ON THE NUMBER OF EIGENVALUES OF THE DIRAC OPERATOR IN A BOUNDED INTERVAL

A BSTRACT . Let H 0 be the free Dirac operator and V ⩾ 0 be a positive potential. We study the discrete spectrum of H ( α ) = H 0 − αV in the interval ( − 1 , 1) for large values of the coupling constant α > 0 . In particular, we obtain an asymptotic formula for the number of eigenvalues of H ( α ) situated in a bounded interval [ λ, µ ) as α → ∞ .


STATEMENT OF THE MAIN THEOREM
Let H 0 be the free Dirac operator where γ j are 4 × 4 selfadjoint matrices obeying the conditions The operator H 0 is selfadjoint in the space L 2 (R 3 ; C 4 ), consisting of functions on R 3 that take values in C 4 .The spectrum of H 0 is the set σ(H 0 ) = (−∞, −1] ∪ [1, ∞).Let V ⩾ 0 be a bounded potential on R 3 .Define H(α) to be the operator We will always assume that V ∈ L 3 (R 3 ).In this case, besides having a continuous spectrum that coincides with σ(H 0 ), the operator H(α) might have a discrete spectrum in the interval (−1, 1).Choose λ and µ so that −1 < λ < µ < 1.We define N (α) to be the number of eigenvalues of H(α) inside [λ, µ).
Our main result is the theorem below which establishes the rate of growth of N (α) at infinity.The symbol f + denotes the positive part f + = (|f | + f )/2 of f , which can be either a real number or a real-valued function.
The question about the number of eigenvalues of the Dirac operator in a bounded interval is considered here for the first time.This theorem is new.Perturbations V ∈ L 3 (R 3 ) were studied in [16] by M. Klaus and,later,in [4] by M. Birman and A. Laptev.However, the object of the study in [16] and [4] was completely different from N (α), considered in this article.The main results of [16] and [4] imply that if V ∈ L 3 (R 3 ), then the number N (λ, α) of eigenvalues of H(t) passing a regular point λ ∈ (−1, 1) as t increases from 0 to α satisfies In addition, M. Klaus proved in [16] that if V ∈ L 3 ∩ L 3/2 , then the asymptotic formula (1.2) holds even for λ = 1.In this case, N (λ, α) is interpreted as the number of eigenvalues of H(t) that appear at the right edge of the gap as t increases from 0 to α.
The crux of the problem.Observe that N (α) = N (µ, α)−N (λ, α).However, since the expression on the right hand side of (1.2) does not depend on λ, this formula only implies that In order to obtain an asymptotic formula for N (α) one would need to know the second term in the asymptotics of N (λ, α).The second term in (1.2) has never been obtained.This explains why the problem is challenging.Another reason why the problem is challanging is that the Dirichlet-Neumann bracketing that is often used for Schrödinger operators cannot be applied to Dirac operators.To prove Theorem 1.1, one needs to develop a new machinery rich in tools that allow us to obtain the estimate of N (α) stated below.
Theorem 1.2.Let 9/4 < q ⩽ 3 and let N (α) be the number of eigenvalues of Theorems 1.1 and 1.2 involve averaging of the function N (α).Averaging of eigenvalue counting functions also appeared in the papers [27] and [28].However, the operators that were studied in these two papers are Schrödinger operators.These are the publications in which one discusses a periodic Schrödinger operator perturbed by a decaying potential αV .The same elliptic model is discussed in [25] , [26], and [29], but the asymptotics of N (α) is established in [25] , [26], and [29] without any averaging.To obtain such strong results, one has to impose very restrictive conditions on the derivatives of V .The remaining papers [1]- [3] [5], [6] , [9], [11] - [15] , [20], [24], that are devoted to Schrödinger operators, do not even deal with N (α).Instead of that, they deal with the number N (λ, α) of eigenvalues passing the point λ.
Finally, we would like to mention the paper [10] which is related to the spectral theory of Dirac operators.However, the problems discussed in [10] are very different from the questions studied here.

