Note on Lieb Thirring Type Inequalities for a Complex Perturbation of Fractional Laplacian

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1 Journal of Mathematical Physics, Analysis, Geometry 5, vol., No. 3, Note on Lieb Thirring Tye Inequalities for a Comlex Perturbation of Fractional Lalacian C. Dubuisson Institut de Mathématiques de Bordeaux Université de Bordeaux 35, cours de la Libération F-3345 Talence cedex clement.dubuisson@math.u-bordeaux.fr Received October 4, 4, revised May 4, 5 For s >, let H = ( ) s be the fractional Lalacian. In this aer, we obtain Lieb Thirring tye inequalities for the fractional Schrödinger oerator defined as H = H + V, where V L (R d ),, d, is a comlex-valued otential. Our methods are based on the results of articles by Borichev Golinskii Kuin [BGK9] and Hansmann [Han]. Key words: fractional Schrödinger oerator, comlex erturbation, discrete sectrum, Lieb Thirring tye inequality. Mathematics Subject Classification : 35P5 (rimary); 3C35, 47A75, 47B (secondary).. Introduction The article by Abramov Aslanyan Davies [AAD] gave rise to many aers devoted to the study of eigenvalues for comlex erturbations of various self-adjoint oerators. In recent years, the fractional Lalacian ( ) s, s >, has received an increasing interest due to its numerous alications in alied mathematics and hysics (see [DPV] for references). For s >, the fractional Lalacian ( ) s is defined with the hel of the functional calculus alied to the nonnegative self-adjoint oerator. That is, ( ) s is essentially self-adjoint on Cc (R d ; C), and the domain of its closure is the fractional Sobolev sace W s, (R d, C) := {f L (R d ), ( + ζ s ) f(ζ) dζ < + }, R d where f is the Fourier transform of f (see [DPV, Sec. 3.]). By the sectral maing theorem, the sectrum of ( ) s is R + = [; + [. c C. Dubuisson, 5

2 C. Dubuisson We consider the fractional Schrödinger oerator H = ( ) s + V, (.) where V is the oerator of multilication by the comlex-valued function V, and we note H := ( ) s. In articular, the erturbed oerator H is not suosed to be self-adjoint. We assume that V is a relatively comact erturbation of H, i.e., dom(h ) dom(v ) and V (λ H ) is comact for λ C\σ(H ). The sectrum, the essential sectrum, and the discrete sectrum of H will be denoted by σ(h), σ ess (H), and σ d (H), resectively. Here, the discrete sectrum is the set of all eigenvalues which are discrete oints of the sectrum whose corresonding eigensaces (or rootsaces) are finite dimensional. Throughout the aer, eigenvalues are counted according to their algebraic multilicity. The essential sectrum of H is defined by σ ess (H) = {λ C, λ H is not Fredholm}, where a closed oerator is a Fredholm oerator if it has a closed range and both its kernel and cokernel are finite dimensional. In our situation, σ ess (H) := σ(h)\σ d (H). For more details on these definitions, see [EE89, Chater IX]. By Weyl s theorem on essential sectrum (see [RS78, Theorem XIII.4]), we have σ ess (H) = σ ess (H ) = σ(h ) = R +, and the only ossible accumulation oints of σ d (H) lie on σ ess (H). Our interest in the resent toic was motivated by the article of Frank Lieb Seiringer [FLS8] on Hardy Lieb Thirring inequalities for fractional Schrödinger oerator with real-valued otential V. As an alication, the authors give a roof of the stability of relativistic matter. In articular, for < s < min{; d }, γ >, and V L γ+d/s (R d ), formula (5.) in [FLS8] says that λ γ C d,s,γ V γ+d/s L γ+d/s, (.) where V = max{; V } and C d,s,γ is defined at [FLS8, formula (5.)]. In this aer, we obtain Lieb Thirring tye inequalities for the fractional Schrödinger oerator H with comlex-valued V. These inequalities give information on the rate of convergence of oints from the discrete sectrum σ d (H) to the essential sectrum of H. The ertaining references on the subject are [FLLS6], [HS], and [DHK3]. We will assume a little more than V being a relatively comact erturbation of H. Actually, we will suose that V is a relatively Schatten von Neumann erturbation of the fractional Lalacian. Namely, let S,, be the Schatten von Neumann class of comact oerators (see Section.3. for further references on the subject). Saying that the otential V defined on R d is a relatively Schatten von Neumann erturbation of H means that dom(h ) dom(v ) and V (λ H ) S, (.3) 46 Journal of Mathematical Physics, Analysis, Geometry, 5, vol., No. 3

