , k ∂ k R − ∆ ( z ) k X → X ∗ ≤ C. Lemma 2.
For every z ∈ ρ ( D ) = C \ { ζ ∈ R : | ζ | ≥ m } , f ∈ L x j L b x j and j ∈ { , . . . , n } we have that (cid:13)(cid:13) χ j (cid:0) | z − m | − / D (cid:1) R D ( z ) f (cid:13)(cid:13) L ∞ xj L b xj ≤ C " n + (cid:12)(cid:12)(cid:12)(cid:12) z + mz − m (cid:12)(cid:12)(cid:12)(cid:12) sgn( ℜ z ) / k f k L xj L b xj , where { χ j } j ∈{ ,...,n } are defined in (2.1) and C > is the same as in Lemma 1. Inparticular, it follows that k R D ( z ) k X → X ∗ ≤ C " n + (cid:12)(cid:12)(cid:12)(cid:12) z + mz − m (cid:12)(cid:12)(cid:12)(cid:12) sgn( ℜ z ) / . Remark 2.1.
Before to proceed further, we give an heuristic explanation for the choice ofthe localization in the frequency domain via the Fourier multiplier χ j ( | z | − / D ) . Since thesymbol ( | ξ | − z ) − of the resolvent R − ∆ ( z ) blows-up as z → ζ ≥ , our trick is to use thenorms L ∞ x j L b x j for j ∈ { , . . . , n } , which allows us to restrict the problem from the sphericalsurface { ξ ∈ R n : | ξ | = | z | − / } to the “equator” given by { ξ ∈ R n : ξ j = 0 , | b ξ j | = | z | − / } .We then avoid these regions thanks to the smooth functions χ j . roof of Lemma 1. The “in particular” part trivially follows from (2.2) and from thedefinitions of the norms on X and X ∗ .For the simplicity, from now on C > z and which can change from line to line. Clearly, by scaling, we canconsider only unitary z ∈ C \ { } . Thus it is sufficient to prove that k χ j ( D ) ∂ sk R − ∆ ( z ) f k L ∞ xj L b xj ≤ C k f k L xj L b xj , where | z | = 1, s ∈ { , } , ∂ k = 1 , ∂ k = ∂ k and j, k ∈ { , . . . , n } . This is equivalent to (cid:13)(cid:13)(cid:13)(cid:13) F − ξ (cid:18) ξ sk χ j ( ξ ) | ξ | − l − iε F x f (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) L ∞ xj L b xj ≤ C k f k L xj L b xj , (2.3)where we have settled z = l + iε , l + ε = 1 and z = 1. We proceed splitting χ j in thefunctions which appear in its definition (2.1), localizing ourselves in the regions of thefrequency domain near the unit sphere, i.e. S ± j , and far from it, i.e. S ∞ . Estimate on S ± j . We want to prove (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) F − ξ ξ sk χ ± j ( ξ ) | ξ | − l − iε F x f !(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ∞ xj L b xj ≤ C k f k L b xj L xj . (2.4)Let us define the family of operators T ± j : L px j L b x j → L px j L b x j , f T ± j f := F − ξ (cid:16) ˆ f ◦ Φ (cid:17) , where Φ( ξ ) := ( ξ j + ϕ ( b ξ j ) , b ξ j ) , ϕ ( b ξ j ) := ± q | − | b ξ j | | . Roughly speaking, the operators T ± j “flatten” in the frequency domain the hemisphereof the unitary sphere, namely { ξ ∈ R n : | ξ | = 1 , ± ξ j > } . Writing more explicitly theintroduced operators, we have that T ± j f ( x ) = 1(2 π ) n/ Z R n e ix · ξ ˆ f ( ξ j + ϕ ( b ξ j ) , b ξ j ) dξ = 1(2 π ) n Z R n e ix · ξ Z R n f ( y ) e − iy · ( ξ j + ϕ ( b ξ j ) , b ξ j ) dydξ = 1(2 π ) n Z R n − e i b x j · b ξ j Z R n − e − iy ′ · b ξ j Z R Z R f ( y ) e i ( x j − y j ) ξ j − iy j ϕ ( b ξ j ) dy j dξ j dy ′ d b ξ j = 12 π F − b ξ j F y ′ (cid:18) e − ix j ϕ ( b ξ j ) Z R Z R f ( y ) e i ( x j − y j ) ξ j dy j dξ j (cid:19) = F − b ξ j F y ′ (cid:16) e − ix j ϕ ( b ξ j ) f ( x j , y ′ ) (cid:17) where we exploited the substitution ξ j ξ j − ϕ ( b ξ j ) in the fourth equality. Hence, applyingthe Plancherel Theorem two times, we obtain that T ± j are isometries respect to the norm L px j L b x j , i.e. for p ∈ [1 , + ∞ ] it holds the relation (cid:13)(cid:13) T ± j f (cid:13)(cid:13) L pxj L b xj = k f k L pxj L b xj . (2.5)8hanks to this equality, we can write (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) F − ξ ξ sk χ ± j ( ξ ) | ξ | − l − iε F x f !(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ∞ xj L b xj = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) T ± j F − ξ ξ sk χ ± j ( ξ ) | ξ | − l − iε F x f !(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ∞ xj L b xj = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) F − ξ ( ξ sk χ ± j ) ◦ Φ | Φ | − l − iε d T ± j f !(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ∞ xj L b xj = 1 √ π (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) a l,ε ( D ) ψ ∗ x j F − ξ j d T ± j fξ j − i | ε | !(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ∞ xj L b ξj ≤ √ π k a l,ε ( D ) ψ k L xj L ∞ b ξj (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) F − ξ d T ± j fξ j − i | ε | !(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ∞ xj L b ξj where the last inequality follows from the Young’s Theorem and a l,ε ( D ) ψ = F − ξ j (cid:0) a l,ε F x j ( ψ ) (cid:1) ,a l,ε ( ξ ) := ( ξ j − i | ε | ) (cid:18) ξ k ± δ k,j q − | b ξ j | (cid:19) s ξ j (cid:18) ξ j ± q − | b ξ j | (cid:19) + 1 − l − iε q ( χ ± j ◦ Φ)( ξ ) ,ψ ( x j , b ξ j ) = F − ξ j (cid:16)q ( χ ± j ◦ Φ)( ξ ) (cid:17) . Observe that we dropped the absolute value in the definition of ϕ , namely it results q | − | b ξ j | | = q − | b ξ j | , because supp { χ ± j ◦ Φ } ⊂ { ξ ∈ R n : | b ξ j | ≤ } , thanks to thedefinition of S ± j and from the assumption δ ≥ √ R −
1. Despite of the involute definition,it is simple to see that a l,ε ( D ) ψ ∈ S , where S is the space of Schwartz functions, since a l,ε ( D ) ψ is the inverse Fourier transform of a smooth compactly supported function.Moreover, considering a l,ε ( D ) ψ as a pseudo-differential operator of symbol a l,ε applied tothe Schwartz function ψ , let us to observe that, since letting l + iε → l + iε → a l,ε ( ξ ) = (cid:18) ξ k ± δ k,j q − | b ξ j | (cid:19) s ξ j ± q − | b ξ j | s χ ± j (cid:18) ξ j ± q − | b ξ j | , b ξ j (cid:19) =: a ( ξ ) ∈ S pointwisely, then a l,ε ( D ) ψ → a ( D ) ψ in S , and solim l + iε → k a l,ε ( D ) ψ k L xj L ∞ b ξj = k a ( D ) ψ k L xj L ∞ b ξj < + ∞ . Thus, k a l,ε ( D ) ψ k L xj L ∞ b ξj is uniformly bounded respect to z ∈ C unitary.Hence we have obtained (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) F − ξ ξ sk χ ± j ( ξ ) | ξ | − l − iε F x f !