COMPACT OPERATORS
For a compact operator T , the symbols s k (T ) denote the singular values of T enumerated in the non-increasing order (k ∈ N) and counted in accordance with their multiplicity.Observe that s 2 k (T ) are eigenvalues of T * T .We set n(s, T ) = #{k : s k (T ) > s}, s > 0.
For a self-adjoint compact operator T we also set where λ k (T ) are eigenvalues of T .Observe that A similar inequality holds for the function n.Also, Theorem 2.1.Let A and B be two compact operators on the same Hilbert space.Then for any r ∈ N, ) The first inequality was discovered by S. Rotfeld [22].The second estimate is called Horn's inequality (see Section 11.6 of the book [7]).
Below we use the following notation for the positive and negative part of a self-adjoint operator T : Theorem 2.2.Let 0 < p ⩽ 1.Let q ⩾ p.Let A and B be two compact selfadjoint operators.Then for any s > 0, A proof of Theorem 2.2 can be found in [28].
Let H 0 and V ⩾ 0 be two selfadjoint operators acting on the same Hilbert space.Assume that V is bounded.For λ ∈ R \ σ(H 0 ), define the operator X λ by Two points λ and µ are said to be in the same spectral gap of Proposition 2.3.Let 0 < p ⩽ 1.Let q ⩾ p. Suppose the operators X λ , X µ are compact for the two points λ < µ that belong to the same spectral gap of H 0 .Then for any s > 0, Proof.Here one needs to use the fact that X λ ⩽ X µ . 2 Let S ∞ be the class of compact operators.Note that the condition implies that operators (2.4) are compact for all λ ∈ R\σ(H 0 ).Moreover, (2.5) implies that, for each α > 0, the spectrum of The following proposition is called the Birman-Schwinger principle: Proposition 2.4.Let H 0 and V ⩾ 0 be self-adjoint operators in a Hilbert space.Assume that (2.5) holds for some λ 0 .Let N (λ, α) be the number of eigenvalues of The idea of the proof of (2.6) is the following.First, one shows that ).This relation holds with multiplicities taken into account.After that, one simply uses the definition of the distribution function n + (s, X λ ).
Corollary 2.5.Let H 0 and V ⩾ 0 be self-adjoint operators in a Hilbert space.Assume that (2.5) holds for some λ 0 .Let N (α) be the number of eigenvalues of the operator H(α) in [λ, µ) contained in a gap of the spectrum σ(H 0 ).Then Let p > 0. The class of compact operators T whose singular values satisfy The following statement provides a Hölder type inequality for products of compact operators that belong to different Schatten classes.Proposition 2.6.Let T 1 ∈ S p and T 2 ∈ S q where p > 0 and q > 0. Then T 1 T 2 ∈ S r , where 1/r = 1/p + 1/q, and A proof of this proposition can be found in [7].

Consider the following important example of an integral operator on
If F is the Fourier transform operator, [a] and [b] are operators of multiplication by the functions a and b, then The constants in both inequalities depend only on d and p.
The proof of this theorem can be found in [6].
Let p > 0.Besides the classes S p , we will be dealing with the so-called weak Schatten classes Σ p of compact operators T obeying the condition It turns out that Y defined by (2.8) belongs to Σ p if a ∈ L p and the other factor b satisfies the condition Such functions b are said to belong to the space L p w (R d ).The following result is the socalled Cwikel inequality (see [8]).

PRELIMINARY ESTIMATES
For the sake of brevity, the norms in the spaces S p and L p will be often denoted by the symbol ∥ • ∥ p .Theorem 3.1.Let l ⩾ 1 be an integer number.Let p satisfy the condition p > 6l l+1 .Let also be the family of Birman-Schwinger operators with H 0 being the Dirac operator.Assume that W ∈ L p (R 3 ).Then the operator with a constant C > 0 that does not depend on W .
Proof.It is easy to see that T If the factors W were written before the factors R λ and R µ , then this term would be the operator and Theorem 2.7 would imply an estimate that is similar to (3.1).We have to show that the position of the factors does not matter too much.
For that purpose, we observe that where l+1 with q 1 = 1 and q j = q j−1 + 2 l+1 = l−1+2j l+1 .Therefore, . This leads to the estimate Similarly, since with p m = 1 and p j−1 = p j + 2 l+1 , we obtain that where 1 τ = 2lm p(l+1) .It remains to estimate the Schatten norm of the operator For that purpose, we write it as is a bounded operator.Obviously, = 2l p(l+1) .Therefore, Combining the relations (3.2) -(3.5), we obtain that If l is an odd number, then In this case, if p ⩽ 2l, then it follows from Proposition 2.3 that As a consequence, applying Theorem 3.1 with l = 3, we obtain Theorem 1.2, saying that

SPLITTING
For ε > 0, we introduce two parts V 1 and V 2 of the potential V by setting Let N j (t) be the number of eigenvalues of the operator H 0 − tV j in the interval [λ, µ), j = 1, 2. We want to show that We introduce Xλ by where As we know from (2.3), for s = α −1 , Here, the value of the parameter q is the same as in Theorem 1.2.The next proposition and its corollaries show that the right hand side is of order o(α 3/ν−q ) as α → ∞.That allows us to replace X λ and X µ by the operators Xλ and Xµ and claim that Proposition 4.1.Let p > 9/4 and γ ⩾ 2. Assume that the support of the function W 2 is contained in the set {x ∈ R 3 : |x| > εα 1/ν + 1}.Let also q = 3γp 6γ + p .
Then there is an α 0 > 0 such that with a constant C > 0 independent of α and W .
Proof.Let θ be a smooth function on the real line R such that Define θ α on R 3 by θ α (x) = θ(|x| − εα 1/ν ).Then, obviously, θ α W 1 = W 1 and θ α W 2 = 0. Using the identity is an operator of multiplication by a bounded matrix-valued function supported in the layer Therefore, the operator belongs to the Schatten class S γ at least for γ ⩾ 2 and The operator X 3 λ − X3 λ is a finite linear combination of operators of the form Repeating the arguments that lead to the estimate (3.2), we obtain with r = 2p 3n .Similarly, we obtain that with τ = 2p 3m .It remains to estimate Schatten norms of the operators .
Obviously, it is enough to estimate only the norm of B 1,2 .For that purpose, we write it as where Obviously, Combining the relations (4.2) -(4.5), we obtain that In fact we proved more: exactly the same arguments can be used to justify the following statement.
Let the operator T (α) be a finite linear combination of products of three factors of the form where χ ± j are characteristic functions of some subsets of R 3 (that might depend on α).Assume that, at least for one of the three factors (4.6) in each product, the supports of χ − j and χ + j are separated from each other by a spherical layer of the form {x ∈ R 3 : Then there is an α 0 > 0 such that with a constant C > 0 independent of α and W .
As a consequence, we immediately obtain the next result, in which we use the notation Corollary 4.3.Let p > 9/4, γ ⩾ 2 and q = 3γp 6γ + p .
Let χ j be the characteristic functions of the sets and Y λ,α be the operator defined by Then there is an α 0 > 0 such that with a constant C > 0 independent of α and W .
Proof.One only needs to realize that the operator On the other hand, applying Cwikel's inequality, one can easily show that In other words, Consequently, Proof.Choose γ > q 3−νq and define p by Then p > 6/ν.Therefore W ∈ L p (R 3 ).Using Rotfeld's inequality, we obtain The inequality (4.9) estimates the last term on the right hand side of (4.1).It remains to apply Corollary 4.3 and the relation (4.8). 2