3 Lieb Thirring Inequalities for Fractional Schrödinger Oerator for one (and hence for all) λ C\σ(H ). Hyothesis (.3) is fulfilled rovided V L (R d ) and > max{; d s } (see Proosition.3). We denote by d(z, Ω) := inf z w the distance between z C and Ω C. As w Ω usual, x + = max{; x}. The main results of the resent article are the following theorems. The constants and C are defined in (4.). Theorem.. Let H be the fractional Schrödinger oerator defined by (.) with < s d and V L (R d ) with > d s. Then, for τ > small enough, we have d(λ, σ(h )) +τ λ α ( + λ ) β K C β τ V L, (.4) τ where the owers are () α = min{ +τ ; d s }, () β = τ + ( d s τ) +. The constant K deends on d,, s, and τ. Theorem.. Let H be the fractional Schrödinger oerator defined by (.) with s > d and V L (R d ) with >. Then, for τ > small enough, we have d(λ, σ(h )) + d s +τ λ α ( + λ ) β K C β τ τ V L, (.5) where the owers are () α = + min{ d s + τ; }, () β = τ + ( d s + τ) +. The constant K deends on d,, s, and τ. The above theorems essentially rely on comlex analysis methods resented in [BGK9], while Theorem.3 is based on results of [Han], obtained with the hel of tools of functional analysis and oerator theory. Theorem.3. Let H be the fractional Schrödinger oerator defined in (.) with s > and V L (R d ), with > max{; d s }. Then, for τ >, the following inequality holds: with K deending on d,, s, and τ. d s d(λ, σ(h )) C ( + λ ) d K +τ s τ V L, (.6) Journal of Mathematical Physics, Analysis, Geometry, 5, vol., No. 3 47

4 C. Dubuisson Let us assume that V is real-valued and V L (R d ) for > max{; d s }. Then H = ( ) s + V is a self-adjoint oerator and σ d (H) lies on the negative real half-axis. In [FLS8], the values of the arameter s are restricted to the range < s < min{, d } due to the resence of the magnetic otential (see [FLS8, Sec..]). Setting γ = d s >, (.) becomes λ d s C,d,s V L, (.7) where, as always, V = min{v, }. Theorems. and. give Lieb Thirring tye inequalities for all ositive values of s. In articular, for < s d, (.4) becomes +τ max{ λ ; d s +τ} C d,,s, V L. (.8) Rather exectedly, we see that (.8) is slightly weaker than (.7), but our results aly to a considerably larger class of otentials. We continue with a few words on the notation. The generic constants will be denoted by C, that is, they will be allowed to change from one relation to another. For two ositive functions f, g defined on a domain Ω of the comlex lane C, we write f(λ) g(λ) if there are constants C, C > such that C f(λ) g(λ) C f(λ) for all λ Ω. We write f(λ) g(λ) ( f(λ) g(λ)) if there is a ositive constant C such that f(λ) Cg(λ) (f(λ) Cg(λ), resectively) for λ Ω. The choice of the domain Ω will be clear from the context. To comare Theorem. and Theorem.3, it is convenient to consider a sequence (λ n ) σ d (H) which converges to λ σ ess (H). Recall that σ ess (H) = σ(h ). We give details on the comarison between (.4) and (.6), the comarison between (.5) and (.6) being similar. Without loss of generality, we assume d(λ n, σ(h )). In the case λ ]; + [, (.6) is better than (.4). In the case λ =, the term in (.4) becomes d(λ n, σ(h )) +τ, for n large λ n d s +τ enough, and (.6) is again better than (.4). When λ =, the situation is slightly more comlicated. Choose τ > small enough to guarantee α/τ. Then (.4) is better than (.6) rovided Re(λ n ) or Im(λ n ) τ Re(λ n ) α for Re(λ n ) >. Inequality (.6) is better than (.4) in the oosite case, i.e., when Re(λ n ) > and Im(λ n ) τ Re(λ n ) α. To sum u, we see that neither Theorem. nor Theorem.3 has an advantage over each other. 48 Journal of Mathematical Physics, Analysis, Geometry, 5, vol., No. 3

