Cwikel Estimates and Negative Eigenvalues of Schroedinger Operators on Noncommutative Tori
aa r X i v : . [ m a t h . OA ] F e b CWIKEL ESTIMATES AND NEGATIVE EIGENVALUES OF SCHR ¨ODINGEROPERATORS ON NONCOMMUTATIVE TORI
EDWARD MCDONALD AND RAPHA¨EL PONGE
Abstract.
In this paper, we establish Cwikel-type estimates for noncommutative tori for anydimension n ≥
2. We use them to derive a Cwikel-Lieb-Rozenblum inequality for the num-ber of negative eigenvalues of fractional Schr¨odinger operators on noncommutative tori in anydimension n ≥
2. In order to do so we establish a “borderline version” of the abstract Birman-Schwinger principle for the number of negative eigenvalues of relatively compact form pertur-bations of a non-negative semi-bounded operator with isolated 0-eigenvalue. Introduction
The celebrated estimates of Cwikel [17] are a landmark application of trace-ideal techniques inmathematical physics. They assert that if f ∈ L p ( R n ) and g is in the weak L p -space L p, ∞ ( R n )with p >
2, then the operator f ( X ) g ( − i ∇ ) on L ( R n ) is in the weak Schatten class L p, ∞ , and wehave(1.1) (cid:13)(cid:13) f ( X ) g ( − i ∇ ) (cid:13)(cid:13) L p, ∞ ≤ c np (cid:13)(cid:13) f (cid:13)(cid:13) L p (cid:13)(cid:13) g (cid:13)(cid:13) L p, ∞ , where the constant c np depends only on n and p . Here f ( X ) is the multiplication by f in positionspace and g ( − i ∇ ) is the multiplication by g in momentum space. We refer to Section 4 forbackground on Schatten classes and weak Schatten classes. Cwikel’s estimates were extended to p ∈ (0 ,
2) by Birman-Solomyak [6] and to p = 2 by Birman-Karadzhov-Solomyak [4] (who also dealtwith more general Lorentz ideals L p,q ). Solomyak [67] substantially improved the understandingof the p = 2 case in even dimension.Cwikel’s estimates in the form stated above were conjectured by Simon [64]. He pointed outthat combining them with the Birman-Schwinger principle [3, 59] would give an upper-bound forthe number of negative eigenvalues (i.e., the number of bound states) of Schr¨odinger operators∆ + V with V ∈ L n ( R n ), n ≥
3. Namely, there exists a constant c n > n such that(1.2) N − (∆ + V ) ≤ c n Z | V − ( x ) | n dx, where N − (∆ + V ) is the number of negative eigenvalues of ∆ + V counted with multiplicity and V − = ( | V | − V ) is the negative part of V .The main motivation for the CLR inequality (1.2) is establishing a Weyl’s law for Schr¨odingeroperators ~ ∆ + V with a non-smooth potential V under the under the semi-classical limit h → n/ p + V (see, e.g., [18,35, 55, 56, 57, 67]). Such operators naturally appear in the framework of fractional quantummechanics [34]; for p = n we get the ultra-relativistic Schr¨odinger operator |∇| + V (see [18]).CLR inequalities have widespread applications in mathematical physics and geometric analysis(see [24] for a recent survey). Date : February 25, 2021.2020
Mathematics Subject Classification.