(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ∞ xj L b xj ≤ C (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) F − ξ d T ± j fξ j − i | ε | !(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ∞ xj L b ξj . (2.6)9y Plancherel and Young Theorems, and by equality (2.5), we have √ π (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) F − ξ j d T ± j fξ j − i | ε | !(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ∞ xj L b ξj = (cid:13)(cid:13)(cid:13)(cid:13) F − ξ j (cid:18) ξ j − i | ε | (cid:19) ∗ x j F b x j ( T ± j f ) (cid:13)(cid:13)(cid:13)(cid:13) L ∞ xj L b ξj = (cid:13)(cid:13) ie −| ε | x j θ ∗ x j F b x j ( T ± j f ) (cid:13)(cid:13) L ∞ xj L b ξj ≤ (cid:13)(cid:13)(cid:13)(cid:13) e −| ε | x j θ ∗ x j (cid:13)(cid:13) T ± j f (cid:13)(cid:13) L b xj (cid:13)(cid:13)(cid:13)(cid:13) L ∞ xj ≤ (cid:13)(cid:13) e −| ε | x j θ (cid:13)(cid:13) L ∞ xj k f k L xj L b xj = k f k L xj L b xj , where θ ≡ θ ( x j ) is the Heaviside function. Thus, inserting in (2.6), we get (2.4). Estimate on S ∞ . We want to prove now (cid:13)(cid:13)(cid:13)(cid:13) F − ξ (cid:18) ξ sk χ ∞ ( ξ ) | ξ | − l − iε F x f (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) L ∞ xj L b xj ≤ C k f k L b xj L xj . (2.7)We need to distinguish three cases depending on whether we are localized in the regionsdefined by C R,j := { ξ ∈ R n : | b ξ j | > R } , C R,j := { ξ ∈ R n : | b ξ j | ≤ R, | ξ j | ≤ R } , C R,j := { ξ ∈ R n : | b ξ j | ≤ R, | ξ j | > R } . Let us set then χ ∞ ( ξ ) := ( | b ξ j | > R ,0 otherwise, χ ∞ ( ξ ) := ( χ ∞ ( ξ ) if | b ξ j | ≤ R and | ξ j | ≤ R ,0 otherwise, χ ∞ ( ξ ) := ( | b ξ j | ≤ R and | ξ j | > R ,0 otherwise,and observe that χ ∞ = χ ∞ + χ ∞ + χ ∞ , since χ ∞ ≡ | ξ | > R , from the requirementson the cover S and the partition χ .By Plancherel Theorem and H¨older’s, Young’s and Minkowski’s integral inequalities,for h ∈ { , , } we infer (cid:13)(cid:13)(cid:13)(cid:13) F − ξ (cid:18) ξ sk χ h ∞ ( ξ ) | ξ | − l − iε F x f (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) L ∞ xj L b xj ≤ C h k f k L xj L b xj with C h := 1 √ π (cid:13)(cid:13)(cid:13)(cid:13) F − ξ j (cid:18) ξ sk χ h ∞ ( ξ ) ξ j + σ (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) L ∞ xj L ∞ b ξj , σ := q | b ξ j | − l − iε. (2.8)10ere and below, we always consider the principal branch of the complex square rootfunction. Clearly, if we prove that C h for h ∈ { , , } is bounded uniformly respect to l, ε , we recover (2.7). Estimate on C R,j . Observing that χ ∞ ( ξ ) ≡ χ ∞ ( b ξ j ) and noting that ℜ σ = s | σ | + | b ξ j | − l > , we can compute explicitly the Fourier transforms: • if k = j , then C = (cid:13)(cid:13)(cid:13)(cid:13) χ ∞ ( b ξ j ) ξ sk e − σ | x j | σ (cid:13)(cid:13)(cid:13)(cid:13) L ∞ xj L ∞ b ξj ≤ (cid:13)(cid:13)(cid:13)(cid:13) χ ∞ ( b ξ j ) | b ξ j | s e −ℜ σ | x j | | σ | (cid:13)(cid:13)(cid:13)(cid:13) L ∞ xj L ∞ b ξj ≤ sup | b ξ j | >R | b ξ j | s | b ξ j | − l | b ξ j | + 1) / ≤ ( R s √ R − if l > / l ≤ • if s = 1, k = j , then C = (cid:13)(cid:13)(cid:13)(cid:13) χ ∞ ( b ξ j ) i { x j } e − σ | x j | (cid:13)(cid:13)(cid:13)(cid:13) L ∞ xj L ∞ b ξj ≤ . Estimate on C R,j . Expliciting the definition of inverse Fourier transform in (2.8) andfrom the fact that χ ∞ ( ξ ) = 0 when | ξ | < r , we can trivially see that C ≤ π (cid:13)(cid:13)(cid:13)(cid:13)Z + ∞−∞ | e ix j ξ j | | ξ sk | χ ∞ ( ξ ) || ξ | − l | dξ j (cid:13)(cid:13)(cid:13)(cid:13) L ∞ xj L ∞ b ξj ≤ (2 R ) s π (cid:13)(cid:13)(cid:13)(cid:13) χ ∞ ( ξ ) | ξ | − (cid:13)(cid:13)(cid:13)(cid:13) L ∞ b ξj L ξj which is finite since χ ∞ is compactly supported due to its definition. Estimate on C R,j . Expliciting the inverse Fourier transform in (2.8), recalling thedefinition of χ ∞ and exploiting the substitution ξ j sgn { x j } ξ j , we have C = 12 π (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (1 − χ ∞ )( b ξ j ) Z | ξ j | >R e i | x j | ξ j ξ sk ξ j + σ dξ j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ∞ xj L ∞ b ξj = 12 π (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z | ξ j | >R ψ ( x j , b ξ j , ξ j ) dξ j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ∞ xj L ∞ b ξj where, for fixed b ξ j , x j , the complex function ψ ( x j , b ξ j , · ) : C → C is defined by ψ ( x j , b ξ j , w ) := (1 − χ ∞ )( b ξ j ) ξ sk w + σ e i | x j | w , if k = j ,(1 − χ ∞ )( b ξ j ) ww + σ e i | x j | w , if s = 1, k = j ,11hich is holomorphic in C \ { w − , w + } , where w ± = ± iσ . Observe that ψ ≡ | b ξ j | > R ,and if | b ξ j | ≤ R we have | w ± | = | σ | = q ( | b ξ j | − l ) + ε < √ R. (2.9)Define, for a radius A >
0, the semicircle γ A := { Ae iθ : θ ∈ [0 , π ] } in the upper halfcomplex plane. Fixing ρ > R , by the Residue Theorem, we get Z [ − ρ, − R ] − Z γ R + Z [2 R,ρ ] + Z γ ρ ! ψ ( x j , b ξ j , w ) dw = 0 . Observing that we can consider x j = 0, letting ρ → + ∞ we can apply the Jordan’s lemmato the integral on the curve γ ρ , finally getting C = 12 π (cid:13)(cid:13)(cid:13)(cid:13)Z γ R ψ ( x j , b ξ j , w ) dw (cid:13)(cid:13)(cid:13)(cid:13) L ∞ xj L ∞ b ξj ≤ (2 R ) s π (cid:13)(cid:13)(cid:13)(cid:13) (1 − χ ∞ )( b ξ j ) Z π dθ | R e iθ + σ | (cid:13)(cid:13)(cid:13)(cid:13) L ∞ b ξj ≤ (2 R ) s − where we used the relation (2.9).Summing all up, we can finally recover the desired estimate (2.3), where the positiveconstant C does not depend on l, ε , but only on R and the partition χ .We can prove now Lemma 2, as a straightforward corollary of Lemma 1. Proof of Lemma 2.
Again, the “in particular” part follows from (2.2) and the definitionsof the X and X ∗ norms.From the anticommutation relations (1.1) we infer, for every z ∈ C ,( D − zI N )( D + zI N ) = ( − ∆ + m − z ) I N . So, thanks to this well-known trick, for z ∈ ρ ( D ) we can write R D ( z ) = ( D + zI N ) R − ∆ ( z − m ) I N . Set f j = χ j ( | z − m | − / D ) f for the simplicity. Exploiting Lemma 1, it is easy to get k R D ( z ) f j k L ∞ xj L b xj ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X k =1 α k ∂ k R − ∆ ( z − m ) f j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ∞ xj L b xj + (cid:13)(cid:13) ( mα + zI N ) R − ∆ ( z − m ) f j (cid:13)(cid:13) L ∞ xj L b xj ≤ n X k =1 (cid:13)(cid:13) ∂ k R − ∆ ( z − m ) f j (cid:13)(cid:13) L ∞ xj L b xj + max {| z + m | , | z − m |} (cid:13)(cid:13) R − ∆ ( z − m ) f j (cid:13)(cid:13) L ∞ xj L b xj ≤ C " n + (cid:12)(cid:12)(cid:12)(cid:12) z + mz − m (cid:12)(cid:12)(cid:12)(cid:12) sgn( ℜ z ) / k f k L xj L b xj . as claimed. 12 The Birman-Schwinger principle
In this section, working in great generality, we precisely define the closed extension of aperturbed operator with a factorizable potential, formally defined as H + B ∗ A . Next, westate an abstract version of the Birman-Schwinger principle. We will follow the approachof Kato [29] and Konno and Kuroda [31].Let H , H ′ be Hilbert spaces and consider the densely defined, closed, linear operators H : dom( H ) ⊆ H → H , A : dom( A ) ⊆ H → H ′ , B : dom( B ) ⊆ H → H ′ , such that ρ ( H ) = ∅ anddom( H ) ⊆ dom( A ) , dom( H ∗ ) ⊆ dom( B ) . For the simplicity, we assume also that σ ( H ) ⊂ R and σ p ( H ) = ∅ . For z ∈ ρ ( H ),denote by R H ( z ) = ( H − z ) − the resolvent operator of H .As a warm-up, to sketch the idea of the principle, let us firstly consider the case ofbounded operators A and B . Thus, H = H + B ∗ A is well-defined as a sum operator.Moreover, if z ∈ ρ ( H ), we can define the bounded operator Q ( z ) = A ( H − z ) − B ∗ . Itis easy to check that z ∈ σ p ( H ) ∩ ρ ( H ) implies − ∈ σ p ( Q ( z )), and so k Q ( z ) k H ′ → H ′ ≥ Q ( z ) give us informationon the localization of the non-embedded eigenvalues of H .Let us return to the general setting of an unbounded potential B ∗ A . Furthermore, westudy also the case of embedded eigenvalues. To afford these, we assume a stronger setof hypotheses respect to the ones in [31], which instead will be proved here in Lemma 3. Hypothesis ( Hyp ) . Let X a complex Banach space of function on R n such that thefollowing estimates hold true: k AR H ( z ) k X → H ′ ≤ α Λ( z ) , k B ∗ k H ′ → X ≤ β, where α, β > , Λ( z ) is finite for z ∈ ρ ( H ) and continue up to σ ( H ) \ Ω , for some set Ω ⊆ σ ( H ) , and there exists z ∈ ρ ( H ) such that Λ( z ) < ( αβ ) − . Lemma 3 (Birman-Schwinger operator) . Assume (Hyp). Then, for z ∈ ρ ( H ) , the oper-ator AR H ( z ) B ∗ , densely defined on dom( B ∗ ) , has a closed extension Q ( z ) in H ′ , Q ( z ) = AR H ( z ) B ∗ with norm bounded by k Q ( z ) k H ′ → H ′ ≤ αβ Λ( z ) . (3.1) Moreover, there exists z ∈ ρ ( H ) such that − ∈ ρ ( Q ( z )) . Proof.
For z ∈ ρ ( H ) and ϕ ∈ dom( B ∗ ), we have k AR H ( z ) B ∗ ϕ k H ′ → H ′ ≤ k AR H ( z ) k X → H ′ k B ∗ k H ′ → X k ϕ k H ′ and so by density (3.1). In particular, for z ∈ ρ ( H ) such that Λ( z ) < ( αβ ) − , weget k Q ( z ) k H ′ → H ′ <
1. Hence, from Neumann series there exists (1 + Q ( z )) − , and so − ∈ ρ ( Q ( z )). 13et us collect some useful facts in the next lemma. Lemma 4.