5 Lieb Thirring Inequalities for Fractional Schrödinger Oerator As a concluding remark, we would like to mention that it is ossible to consider the comlex matrix-valued otential V as in [Dub4a], devoted to the study of non-self-adjoint Dirac oerators. Unlike the latter aer, the study of matrixvalued otentials is neither natural nor comlicated in the resent framework. The only difference with the scalar-valued case will be the resence of the constant n / in (3.) and (3.), n being the size of the square matrix giving the matrixvalued otential. At last, we say a few words on the structure of the aer. We recall some known results and give references in Sec.. The key oint of the roofs is the bound on the resolvent of H, and it is roved in Sec. 3. In Secs. 4 and 5, we rove Theorems. and., resectively. In Sec. 6, we deal with Theorem.3.. Preliminaries.. Theorem of Borichev Golinskii Kuin. The following theorem, roved in [BGK9, Theorem.], gives a bound on the distribution of zeros of a holomorhic function on the unit disc D := { z < } in terms of its growth towards the boundary T := { z = }. Theorem.. Let h be a holomorhic function on D with h() =. Assume that h satisfies a bound of the form h(z) ex K N ( z ) α z ζ j β, (.) j where ζ j = and α, β j, j =,..., N. Then, for any < τ <, the zeros of h satisfy the inequality h(z)= where C deends on α, β j, ζ j and τ. Above, x + = max{x, }... Conformal mas j= N ( z ) α++τ z ζ j (β j +τ) + C K, j= Let ϕ a be a conformal ma sending D to the resolvent set of the oerator H, ρ(h ) = C\R +. For a >, it is given by the relation ϕ a : z λ := a ( ) z +, (.) z Journal of Mathematical Physics, Analysis, Geometry, 5, vol., No. 3 49

6 C. Dubuisson and the inverse ma going from C\R + to D is λ i a ϕ a : λ z :=. λ + i a Later in the aer, we will have to comare the distance from λ = ϕ a (z) to the boundary of ρ(h ), ρ(h ) = R +, and the distance from z to D = T. The results of this kind are called distortion theorems. Proosition.. (Distortion between C\R + and D). Let, as above, λ = ϕ a (z). We have z + a d(z, T) z 3 d(λ, z + R+ ) 8a d(z, T) z 3, (.3) and a d(λ, R + ) d(z, T) 4 d(λ, R + ) a. (.4) 4 λ (a + λ ) λ (a + λ ) P r o o f. The first inequality is a direct alication of Koebe distortion theorem [Pom9, Corollary.4] to the ma ϕ a, so the roof is omitted. For the second one, we have z + = λ λ + i a, z = a λ + i a. (.5) On the other hand, λ + i a = λ + a + a Im( λ), and, since Im( λ), we obtain λ + a λ + i ( a λ + a ) (a + λ ). Going back to inequalities (.3), we get (.4)..3. Schatten classes and determinants One can find the definitions and roerties of Schatten classes and regularized determinants related to these classes in [DHK9] or [Dub4a]. For detailed discussion and roofs, see the monograhs by Gohberg Krein [GK69] and Simon [Sim77]. Let us consider the oerator F (λ) := (λ + a)(a + H) V (λ H ), (.6) 5 Journal of Mathematical Physics, Analysis, Geometry, 5, vol., No. 3

7 Lieb Thirring Inequalities for Fractional Schrödinger Oerator where a is large enough to guarantee that (a + H) is invertible. The coming Proosition.3 imlies that V (λ H ) S for λ ρ(h ), rovided V L (R d ) and > max{; d s }. For λ ρ(h ), we have F (λ) S and F holomorhic in ρ(h ), therefore the holomorhic function of interest is, for all λ ρ(h ), f(λ) := det (Id F (λ)), (.7) where = min{n N, n }. In articular, the zeros of f are the eigenvalues of H (counted with algebraic multilicities), and, for A S, we have ) det (Id A) ex (Γ A. (.8) S We also use the well-known inequality from [Sim5, Theorem 4.], which we call Birman Solomyak inequality; some authors refer to call it Kato Seiler Simon inequality. Observe that this inequality holds true for < by duality of the case. Proosition.3. Let V L (R d ) be comlex-valued with > max{; d s }, and s >. Assume that λ ρ(h ). Then V (λ H ) S, and V (λ H ) S (π) d V L (λ s ) L. 3. Bound on the Resolvent In this section, we bound the exression (λ s ) L aearing in Proosition.3. The difficulty is to obtain the right bound when s > d. Indeed, when < s d, we simly adat the roof from [DHK9] to the dimensions d. The roof of the bound when s > d requires more work. We will reeatedly use the following elementary inequalities. Lemma 3.. () Let a, b and α >, then min{; α }(a α + b α ) (a + b) α max{; α }(a α + b α ). () In articular, with α =, for a, b, we have a + b a + b a + b. d =. We recall that v d = d π Γ( d ) for d and it is convenient to ut v = for Journal of Mathematical Physics, Analysis, Geometry, 5, vol., No. 3 5