Key words and phrases. noncommutative geometry; Cwikel estimates; Schr¨odinger operators. ecently, Cwikel estimates have been experiencing a revival of interest. A general abstract oper-ator theoretic framework for Cwikel-type estimates has been emerging (see [23, 30, 38]). In a recentarticle [30] Hundertmark et al . used a refinement of Cwikel’s estimates to get some of the best CLRbounds to date. Furthermore, applications of Cwikel-type estimates to Connes’s noncommutativegeometry program were found [25, 36, 37, 38, 44, 46, 70, 72]. In particular, Cwikel-type estimatesfor noncommutative Euclidean spaces were established by Levitina-Sukochev-Zanin [38].The aim of this paper is to obtain general Cwikel-type estimates and establish CLR inequalitieson noncommutative tori (a.k.a. quantum tori). Noncommutative tori are arguably the most wellknown examples of noncommutative spaces. In particular, they naturally appear in the noncom-mutative geometry approach to the quantum Hall effect [1] and to topological insulators [10, 50].In addition, noncommutative tori have been considered in the context of string theory (see, e.g.,[16, 61]). Noncommutative 2-tori naturally arise from actions of Z on the circle S by irrational ro-tations. More generally, a noncommutative n -tori T nθ is generated by unitaries U , . . . , U n subjectto the relations, U l U j = e iπθ jl U j U l , j, l = 1 , . . . , n, where θ = ( θ jl ) is a given real anti-symmetric matrix. We refer to Section 2 for more backgroundon noncommutative tori.We establish Cwikel-type estimates on T nθ for operators λ ( x ) g ( − i ∇ ), where x is in L p ( T nθ ) with p ≥
2. Here ∇ = ( ∂ , . . . , ∂ n ), where ∂ , . . . , ∂ n are the canonical derivations of T nθ , and λ isthe extension to L p -spaces of the left-regular representation of L ∞ ( T nθ ) in L ( T nθ ) (see Section 3).More precisely, we show that if x ∈ L q ( T nθ ) and g ∈ ℓ p, ∞ ( Z n ), we have(1.3) k λ ( x ) g ( − i ∇ ) k L p, ∞ ≤ c pq k x k L q k g k ℓ p, ∞ , provided that, either p < q , or p = 2 < q . We also have L p -estimates for p <
2. Namely, if x ∈ L ( T nθ ) and g ∈ ℓ p ( Z n ) with p <
2, then k λ ( x ) g ( − i ∇ ) k L p ≤ c p k x k L k g k ℓ p . Note that L p, ∞ -estimates with p > L p -estimate with p ≥ L , ∞ -estimate is deduced from the estimates in the p > p < p < λ ( x )∆ − n/ p ,where ∆ = ∇ ∗ ∇ is the (positive) Laplacian on T nθ . If x ∈ L q ( T nθ ), and either p = 2 and q =max( p, p = 2 < q , then(1.4) (cid:13)(cid:13) λ ( x )∆ − n p (cid:13)(cid:13) L p, ∞ ≤ c npq k x k L q . This allows us to get estimates for operators of the form ∆ − n/ p λ ( x )∆ − n/ p . More precisely, if x ∈ L q ( T mθ ), and either p = 1 and q = max( p, p = 1 < q , then(1.5) (cid:13)(cid:13) ∆ − n p λ ( x )∆ − n p (cid:13)(cid:13) L p, ∞ ≤ c npq k x k L q . In the Euclidean space setting the CLR inequality (1.2) is deduced from Cwikel’s estimates (1.1)by using the Birman-Schwinger principle [3, 59]. In its abstract form due to Birman-Solomyak [9](see also Proposition 7.6 and Corollary 7.8) the Birman-Schwinger principle implies that if H is a (semi-bounded) non-negative selfadjoint operator on Hilbert space and V is a non-positiverelatively form-compact perturbation such that ( H + 1) − / V ( H + 1) − / ∈ L p, ∞ , p >
0, then(1.6) N ( H + V ; λ ) ≤ (cid:13)(cid:13) ( H + λ ) − V ( H + λ ) − (cid:13)(cid:13) p L p, ∞ ∀ λ < . When H is the Laplacian on R n , n ≥