Suppose (Hyp) and fix z, z , z ∈ ρ ( H ) . Then the following relations holdtrue:(i) AR H ( z ) ∈ B ( H , H ′ ) , R H ( z ) B ∗ = [ B ( H ∗ − z ) − ] ∗ ∈ B ( H ′ , H ) ,(ii) R H ( z ) B ∗ − R H ( z ) B ∗ = ( z − z ) R H ( z ) R H ( z ) B ∗ = ( z − z ) R H ( z ) R H ( z ) B ∗ ,(iii) Q ( z ) = AR H ( z ) B ∗ , Q ( z ) ∗ = BR H ( z ) ∗ A ∗ ,(iv) ran( R H ( z ) B ∗ ) ⊆ dom( A ) , ran( R H ( z ) ∗ A ∗ ) ⊆ dom( B ) ,(v) Q ( z ) − Q ( z ) = ( z − z ) AR H ( z ) R H ( z ) B ∗ = ( z − z ) AR H ( z ) R H ( z ) B ∗ .Proof. See Lemma 2.2 in [28].We can construct now the extension of the perturbed operator H + B ∗ A . Lemma 5 (Extension of operators with factorizable potential) . Suppose (Hyp). Fix z ∈ ρ ( H ) , given by Lemma 3, such that − ∈ ρ ( Q ( z )) . The operator R H ( z ) = R H ( z ) − R H ( z ) B ∗ (1 + Q ( z )) − AR H ( z ) (3.2) defines a densely defined, closed, linear operator H in H , which have R H ( z ) as resolvent, R H ( z ) = ( H − z ) − , and which is an extension of H + B ∗ A .Proof. We refer to Theorem 2.3 in [28]. See also the work by Kato [29].Finally, we enunciate the Birman-Schwinger principle.
Lemma 6 (Birman-Schwinger principle) . Suppose (Hyp). Let z ∈ ρ ( H ) , given byLemma 3, such that − ∈ ρ ( Q ( z )) and H be the extension of H + B ∗ A , given byLemma 5. Fix z ∈ σ p ( H ) with eigenfunction = ψ ∈ dom( H ) , i.e. Hψ = zψ .Setted φ := Aψ , we have that φ = 0 and(i) if z ∈ ρ ( H ) then Q ( z ) φ = − φ and in particular ≤ k Q ( z ) k H ′ → H ′ ≤ αβ Λ( z ); (ii) if z ∈ σ ( H ) \ Ω and ψ ∈ X loc , i.e. K ψ ∈ X for every compact set K ⊂ R n where K is the indicator function of K , then lim ε → ± ( ϕ, Q ( z + iε ) φ ) H ′ = − ( ϕ, φ ) H ′ (3.3) for every ϕ ∈ H ′ compactly supported, and in particular ≤ lim inf ε → ± k Q ( z + iε ) k H ′ → H ′ ≤ αβ lim ε → ± Λ( z + iε ) . (3.4)14 roof. Let us prove only case (ii) for the embedded eigenvalues, being case (i) similar andeasier, adapting the argument of Lemma 1 in [31] for the non-embedded eigenvalues.Noted that Hψ = zψ is equivalent to ψ = ( z − z ) R H ( z ) ψ, (3.5)we obtain from (3.2) that( H − z − iε ) R H ( z ) ψ = − ( z − z ) R H ( z ) B ∗ (1+ Q ( z )) − AR H ( z ) ψ − iεR H ( z ) ψ. (3.6)Define e ψ = (1 + Q ( z )) − AR H ( z ) ψ . If e ψ = 0, by (3.6) follows ( H − z ) R H ( z ) ψ = 0.Since 0 = R H ( z ) ψ ∈ dom( H ), we get z ∈ σ p ( H ), which however is empty by ourassumptions on H . Thus, we proved e ψ = 0. Moreover, we can show the identity φ = Aψ = ( z − z )(1 + Q ( z )) − AR H ( z ) ψ = ( z − z ) e ψ, (3.7)from which in particular φ = 0. Indeed, by (3.2) and (iii) of Lemma 4, it follows that AR H ( z ) = (1 + Q ( z )) − AR H ( z )which combined with (3.5) gives us (3.7).Let us prove the limit (3.3). Multiplying by (1 + Q ( z )) − AR H ( z + iε ) both sides of(3.6), we obtain e ψ = − ( z − z )(1 + Q ( z )) − AR H ( z + iε ) R H ( z ) B ∗ e ψ − iε (1 + Q ( z )) − AR H ( z + iε ) R H ( z ) ψ and so, by (v) of Lemma 4 and by the resolvent identity, we have e ψ = − z − z z − z + iε (1 + Q ( z )) − [ Q ( z + iε ) − Q ( z )] e ψ − iεz − z + iε (1 + Q ( z )) − A [ R H ( z + iε ) − R H ( z )] ψ = e ψ − z − z z − z + iε (1 + Q ( z )) − (1 + Q ( z + iε )) e ψ − iεz − z + iε (1 + Q ( z )) − AR H ( z + iε ) ψ, from which, using identity (3.7), we finally arrive to Q ( z + iε ) φ = − φ − iεAR H ( z + iε ) ψ. Fixed ϕ ∈ H ′ with compact support K , we have that( ϕ, Q ( z + iε ) φ ) H ′ = − ( ϕ, φ ) H ′ − iε ( ϕ, AR H ( z + iε ) ψ ) H ′ . (3.8)Since ( ϕ, AR H ( z + iε ) ψ ) H ′ = Z K ϕ † · AR H ( z + iε ) ψ = Z K R H ( z − iε ) A ∗ ϕ † · ψ = ( ϕ, AR H ( z + iε )[ K ψ ]) H ′ ,
15y (Hyp) we get | ( ϕ, AR H ( z + iε ) ψ ) H ′ | ≤ k ϕ k H ′ k AR H ( z + iε )[ K ψ ] k H ′ ≤ α Λ( z + iε ) k ϕ k H ′ k K ψ k X , and so the last term in (3.8) vanishes as ε → ± , proving (3.3).Finally, we prove the first inequality in (3.4), being the second one given by Lemma 3.For n ∈ N , let χ n ( x ) := χ ( x/n ) where χ ∈ C ∞ ( R n ) is a cut-off function such that χ ( x ) = 1for | x | ≤ χ ( x ) = 0 for | x | ≥
2. Since, by (3.3), | ( χ n φ, φ ) H ′ | = lim ε → ± | ( χ n φ, Q ( z + iε ) φ ) H ′ | ≤ k χ n φ k H ′ k φ k H ′ lim inf ε → ± k Q ( z + iε ) k H ′ → H ′ we get (3.4) letting n → + ∞ . Here we simply apply the results of the last section to our problem. In our case, H = H ′ = L ( R n ; C N ), X = T nj =1 L x j L e x j , H is the free Dirac operator D , and the factorization of V is reached considering the polar decomposition V = U W , i.e. W = ( V ∗ V ) / and theunitary matrix U is a partial isometry, and setting A = W / and B = W / U ∗ .It is easy to see that hypothesis (Hyp) holds thanks to Lemma 2 with α = β = k V k / Y , Λ( z ) = nC " n + (cid:12)(cid:12)(cid:12)(cid:12) z + mz − m (cid:12)(cid:12)(cid:12)(cid:12) sgn( ℜ z ) / , Ω = ( {− m, m } if m = 0, ∅ if m = 0.Indeed, for ϕ ∈ H , k AR D ( z ) ϕ k H ≤ n X j =1 (cid:13)(cid:13) A χ j ( | z − m | − / D ) R D ( z ) ϕ (cid:13)(cid:13) H ≤ C " n + (cid:12)(cid:12)(cid:12)(cid:12) z + mz − m (cid:12)(cid:12)(cid:12)(cid:12) sgn( ℜ z ) / n X j =1 k A k L xj L ∞ b xj k ϕ k L xj L b xj ≤ nC " n + (cid:12)(cid:12)(cid:12)(cid:12) z + mz − m (cid:12)(cid:12)(cid:12)(cid:12) sgn( ℜ z ) / k V k / Y k ϕ k X , k B ∗ ϕ k X ≤ k V k Y k ϕ k H , where we used the relation k A k L xj L ∞ b xj = k B ∗ k L xj L ∞ b xj = (cid:13)(cid:13) W / (cid:13)(cid:13) L xj L ∞ b xj = k V k / L xj L ∞ b xj . To see that there exists z ∈ ρ ( D ) such that Λ( z ) < ( αβ ) − , define C = [ n ( n + 1) C ] − , V = [( n + 1) C / k V k Y − n ] . Since from the hypothesis of Theorems 1 & 2 we have k V k Y < C and so V >
1, thecondition 1 ≤ αβ Λ( z ) is equivalent to V ≤ | z/z | if m = 0, and to (cid:18) ℜ z − sgn( ℜ z ) m V + 1 V − (cid:19) + ℑ z ≤ (cid:18) m VV − (cid:19) (4.1)16f m >