8 C. Dubuisson Proosition 3.. Let λ = λ + iλ C\R +. Set δ = d s. For < s d and > d s, we have where M = max on d,, and s. For s > d where N = max (λ s ) v d L s M λ d s, (3.) d(λ, σ(h )) { } + t δ dt K ;, and K (t + ) is defined in (3.7) and deends and >, we have (λ s ) L v d s { + t δ (t + ) dt; N, (3.) d(λ, σ(h )) d s + t δ dt + dt (t + ) }. P r o o f. We start with the olar change of variables (λ x s ) = v L d r d r s λ dr, and we ut I = r d r s λ dr = r d (r s λ ) + λ dr. (3.3) First, we assume that λ <, that is, d(λ, σ(h )) = λ. In (3.3), we use (r s λ ) r 4s + λ rs, and we make the change of variables t = λ, so I r d (r 4s + λ ) dr = λ d s s λ t d s (t + ) dt. (3.4) The integrals in (3.4) converge since > d s >. Hence, for λ <, I λ d s s d(λ, σ(h )) t d s (t + ) dt. (3.5) 5 Journal of Mathematical Physics, Analysis, Geometry, 5, vol., No. 3

9 Lieb Thirring Inequalities for Fractional Schrödinger Oerator Second, we assume λ and λ > (since (λ x s ) = ( λ x s ) ). In (3.3), with the hel of the change of variables t = rs λ λ I = = sλ sλ λ λ λ λ (λ t + λ ) d s (λ t + λ ) s (t + ) we obtain (λ t + λ ) d s dt. (3.6) (t + ) We distinguish two cases: < s d and s > d. If s = d, for Re(λ), bound (3.) is obvious from (3.6). Now assume that < s < d and ut δ = d s. Since λ >, λ λ <, we have λ λ (λ t + λ ) δ dt = (t + ) λ λ (λ t + λ ) δ dt + (t + ) dt (λ t + λ ) δ dt. (t + ) In the first integral on the right-hand side of this equality, we use that λ t + λ ( λ. As for the second term, we observe that, by Lemma 3., (λ t + λ ) δ C d,s (λ t) δ + λ δ ). Here, Cd,s = max{; δ }. Hence, we have I sλ [ λ δ λ λ (t + ) t δ dt + C d,s λ δ dt + C (t + ) d,s λ δ K [ ] λ δ + λ δ, sλ { where K = max ( + C d,s ) dt; C R + (t + ) d,s C d,s = max{; δ }, we have by Lemma 3., I K sλ K sλ C d,s (λ + λ ) δ C d,s ( ) δ λ δ. R + (t + ) t δ (t + ) dt dt }. Then, utting Journal of Mathematical Physics, Analysis, Geometry, 5, vol., No. 3 53

10 C. Dubuisson Consequently, for λ, I K s λ d s, (3.7) d(λ, σ(h )) where K = K C d,s δ/. Recalling (3.5), we obtain inequality (3.) in the case s d. Let us turn now to the case s > d. Suose again that λ >. We recall (3.6) I = sλ λ λ (λ t + λ ) δ dt. (t + ) Since < δ = d s <, we cannot use the revious bound. Making the change of variables u = t + λ λ, we obtain λ λ (λ t + λ ) δ dt = λ δ (t + ) = λ δ λ λ (t + λ λ ) δ dt (t + ) u δ ( (u λ λ ) + The last integral is bounded indeendently of λ, i.e., u δ ( (u λ λ ) + ) du u δ du + u δ du + R ) du. ( (u λ λ ) + (u + ) Indeed, in the first inequality, we use (u λ λ ) + when u, and u δ when u (since δ < and > ). Hence, for λ, with K 3 = (3.). I K 3 s + u δ du + λ δ λ = K 3 s du. ) du, (3.8) d(λ, σ(h )) d s du. Recalling (3.5), we finish the roof of (u + ) 54 Journal of Mathematical Physics, Analysis, Geometry, 5, vol., No. 3

11 Lieb Thirring Inequalities for Fractional Schrödinger Oerator 4. Proof of Theorem. Reminding (.7), we have f(λ) = det (Id F (λ)) for λ ρ(h ) = C\R +, where F (λ) := (λ + a)(a + H) V (λ H ) S,. Inequality (.8) imlies that log( f(λ) ) Γ (λ + a)(a + H) V (λ H ) S. From [Han, Lemma 3.3.4], we know the following bound on ( a H) for some a >. Then, there exists, deending on d,, s, and V, such that for any a, ( a H) C a, (4.) where C = ( V ( H ) ). By Proosition.3, we get the next inequality for > and λ C\R +, where log f(λ) Since < s d, from (3.), we have Γ C (π) d a λ + a V L (λ s ) L. (4.) d s log f(λ) K C a V λ + a λ L, (4.3) d(λ, σ(h )) K = Γ (π) d v d s M (4.4) and M is defined in (3.). We now transfer the above inequality on D in order to aly Theorem.. That is, we consider the function g(z) = f ϕ a (z), where ϕ a is defined by (.). It is clearly holomorhic on D. By definition (.), we see λ + a = 4a z z. So, Proosition. alied to inequality (4.3) gives log g(z) K C a V (4a) z d a s z + d s z 3( ) L z z d s a d(z, T) z + with K = 4 K C. K a d s a V L z d(z, T) z + d s + z d s + (4.5) Journal of Mathematical Physics, Analysis, Geometry, 5, vol., No. 3 55