3, the inequality (1.6) continues to hold for λ = 0, therebyproviding an estimate for N − ( H + V ). In various examples, including the Laplacians on NC tori,the origin is in the discrete spectrum, and so the resolvent ( H − λ ) − has a pole singularity at = 0. This prevents us from letting λ → − in (1.6). To remedy we derive a “borderline”Birman-Schwinger principle. Namely, we show that if 0 is in the discrete spectrum of H and V is a non-positive relatively form-compact perturbation such that H − / V H − / ∈ L p, ∞ , p > ≤ N − ( H + V ) − N − (Π V Π ) ≤ (cid:13)(cid:13) H − V H − (cid:13)(cid:13) p L p, ∞ . This result seems to be new, at least at this level of generality. Its scope of validity goes beyondthe scope of this paper. For instance, it also encompasses (fractional) Schr¨odinger operators onclosed manifolds or compact manifolds with boundary under Neumann boundary conditions, aswell as Schr¨odinger operators on hyperbolic manifolds with infinite volume.Combining the specific Cwikel estimates (1.5) and the borderline Birman-Schwinger princi-ple (1.7) allows us to get CLR-type inequalities for fractional Schr¨odinger operators ∆ n/ p + λ ( V )on NC tori. Namely, if V = V ∗ ∈ L q ( T nθ ) and, either p = 1 and q = max( p, p = 1 < q , then(1.8) N − (cid:0) ∆ n p + λ ( V ) (cid:1) − ≤ c npq τ (cid:2) | V − | q (cid:3) pq , where V − = ( | V |− V ) is the negative part of V and τ is the standard normalized trace of L ∞ ( T nθ ).This inequality is consistent with Lieb’s version of the CLR inequality for closed manifolds [39, 40].For p = n/ λ ( V ) with V = V ∗ ∈ L n/ ( T nθ )if n ≥ V = V ∗ ∈ L q ( T nθ ), q >
1, if n = 2. The latter condition is consistent with theCLR inequality for Schr¨odinger operators on bounded regions of R by Birman-Solomyak [5] (seealso [7]). In particular, unlike in the Euclidean space setting we do get an inequality in dimension 2.Under the semiclassical limit h → + the CLR inequality (1.8) implies that(1.9) N − (cid:0) h np ∆ n p + λ ( V ) (cid:1) ≤ c npq h − n τ (cid:2) | V − | q (cid:3) pq + O(1) . This leads us to conjecture that if if V = V ∗ ∈ L q ( T nθ ) and, either p = 1 and q = max( p, p = 1 < q , then we have the semi-classical Weyl’s law,(1.10) N − (cid:0) h np ∆ n p + λ ( V ) (cid:1) = c n h − n τ (cid:2) | V − | p (cid:3) + o (cid:0) h − n (cid:1) . Here, c n is the volume of the Euclidean unit n -ball. In the same way as in the Euclidean spacesetting, the semiclassical CLR inequality (1.9) allows us to reduce the proof of (1.10) for L q -potentials to that for smooth potentials. We believe a semiclassical Weyl law for smooth potentialscan be established by using semiclassical pseudodifferential calculus on NC tori. However, such apseudodifferential calculus has yet to be set up. As this project falls out of the scope of this paperwe leave (1.10) as a conjecture.We refer to [45] for further applications of the Cwikel estimates (1.4)–(1.5) to curved noncom-mutative tori, i.e., noncommutative tori equipped with a Riemannian metric. In this setting therole of the flat Laplacian is played by the corresponding Laplace-Beltrami operator. In particular,we get “curved” analogues of the CLR inequalities (1.8) and obtain L -versions of the Connes’integration formulas of [47, 49].The paper is organized as follows. In Section 2, we review the main background on noncom-mutative tori. Section 3 contains some technical preliminaries, including sufficient conditions forthe operators appearing in Cwikel-type estimates to be bounded. In Section 4, we present ourmain Cwikel-type estimates for noncommutative tori and show how to deduce the p = 2 casefrom the p < p < Acknowledgements.