0. Then, if m = 0 it is sufficient to choose z ∈ C \ R , whereas if m > z ∈ ρ ( D ) outside the disks in the statement of Theorem 1. Finally, it is trivialto see that if ψ ∈ H , then ψ ∈ X loc , since, for every compact set K ∈ R n , we have k K ψ k L xj L b xj ≤ κ k ψ k L , for some constant κ depending on K .Thus, we can apply Lemma 6, 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 thepotential V ∈ Y = T nj =1 L x j L ∞ b x j ( R n ) leaves the essential spectrum invariant and that theresidual spectrum of D V is absent. To get the invariance of the essential spectrum, it issufficient to prove that, fixed z ∈ ρ ( D ) such that − ∈ ρ ( Q ( z )), the operator AR D ( z ) isan Hilbert-Schmidt operator, and hence compact. Thus, identity (3.2) gives us R D V ( z ) − R D ( z ) = − R D ( z ) B ∗ (1 + Q ( z )) − AR D ( z )from which follows that R D V ( z ) − R D ( z ) is compact and so, by Theorem 9.2.4 in [17], σ e ( D V ) = σ e ( D ) = ( −∞ , m ] ∪ [ m, ∞ ) . To see that AR D ( z ) is an Hilbert-Schmidt operator, we need to prove that its kernel A ( x ) K ( z, x − y ) is in L ( R n × R n ; C N ), where K ( z, x − y ) is the kernel of the resolvent( D − z ) − . By the Young inequality (cid:13)(cid:13) A ( D − z ) − (cid:13)(cid:13) HS = Z R n Z R n | A ( x ) | | K ( z, x − y ) | dxdy ≤ k V k L p k K k L q (4.2)where 1 /p + 1 /q = 2. Hence we need to find in which Lebesgue space L q ( R n ; C N ) thekernel K ( z, x ) lies. For z ∈ ρ ( − ∆) = C \ [0 , ∞ ), it is well-know (see e.g. [26]) that thekernel K ( z, x − y ) of the resolvent operator ( − ∆ − z ) − is given by K ( z, x − y ) = 1(2 π ) n/ (cid:18) √− z | x − y | (cid:19) n − K n − ( √− z | x − y | )where K ν ( w ) is the modified Bessel function of second kind and we consider the principalbranch of the complex square root. Fixed now z ∈ ρ ( D ) = C \ { ζ ∈ R : | ζ | ≥ m } , fromthe identity ( D − zI N ) − = ( D + zI N )( − ∆ + m − z ) − I N and relations 9.6.26 in [1] for the derivative of the modified Bessel functions, we obtain K ( z, x − y ) = 1(2 π ) n/ (cid:18) k ( z ) | x − y | (cid:19) n α · ( x − y ) K n ( k ( z ) | x − y | )+ 1(2 π ) n/ (cid:18) k ( z ) | x − y | (cid:19) n − ( mα + z ) K n − ( k ( z ) | x − y | )where for the simplicity k ( z ) = √ m − z . From the limiting form for the modified Besselfunctions K ν ( w ) ∼
12 Γ( ν ) (cid:16) w (cid:17) − ν for ℜ ν > w → K ( w ) ∼ − ln w for w → K ν ( w ) ∼ r π w e − w for z → ∞ in | arg z | ≤ π/ − δ ,17e obtain that k K ( z, x ) k ≤ C ( n, m, z ) | x | n − if | x | ≤ x ( n, m, z ) | x | − ( n − / e −ℜ k ( z ) | x | if | x | ≥ x ( n, m, z )for some positive constants C ( n, m, z ), x ( n, m, z ) depending on z . Hence is clear that K ( z, x ) ∈ L q ( R n ; C N ) for 2 q < n/ ( n −
1) and, consequently, from equation (4.2) we havethat A ( D − z ) − is an Hilbert-Schmidt operator if V ∈ L p ( R n ; C N ) for p > n/
2. Since,by (1.3), V ∈ L n ( R n ; C N ), the proof of the identity σ e ( D V ) = σ e ( D ) is complete.Finally, to get the absence of residual spectrum, since ρ ( D ) = C \ σ e ( D ) is composedby one, or two in the massless case, connected components which intersect ρ ( D V ) in anon-empty set, by Theorem XVII.2.1 in [27] we have σ ( D V ) \ σ e ( D V ) = σ d ( D V ) . Acknowledgement
The authors are members of the Gruppo Nazionale per L’Analisi Matematica, la Prob-abilit`a e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica(INdAM). The third author is partially supported by