12 C. Dubuisson Now, by Theorem., we have for all τ >, g(z)= ( z ) +τ z ( d s +τ) + z + ( d s +τ) + C K a d s a V L, (4.6) where K is defined above and C deends on d,, s, and τ. There are three cases d to be considered: Case : s < < d s, Case : = d s, and Case 3: > d s. For each case, we will transfer the relation (4.5) back to ρ(h ) = C\R +. Recalling Proosition., we see a d(λ, σ(h )) z = d(z, T) 4 λ / (a + λ ), z + λ a + λ, and z a a + λ. Then we will integrate the resulting inequality with resect to a [; + [ to get to a sharer bound. This integration follows the idea of Demuth, Hansmann, and Katriel (see [DHK9] or [DHK3]. Since the comutations are similar for all above cases, we will give the details of the roof in the first case and resent only the main stes in the remaining cases. 4.. Case : d s < < d s For < τ < d s, relation (4.6) becomes g(z)= ( z ) +τ z d s +τ C K a d s a V L. (4.7) We transfer this inequality back to ρ(h ), i.e., so ( z ) +τ z d s +τ a d s +τ d(λ, σ(h )) +τ λ +τ (a + λ ) d s + + 3τ 5 + 3τ d s d(λ, σ(h )) +τ λ +τ (a + λ ) d s + + 3τ a τ K a 5 + 3τ d s V L. (4.8), We can now erform the integration with resect to a [; + [. All terms in this relation are ositive, so we can ermute the sum and the integral by the Fubini theorem. Consequently, we obtain d(λ, σ(h )) +τ λ +τ a a (a + λ ) d s + + 3τ da da a +τ V L. 56 Journal of Mathematical Physics, Analysis, Geometry, 5, vol., No. 3

13 Lieb Thirring Inequalities for Fractional Schrödinger Oerator + da Obviously, a +τ =. In the left-hand side of this relation, we use the ττ bound a a + λ and we make the change of variables t = a. Hence, we λ + come to a a (a + λ ) d s + + 3τ So, for d s < < d s, da a (a + λ ) d s τ ( λ + ) + ( λ + ) d s τ da t dt (t + ) d s τ. d(λ, σ(h )) +τ λ +τ ( + λ ) d s + 3τ C K C I τ τ δ V L, (4.9) where K is defined in (4.4), C is defined in (4.6), I = and δ = 9 + 3τ d s. + t dt (t + ) d s τ, 4.. Case : = d s Reminding (4.6), we obtain g(z)= Furthermore, we have so, for all < τ <, ( z ) +τ z τ z + τ C K a d s a V L. (4.) ( z ) +τ z τ z + τ a +τ λ σ(h) After integration, we find +τ d(λ, σ d (H )) +τ λ (a + λ ) +τ, d(λ, σ(h )) +τ λ (a + λ ) +τ a τ K a +τ V L. (4.) d(λ, σ(h )) +τ λ ( + λ ) C K C τ I τ τ δ V L, (4.) Journal of Mathematical Physics, Analysis, Geometry, 5, vol., No. 3 57

14 C. Dubuisson where K is defined in (4.4), C is defined in (4.6), I = + t dt, (t + ) ++τ and δ = τ. In (4.), we used τ/ to obtain a clear statement of δ j in Remark Case 3: > d s For < τ < d s, relation (4.6) becomes Then, we get g(z)= ( z ) +τ z + d s +τ C K a d s a V L. (4.3) ( z ) +τ z + d s +τ and, for < τ < d s, d(λ, σ(h )) +τ λ d s (a + λ ) 3 (+τ) d s We integrate the above inequality d(λ, σ(h )) +τ λ d s a +τ 3 (+τ)+ d s a d s +τ d(λ, σ(h )) +τ λ d s (a + λ ) 3 d s + 3τ K a 3 (+τ)+ d s V L. (4.4) a a +τ d s τ (a + λ ) 3 (+τ) d s da V L τ τ. As before, we do the change of variables t = a and we distinguish two cases: λ + if ( d s τ) <, we use the bound [( λ + )t + ] ( λ + )(t + ), and if ( d s τ), we aly the bound from below [( λ + )t + ] ( λ + )t. We resent the case ( d s τ) in details, the other cases are analogous and are omitted. We see a a d s τ (a + λ ) 3 (+τ) d s ( λ + ) + = ( λ + ) 3 (+τ) d s ( λ + ) d s τ ( λ + ) d s + 3τ + da t [( λ + )t + ] d s τ (t + ) 3 (+τ) d s t 3 d s τ (t + ) 3 (+τ) d s dt. dt, 58 Journal of Mathematical Physics, Analysis, Geometry, 5, vol., No. 3