The warmest thanks of the authors go to Fedor Sukochev for his constantsupport and encouragements throughout the preparation of this paper, and to Rupert Frank andGrigori Rozenblum for their careful reading of earlier versions of the manuscript and for sharingtheir insights on the CLR inequality. In addition, we thank Galina Levitina and Dmitriy Zaninfor various discussions related to the subject matter of the article. otation. Throughout the paper we make the convention that c abd and C abd are positive con-stants which depend only on the parameters a , b , d , etc., and may change from line to line. Theydo not depend on the other variables that are floating around.2. Noncommutative Tori
In this section, we review the main definitions and properties of noncommutative n -tori, n ≥ θ = ( θ jk ) be a real anti-symmetric n × n -matrix, and denoteby θ , . . . , θ n its column vectors. We also let L ( T n ) be the Hilbert space of L -functions on theordinary torus T n = R n / (2 π Z ) n equipped with the inner product,(2.1) h ξ | η i = (2 π ) − n Z T n ξ ( x ) η ( x ) dx, ξ, η ∈ L ( T n ) . For j = 1 , . . . , n , let U j : L ( T n ) → L ( T n ) be the unitary operator defined by( U j ξ ) ( x ) = e ix j ξ ( x + πθ j ) , ξ ∈ L ( T n ) . We then have the relations,(2.2) U k U j = e iπθ jk U j U k , j, k = 1 , . . . , n. The noncommutative torus is the noncommutative space whose C ∗ -algebra C ( T nθ ) and vonNeuman algebra L ∞ ( T nθ ) are generated by the unitary operators U , . . . , U n . For θ = 0 we obtainthe C ∗ -algebra C ( T n ) of continuous functions on the ordinary n -torus T n and the von Neumanalgebra L ∞ ( T n ) of essentially bounded measurable functions on T n . Note that (2.2) implies that C ( T nθ ) (resp., L ∞ ( T nθ )) is the norm closure (resp., weak closure) in L ( L ( T n )) of the linear spanof the unitary operators, U k := U k · · · U k n n , k = ( k , . . . , k n ) ∈ Z n . GNS representation.
Let τ : L ( L ( T n )) → C be the state defined by the constant function1, i.e., τ ( T ) = h T | i = (2 π ) − n Z T n ( T x ) dx, T ∈ L ( L ( T n )) . This induces a continuous tracial state on the von Neuman algebra L ∞ ( T nθ ) such that τ (1) = 1 and τ ( U k ) = 0 for k = 0. The GNS construction then allows us to associate with τ a ∗ -representationof L ∞ ( T nθ ) as follows.Let h·|·i be the sesquilinear form on C ( T nθ ) defined by(2.3) h u | v i = τ ( uv ∗ ) , u, v ∈ C ( T nθ ) . Note that the family { U k ; k ∈ Z n } is orthonormal with respect to this sesquilinear form. Welet L ( T nθ ) be the Hilbert space arising from the completion of C ( T nθ ) with respect to the pre-inner product (2.3). The action of C ( T nθ ) on itself by left-multiplication uniquely extends to a ∗ -representation of L ∞ ( T nθ ) in L ( T nθ ). When θ = 0 we recover the Hilbert space L ( T n ) withthe inner product (2.1) and the representation of L ∞ ( T n ) by bounded multipliers. In addition, as( U k ) k ∈ Z n is an orthonormal basis of L ( T nθ ), every u ∈ L ( T nθ ) can be uniquely written as(2.4) u = X k ∈ Z n u k U k , u k := (cid:10) u | U k (cid:11) , where the series converges in L ( T nθ ). When θ = 0 we recover the Fourier series decomposition in L ( T n ). .2. The smooth algebra C ∞ ( T nθ ) . The natural action of R n on T n by translation gives rise toan action on L ( L ( T n )). This induces a ∗ -action ( s, u ) → α s ( u ) on C ( T nθ ) given by α s ( U k ) = e is · k U k , for all k ∈ Z n and s ∈ R n . This action is strongly continuous, and so we obtain a C ∗ -dynamical system ( C ( T nθ ) , R n , α ). We areespecially interested in the subalgebra C ∞ ( T nθ ) of smooth elements of this C ∗ -dynamical system,i.e., u ∈ C ( T nθ ) such that α s ( u ) ∈ C ∞ ( R n ; C ( T nθ )).The unitaries U k , k ∈ Z n , are contained in C ∞ ( T nθ ), and so C ∞ ( T nθ ) is a dense subalgebra of C ( T nθ ). Denote by S ( Z n ) the space of rapid-decay sequences with complex entries. In terms ofthe Fourier series decomposition (2.4) we have C ∞ ( T nθ ) = (cid:26) u = X k ∈ Z n u k U k ; ( u k ) k ∈ Z n ∈ S ( Z n ) (cid:27) . When θ = 0 we recover the algebra C ∞ ( T n ) of smooth functions on the ordinary torus T n andthe Fourier-series description of this algebra.For j = 1 , . . . , n , let ∂ j : C ∞ ( T nθ ) → C ∞ ( T nθ ) be the derivation defined by ∂ j ( u ) = ∂ s j α s ( u ) | s =0 , u ∈ C ∞ ( T nθ ) , When θ = 0 it agrees with the derivation ∂ x j on C ∞ ( T n ). In general, we have ∂ j ( U l ) = (cid:26) iU j if l = j, l = j. L p -Spaces. The L p -spaces of T nθ are special instances of noncommutative L p spaces associ-ated with a semi-finite faithful normal trace on a von Neumann algebra [33, 60] (see also [21, 48]).We refer to [21, 33] for the main background on noncommutative L p -spaces needed in this paper.By definition L ∞ ( T nθ ) is a von Neuman algebra of bounded operators on L ( T n ). Thus, a closeddensely defined operator on L ( T n ) is L ∞ ( T nθ )-affiliated when it commutes with the commutantof L ∞ ( T nθ ) in L ( L ( T n )). Furthermore, as τ is a finite faithful positive trace on L ∞ ( T nθ ) everysuch operator is τ -measurable in the sense of [21, 48]. Therefore, these operators form a ∗ -algebra,where the sum and product of such operators are meant as the closures of their usual sum andproduct in the sense of unbounded operators (see [48]).The space L ( T nθ ) consists of all L ∞ ( T nθ )-affiliated operators x on L ( T n ) such that τ ( | x | ) := Z ∞ λdτ ( E λ ) < ∞ , where E λ = [0 ,λ ] ( | x | ) is the spectral measure of | x | . We obtain a Banach space upon equipping L ( T nθ ) with the norm, k x k L := τ (cid:0) | x | (cid:1) , x ∈ L ( T nθ ) . We have a continuous inclusion with dense range of L ∞ ( T nθ ) into L ( T nθ ). The trace τ uniquelyextends to a continuous linear functional on L ( T nθ ) such that | τ ( x ) | ≤ τ ( | x | ) = k x k ∀ x ∈ L ( T nθ ) . For p >
1, the space L p ( T nθ ) consists of all L ∞ ( T nθ )-affiliated operators x on L ( T n ) such that | x | p ∈ L ( T nθ ). This is a Banach space with respect to the norm, k x k L p := τ ( | x | p ) /p , x ∈ L p ( T nθ ) . We also have a continuous inclusion with dense range of L ∞ ( T nθ ) into L p ( T nθ ). In particular, for p = 2 the above definition is consistent with the previous definition of L ( T nθ ), since in both caseswe get the completion of L ∞ ( T nθ ) with respect to the same norm.We have the following version of H¨older’s inequality. Proposition 2.1 ([21, 33]) . Suppose that p − + q − = r − ≤ . If x ∈ L p ( T nθ ) and y ∈ L q ( T nθ ) ,then xy ∈ L r ( T nθ ) with norm inequality, (2.5) k xy k L r ≤ k x k L p k y k L q . his implies that, for every p ≥
1, the multiplication of L ∞ ( T nθ ) uniquely extends to continuousbilinear maps, L ∞ ( T nθ ) × L p ( T nθ ) −→ L p ( T nθ ) , L p ( T nθ ) × L ∞ ( T nθ ) −→ L p ( T nθ ) . In particular, for p = 2 we recover the GNS representation of L ∞ ( T θ ) associated with τ .If x is an L ∞ ( T nθ )-affiliated operator on L ( L ( T nθ )), its polar decomposition [51, TheoremVIII.32] takes the form x = u | x | , where u is a partial isometry in L ∞ ( T nθ ). Moreover, the equality x ∗ = u | x | u ∗ (see, e.g., [8, Theorem 1.8.3]) and H¨older’s inequality show that x ∈ L p ( T nθ ) = ⇒ x ∗ ∈ L p ( T nθ ) and k x ∗ k L p = k x k L p . We may also define L p -spaces on T nθ for 0 < p < Sobolev spaces.