OTHER CONSEQUENCES
The preceding discussion of the splitting principle, involves a decomposition of the space R 3 into two domains.It is easy to see that the same arguments work for all piecewise smooth domains obtained similarly by scaling by a factor of α 1/ν .In particular, one of the domains that we have alredy considered can be decomposed further into smaller sets.Namely, let Q j be bounded disjoint cubes contained in the region {x ∈ R 3 : |x| ⩾ ε}, 1 ⩽ j ⩽ n − 1.Let {ϕ j } n−1 j=1 be the characteristic functions of the cubes α 1/ν Q j .Define ϕ n to be the characteristic function of the complement as α → ∞.
To prove Theorem 5.1, it is enough to repeat the steps that were needed to prove Corollary 4.4.
Clearly, to obtain an asymptotic formula for dt one has to obtain an asymtotic formula for dt for each j.The latter integral can be written as Observe now that, if then the maximum and minimum values of α V on the cubes α 1/ν Q j do not depend on α: The potential αV can be estimated by constant functions m j ϕ j and M j ϕ j on cubes α 1/ν Q j .So, due to the monotonicity of the counting function n + , for any τ > 0.
Consequently, it remains to obtain an asymptotic formula for the quantity for any fixed t > 0. We are going to prove the following result.
Proposition 5.2.For any fixed t > 0, Taking into account the fact that ± |ξ| 2 + 1 are eigenvalues of the symbol of the operator H 0 , we conclude that we need to prove that for Ψ being the characteristic function of the interval (t, ∞).Since such a function Ψ can be estimated from above and below by continuous functions of the form and the quantity depends on t continuously, we only need to prove (5.1) for Ψ that are continuous and vanishing near zero.Any such function Ψ can be written as where ζ is a continuous function on the real line R. Notice that, in this case, Thus both sides of (5.1) can be estimated by Cα 3/ν ∥ζ∥ ∞ .The functional ∥ζ∥ ∞ is the L ∞ -norm of the function on the interval [−L, L] where L = 1/(1 − |λ|).Since ζ can be uniformly approximated by polynomials, it is enough to prove (5.1) under the assumption that ζ is a polynomial.Put differently, it is enough to prove it for We are going to prove that For that purpose, we write χ + B n χ + as While the norm of the operator χ + Bχ − does not tend to zero, it is still representable in the form where ∥T 1 ∥ → 0, and ∥T 2 ∥ Sn = o(α 3/(nν) ), as α → ∞.
To see that, we define T 2 to be the operator where θ is the operator of multiplication by the characteristic function of the set Then the volume of the support of the function θ does not exceed Cα 5/(2ν) .Therefore, On the other hand, we have the estimate for the integrtal kernel k(x, y) of the operator T 1 : .Thus, we have the following estimate Combining this relation with (5.3) we obtain (5.2). 2 As a consequence, we obtain Proposition 5.3.For any constant M ⩾ 0, we have Proof.Changing the variables αt → t, we obtain (5.5) The integrand on the right hand side can be estimated according to Cwikel's inequality: Consequently, the limit on the right hand side of (5.Proof.It is enough to apply the estimate established in Theorem 1.2 with V replaced by the potential V 1 .2 Corollary 6.2.Let V ⩾ 0 be a bounded real valued potential such that where Φ is a continuous function on the unit sphere.Let 9/4 < q < 3/ν and ν > 1.Let also −1 < λ < µ < 1.Then where Φ is a continuous function on the unit sphere.Let 9/4 < q < 3/ν and ν > 1.Let also −1 < λ < µ < 1.Then

Theorem 2 . 8 .
Let p > 2. Assume that a ∈ L p (R d ) and b ∈ L p w (R d ).Then Y defined by (2.8) belongs to the class Σ p and ∥Y ∥ Σp ⩽ C∥a∥ L p ∥b∥ L p w with a constant C that depends only on d and p.