15 Lieb Thirring Inequalities for Fractional Schrödinger Oerator So, if ( d s τ), we obtain d(λ, σ(h )) +τ λ d s ( + λ ) C K C τ I 3 τ τ δ 3 V L, (4.5) where K is defined in (4.4), C is defined in (4.6), I 3 = and δ 3 = 7 + d s + 3 τ. When ( d s τ) <, we obtain + t 3 d s τ (t + ) 3 (+τ) d s d(λ, σ(h )) +τ λ d s ( + λ ) C K C τ I 4 τ τ δ 4 V L, (4.6) where K is defined in (4.4), C is defined in (4.6), I 4 = + dt, t dt, (t + ) ++τ and δ 4 = δ 3 = 7 + d s + 3 τ. In relations (4.9), (4.), (4.5), and (4.6), we use the bound + λ ( + λ ), because, and we come to inequality (.4). Thus, the roof in the case < s d is finished. R e m a r k 4.. In the above inequalities, one has I j = where j =,..., 4. t + ( d s τ) + (t + ) ++τ+ max{ d s τ;; d s δ j = 7 + 3τ + min{; d s } d s, 5. Proof of Theorem. τ} dt, Taking into account (3.) and (4.), inequality (4.) becomes for λ ρ(h ), log f(λ) K 4 C a V λ + a L, d(λ, σ(h )) d s since s > d and we have K 4 = Γ (π) d v d s N, (5.) where N deends on d,, and s only. As before, we have log g(z) K 4 C 4 d a s z a. d(z, T) d s z 3d s z + d s Journal of Mathematical Physics, Analysis, Geometry, 5, vol., No. 3 59

16 C. Dubuisson We set K 5 = 4 K 4 C. Alying Theorem., we have ( z ) d s ++τ z ( 3d s +τ) + z + ( d s +τ) + g(z)= C K 5 a d s a V L. (5.) The searation in different cases with resect to and d s is clear from the following icture (Fig. ). The x-axis reresents and the y-axis reresents d s. There are four straight lines given by y =, x y =, x + 3y =, and x 3y =. Fig.. The different cases So, we have three different cases to consider: Case : d s and 3d s <, Case : d 3d s < and s <, and Case 3: d s < and 3d s. Below, the comutations are similar to the case s d and thus are omitted. 5.. Case : d s We find and 3d s < d(λ, σ(h )) d s ++τ λ (a + λ ) 3 +τ 3d 4s + a 3d (+ s +τ) a V L. (5.3) It remains to integrate with resect to a on [; + [. We do it in the same way as in the case < s d. 6 Journal of Mathematical Physics, Analysis, Geometry, 5, vol., No. 3

17 Lieb Thirring Inequalities for Fractional Schrödinger Oerator From (5.3), we obtain d(λ, σ(h )) + d s +τ λ a a 3d ( s τ) (a + λ ) 3 +τ 3d 4s + da V L τ τ. Hence, if 3d s >, and τ > is small enough, then d(λ, σ(h )) + d s +τ λ ( + λ ) τ C K 4 C I 5 τ τ δ 5 V L, (5.4) where K 4 is defined in (5.), C is defined in (5.), I 5 = and δ 5 = 3d (7 + 5 s + 3τ). Otherwise, if 3d s, we have + t 3d (3 s τ) (t + ) 3 +τ 3d 4s + dt, d(λ, σ(h )) + d s +τ λ ( + λ ) τ C K 4 C I 6 τ τ δ 6 V L, (5.5) where K 4 is defined in (5.), C is defined in (5.), I 6 = and δ 6 = δ 5 = 3d (7 + 5 s + 3τ). + t dt, (t + ) ++τ 5.. Case : d s We have < and 3d s < d(λ, σ(h )) d s ++τ a λ (+ d s +τ) (a + λ ) + d +τ s Integrating this inequality gives 3d (+ s +τ) a V L. (5.6) d(λ, σ(h )) + d s +τ λ (+ d s +τ) ( + λ ) C K 4 C ( d s ++3τ) I 7 τ τ δ 7 V L, (5.7) + t dt where K 4 is defined in (5.), C is defined in (5.), I 7 = (t + ), (+ d s +3+3τ) and δ 7 = ( + d s + τ). We recall that < d s <, hence d s + >. Journal of Mathematical Physics, Analysis, Geometry, 5, vol., No. 3 6