Given any s ≥
0, the Sobolev space W s ( T nθ ) is defined by(2.6) W s ( T nθ ) := n u = X k ∈ Z n u k U k ∈ L ( T nθ ); X k ∈ Z n (1 + | k | ) s | u k | < ∞ o . This is a Hilbert space with respect to the inner product and norm, h u | v i s = X k ∈ Z n (1 + | k | ) s u k v k , k u k W s = (cid:18) X k ∈ Z n (1 + | k | ) s | u k | (cid:19) . Note that W ( T nθ ) = L ( T nθ ).Equivalently, let ∆ = − ( ∂ + · · · + ∂ n ) be the Laplacian on T nθ . This is a non-negative selfadjointoperator on L (2 T nθ ) with domain W ( T nθ ). We have∆ (cid:0) U k ) = | k | U k , k ∈ Z n . In particular, ∆ is isospectral to the Laplacian on the ordinary torus T n . Set Λ = (1 + ∆) . Givenany s ≥
0, we have W s ( T nθ ) := n u ∈ L ( T nθ ); Λ s u ∈ L ( T nθ ) o , k u k W s = k Λ s u k W . Note also (see [68, 73]) that, given an integer p ≥
0, we have W p ( T nθ ) = n u ∈ L ( T nθ ); δ α u ∈ L ( T nθ ) ∀ α ∈ N n , | α | ≤ p (cid:9) . We mention the following versions of Sobolev’s embedding theorems.
Proposition 2.2 (see [73, Theorem 6.6]) . Let p ∈ [2 , ∞ ) . For every s ≥ n (1 / − p − ) , we have acontinuous embedding W s ( T nθ ) ⊂ L p ( T nθ ) . This embedding is compact when the inequality is strict. Proposition 2.3.
For any s > n/ , we have a compact embedding W s ( T nθ ) ⊂ C ( T nθ ) .Remark . The continuity of the inclusion of the W s ( T nθ ) ⊂ C ( T nθ ) for s > n/ W s ( T nθ ) ⊂ W s ′ ( T nθ ) with s > s ′ > n/ L p -Action and Boundedness of λ ( x ) g ( − i ∇ ) and (1 + ∆) − p n λ ( x )(1 + ∆) − p n In what follows, we denote by λ the left-regular representation of L ∞ ( T nθ ) on L ( T nθ ). In thissection, we shall explain how to extend it to L p ( T nθ ), p ≥
1. We shall then give sufficient conditionsfor the boundedness of operators of the forms λ ( x ) g ( − i ∇ ) and (1 + ∆) − p n λ ( x )(1 + ∆) − p n . .1. Left-multipliers λ ( x ) . First, we observe that H¨older’s inequality (Proposition 2.1) yieldsthe following extension result for the left-regular representation λ : L ∞ ( T nθ ) → L ( L ( T n )). Proposition 3.1 ([33]) . Suppose that p − + q − = r − ≥ . Then the left-regular representationuniquely extends to a continuous linear map, λ : L p ( T nθ ) −→ L (cid:0) L q ( T nθ ) , L r ( T nθ ) (cid:1) . In particular, if p ≥ and p − + q − = 2 , then we get a continuous linear map, λ : L p ( T nθ ) −→ L (cid:0) L q ( T nθ ) , L ( T nθ ) (cid:1) . Remark . The above linear maps are isometries (see [33]).If x ∈ L p ( T n ) with p ≥
2, the above corollary asserts that λ ( x ) a continuous linear operatorfrom L q ( T nθ ) to L ( T nθ ) with p − + q − = 1 /
2. In particular, we may regard λ ( x ) as an unboundedoperator on L ( T nθ ) with domain L q ( T nθ ).Given any q ≥
1, we denote by L q ( T nθ ) ∗ the anti-linear dual of L q ( T nθ ), i.e., the space ofcontinuous anti-linear forms on L q ( T nθ ). We observe that the left-regular regular representationof L ∞ ( T nθ ) can be regarded as a continuous linear map λ : L ∞ ( T nθ ) → L ( L ( T nθ ) , L ( T nθ ) ∗ ) suchthat h λ ( x ) u, v i = h λ ( x ) u | v i = τ (cid:2) v ∗ xu (cid:3) , x ∈ L ∞ ( T nθ ) , u, v ∈ L ( T nθ ) , where h· , ·i : L ( T nθ ) ∗ × L ( T nθ ) → C is the duality pairing. Proposition 3.3.
Suppose that p − + 2 q − = 1 . Then the left-regular representation uniquelyextends to a continuous linear map λ : L p ( T nθ ) → L ( L q ( T nθ ) , L q ( T nθ ) ∗ ) such that (3.1) h λ ( x ) u, v i = τ (cid:2) v ∗ xu (cid:3) ∀ x ∈ L p ( T nθ ) ∀ u, v ∈ L q ( T nθ ) . Proof.
Let x ∈ L p ( T nθ ) and u, v ∈ L q ( T nθ ). By H¨older’s inequality xu ∈ L r ( T nθ ) with r − = p − + q − = 1 − q − , and so v ∗ ( xu ) ∈ L ( T nθ ). Moreover, we have | τ ( v ∗ xu ) | ≤ k v ∗ ( xu ) k L ≤ k v ∗ k L q k xu k L r ≤ k x k L p k u k L q k v k L q . This gives the result. (cid:3)
In particular, if x ∈ L p ( T nθ ) with 1 ≤ p <
2, then the above proposition shows that λ ( x ) makessense as a bounded operator from L q ( T nθ ) to L q ( T nθ ) ∗ , with p − + 2 q − = 1, i.e., q − = (1 − p − ).3.2. The operators (1 + ∆) − s/ λ ( x )(1 + ∆) − s/ . Given any s >
0, we denote by W − s ( T θ )the anti-linear dual of the Sobolev space W s ( T nθ ). Set Λ = (1 + ∆) / . As mentioned aboveΛ s : W s ( T nθ ) → L ( T nθ ) is an isometric isomorphism. By duality we get a continuous isomorphismΛ s : L ( T nθ ) → W − s ( T nθ ) such that h Λ s u, v i = h u | Λ s v i , u ∈ L ( T nθ ) , v ∈ W s ( T nθ ) , where h· , ·i : W − s ( T nθ ) × W s ( T nθ ) → C is the duality pairing. Its inverse Λ − s : W − s ( T nθ ) → L ( T nθ )is given by(3.2) (cid:10) Λ − s u | v (cid:11) = (cid:10) u, Λ − s v (cid:11) , u ∈ W s ( T nθ ) , v ∈ L ( T nθ ) . Lemma 3.4.
Let x ∈ L p ( T nθ ) , p ≥ , and assume that, either s > n/ p , or s = n/ p and p > .Then λ ( x ) uniquely extends to a bounded operator λ ( x ) : W s ( T nθ ) → W − s ( T nθ ) .Proof. Suppose that p − + 2 q − = 1, i.e., q − = (1 − p − ). We know by Proposition 3.3 that λ ( x ) is a bounded operator from L q ( T nθ ) to L q ( T nθ ) ∗ .Suppose that p > s ≥ n/ p . Then q ∈ [2 , ∞ ), and so by Proposition 2.2 we have acontinuous embedding of W s ( T nθ ) into L q ( T nθ ) since s ≥ n/ p = n (1 − q − ). By duality we geta continuous embedding of L q ( T nθ ) ∗ into W − s ( T nθ ). It then follows that λ ( x ) induces a bondedoperator λ ( x ) : W s ( T nθ ) → W − s ( T nθ ).Assume now that p = 1 and s > n/
2. In this case q = ∞ , and so λ ( x ) is a bounded operatorfrom L ∞ ( T nθ ). As s > n/
2, by Proposition 2.3 we have a continuous embedding of W s ( T nθ ) into C ( T nθ ), and hence we get a continuous embedding into L ∞ ( T nθ ). By duality we get a continuous mbedding of L ∞ ( T nθ ) ∗ into W − s ( T nθ ). Thus, as above p > λ ( x ) induces a bounded operator λ ( x ) : W s ( T nθ ) → W − s ( T nθ ). The proof is complete. (cid:3) Combining the above lemma with the boundedness of the operators Λ − s : L ( T nθ ) → W s ( T nθ )and Λ − s : W − s ( T nθ ) → L ( T nθ ) we arrive at the following result. Proposition 3.5.