18 C. Dubuisson 5.3. Case 3: d s This time, we have < and 3d s d(λ, σ(h )) d s ++τ λ ++τ d 4s (a + λ ) + + d 4s + 3τ a τ a V L. (5.8) After integration, the revious inequality becomes d(λ, σ(h )) + d s +τ λ (+ d s +τ) ( + λ ) C K 4 C ( d s ++3τ) I 8 τ τ δ 8 V L, (5.9) + t dt where K 4 is defined in (5.), C in (5.), I 8 =, and (t + ) (+3+ d +3τ) s δ 8 = d 4s + 3 τ. As before, < d s <, and so d s + >. To make the statement of the theorem more transarent in Case, we use ( + λ ) τ. This gives the ower β in relation (.5). Finally, since ( + λ ) 3τ, we bound + λ w( + λ ). The roof of Theorem. is finished. R e m a r k 5.. In the above inequalities, one has I j = (t + ) ++τ+ t + 3d ( s τ) + 3d dt, max{ s τ;; d + τ} s δ j = ( + d s + τ) max{ d s where j = 5,..., 8. + τ; ; 3d s + τ}, 6. Lieb Thirring Bound Using a Theorem from [Han] 6.. Hansmann s theorem and conformal maing The following theorem is the key ingredient for the roof of (.6). It is roved in [Han]. Theorem 6.. Let A be a normal bounded oerator and B be an oerator such that B A S for some. Suose also that σ(a) is convex. Then the following inequality holds: d (λ, σ(a)) B A S. λ σ d (B) 6 Journal of Mathematical Physics, Analysis, Geometry, 5, vol., No. 3

19 Lieb Thirring Inequalities for Fractional Schrödinger Oerator Hansmann has another result in this direction (see [Han3, Cor. ]). Since in our situation the sectrum σ(a) is convex, it gives no imrovement. We will aly Theorem 6. to ( a H) and ( a H ), and so we introduce the arameter a >. As in the revious section, we need a distortion result. Introduce a conformal ma g : C\R + C\[ a, ] defined by g(λ) = a + λ. Below, we denote by λ and µ the variables in C\R + and C\[ a, ], resectively. Proosition 6.. For λ C\R +, we have the bound ( d g(λ), [ a ] ), d (λ, R + ) 5 (a + λ ). P r o o f. We obtain a bound for the function g : C\[a; + [ C\[; a ] defined by g(λ) = λ and then comose it by the translation T : λ λ + a, that is, g = g T. After some technical comutations (see [Dub4b]), we find ( [ d g(λ), ; ] ) d (λ, [a; + [) a 5 λ (a + λ ). The claimed inequality follows. As in the roofs of Theorems. and., we use an integration with resect to the arameter a to imrove the rate of convergence in the left-hand side of inequality (.6). This trick is borrowed from Theorem in [DHK3]. We recall that is defined in (4.). 6.. Proof of Theorem.3 We ut A = ( a H ) which is normal and B = ( a H) which is bounded, for a >, so that A and B exist. We know that B A = BV A S, hence we can aly Theorem 6.. For, it gives d (µ, σ(a)) B A S. (6.) µ σ d (B) For > max{; d s }, we bound the right-hand side of inequality (6.) with the hel of Proosition.3 and the inequalities (4.), (3.5) B A S (π) d ( a H) V L ( a x s ) L a d s K C a V L, Journal of Mathematical Physics, Analysis, Geometry, 5, vol., No. 3 63

20 C. Dubuisson where K = v d s(π) d R + t d s (t dt. (6.) + ) / Then µ = ( a λ) = g(λ) σ d (B) if and only if λ σ d (H), hence d (µ, σ(a)) = d (g(λ), σ(a)) µ σ d (B) {g(λ),} ( 5) d (λ, σ(h )) (a + λ ). The last inequality results from Proosition 6.. Thus we obtain d(λ, σ(h )) (a + λ ) ( 5) K C a V L, (6.3) a d s where K is defined in (6.). The next ste of the roof is the integration with resect to the arameter a. Since the comutations are similar to the integration erformed in Sec. 4, the technical details are omitted. We obtain from (6.3) d(λ, σ(h )) a d s τ a (a + λ ) da V L τ τ. Hence, assuming first that d/s >, we come to a d s τ a da (a + λ ) When d/s <, we have Hence, a d s τ a da (a + λ ) where K is defined in (6.), and I = ( λ + ) d s +τ ( λ + ) d s +τ t d s τ (t + ) dt. t dt (t + ) + d s ++τ. d(λ, σ(h )) ( + λ ) ( 5) K C d s +τ I τ τ V L, t +( d s τ) + dt. (t + ) + d s ++τ+( d s τ) + Using + λ ( + λ ), we comlete the roof of Theorem Journal of Mathematical Physics, Analysis, Geometry, 5, vol., No. 3

21 Lieb Thirring Inequalities for Fractional Schrödinger Oerator Acknowledgment. I thank Stanislas Kuin for his helful comments on the subject. References [AAD] A.A. Abramov, A. Aslanyan, and E.B. Davies, Bounds on Comlex Eigenvalues and Resonances. J. Phys. A, Math. Gen. 34 (), No., [BGK9] A. Borichev, L. Golinskii, and S. Kuin, A Blaschke-Tye Condition and its Alication to Comlex Jacobi Matrices. Bull. Lond. Math. Soc. 4 (9), No., 7 3. [DHK9] M. Demuth, M. Hansmann, and G. Katriel, On the Discrete Sectrum of Non-Selfadjoint Oerators. J. Funct. Anal. 57 (9), No. 9, [DHK3] M. Demuth, M. Hansmann, and G. Katriel, Eigenvalues of Non-Selfadjoint Oerators: a Comarison of Two Aroaches. In: Mathematical Physics, Sectral Theory and Stochastic Analysis, Basel: Birkhäuser/Sringer, 3. [DPV] E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker s Guide to the Fractional Sobolev Saces. Bull. Sci. Math. 36 (), No. 5, [Dub4a] C. Dubuisson, On Quantitative Bounds on Eigenvalues of a Comlex Perturbation of a Dirac Oerator. Integral Equations Oer. Theory 78 (4), No., [Dub4b] C. Dubuisson, Study of the Discrete Sectrum of Comlex Perturbations of Oerators from Mathematical Physics. PhD thesis. University of Bordeaux, 4, soon available at hal.archives-ouvertes.fr. [EE89] D.E. Edmunds and W.D. Evans, Sectral Theory and Differential Oerators. Oxford, Clarendon Press, aerback ed. edition, 89. [FLLS6] R.L. Frank, A. Latev, E.H. Lieb, and R. Seiringer, Lieb Thirring Inequalities for Schrödinger Oerators with Comlex-Valued Potentials. Lett. Math. Phys. 77 (6), No. 3, [FLS8] R.L. Frank, E.H. Lieb, and R. Seiringer, Hardy Lieb Thirring Inequalities for Fractional Schrödinger Oerators. J. Am. Math. Soc. (8), No. 4, [GK69] I.C. Gohberg and M.G. Krein, Introduction to the Theory of Linear Nonselfadjoint Oerators. Translated from the Russian by A. Feinstein. Translations of Mathematical Monograhs, Vol. 8. AMS, Providence, RI, 969. [Han] M. Hansmann, On the Discrete Sectrum of Linear Oerators in Hilbert Saces. PhD thesis, TU Clausthal,. [Han] M. Hansmann, An Eigenvalue Estimate and its Alication to Non-Selfadjoint Jacobi and Schrödinger Oerators. Lett. Math. Phys. 98 (), No., Journal of Mathematical Physics, Analysis, Geometry, 5, vol., No. 3 65

22 C. Dubuisson [Han3] M. Hansmann, Variation of Discrete Sectra for Non-Selfadjoint Perturbations of Selfadjoint Oerators. Integral Equations Oer. Theory 76 (3), No., [HS] M. Hansmann, Lieb Thirring Inequalities for Jacobi Matrices. J. Arox. Theory 8 (), No., 6 3. [Pom9] C. Pommerenke, Boundary Behaviour of Conformal Mas. Verlag, 99. [RS78] [Sim77] Berlin, Sringer- M. Reed and B. Simon, Methods of Modern Mathematical Physics. IV: Analysis of Oerators. Academic Press [Harcourt Brace Jovanovich, Publishers], New York London, 978. B. Simon, Notes on Infinite Determinants of Hilbert Sace Oerators. Adv. Math. 4 (977), [Sim5] B. Simon, Trace Ideals and their Alications. Providence, RI, American Mathematical Society (AMS), nd ed., Journal of Mathematical Physics, Analysis, Geometry, 5, vol., No. 3