Let x ∈ L p ( T nθ ) , p ≥ , and assume that either s > n/ p , or s = n/ p and p > . Then the composition Λ − s λ ( x )Λ − s makes sense as a bounded operator on L ( T nθ ) .Remark . The above result holds verbatim if we replace Λ by √ ∆.3.3. The operators λ ( x ) g ( − i ∇ ) . Recall that, given any p ∈ (0 , ∞ ), the quasi-Banach space ℓ p ( Z n ) consists of p -summable sequences a = ( a k ) k ∈ Z n ⊂ C with quasi-norm, k a k ℓ p := (cid:18) X k ∈ Z n | a k | p (cid:19) /p , a = ( a k ) ∈ ℓ p ( Z n ) . For p ≥ ℓ p ( Z n ) is a Banach space. In addition, we denote by ℓ ∞ ( Z n ) the Banach space of bounded sequences with norm, k a k ℓ ∞ := sup k ∈ Z n | a k | , a = ( a k ) ∈ ℓ ∞ ( Z n ) . For p ∈ (0 , ∞ ), the weak ℓ p -space ℓ p, ∞ ( Z n ) is defined as follows. Given any sequence a =( a k ) k ∈ Z n ∈ ℓ ∞ ( Z n ), let µ ( a ) = ( µ j ( a )) j ≥ be its symmetric decreasing re-arrangement, i.e., µ j ( a ) := sup k ,...,k j ∈ Z n min {| a k | , | a k | , . . . , | a k j |} . In other words, µ ( a ) = ( µ j ( a )) j ≥ is the non-increasing rearrangement of the sequence ( | a k | ) k ∈ Z n .The space ℓ p, ∞ ( Z n ) then consists of sequences a = ( a k ) k ∈ Z n ∈ ℓ ∞ ( Z n ) such that µ j ( a ) = O (cid:0) j − p (cid:1) as j → ∞ . We equip it with the quasi-norm, k a k ℓ p, ∞ := sup j ≥ ( j + 1) /p µ j ( a ) , a ∈ ℓ p, ∞ ( Z n ) . With this quasi-norm ℓ p, ∞ ( Z n ) is a quasi-Banach space. In fact, for p > k a k ′ ℓ p, ∞ := sup N ≥ N − p X j
Suppose that p − + q − = r − and g ∈ ℓ p ( Z n ) with p ≥ and ≤ q, r ≤ . Then g ( − i ∇ ) induces a continuous linear operator g ( − i ∇ ) : ˆ ℓ q ( T nθ ) → ˆ ℓ r ( T nθ ) with norm inequality, (cid:13)(cid:13) g ( − i ∇ ) x (cid:13)(cid:13) ˆ ℓ r ≤ k g k ℓ p k x k ˆ ℓ q ∀ x ∈ ˆ ℓ q ( T nθ ) . Proof.
Let x ∈ ˆ ℓ q ( T nθ ). We have g ( − i ∇ ) x = P g ( k )ˆ x ( k ) U k . By assumption ( g ( k )) k ∈ Z n ∈ ℓ p ( Z n )and ˆ x := (ˆ x ( k )) k ∈ Z n ∈ ℓ q ( Z n ), and so by H¨older’s inequality ( g ( k )ˆ x ( k )) k ∈ Z n ∈ ℓ r ( Z n ), since p − + q − = r − . This means that g ( − i ∇ ) x ∈ ˆ ℓ r ( T nθ ). Moreover, we have the inequalities, (cid:13)(cid:13) g ( − i ∇ ) x (cid:13)(cid:13) ˆ ℓ r = k ( g ( k )ˆ x ( k )) k ℓ r ≤ k g k ℓ p k ˆ x k ℓ q = k g k ℓ p k x k ˆ ℓ q . This proves the result. (cid:3)
The following two lemmas give the relationships of the ˆ ℓ p -spaces with the L p -spaces and Sobolevspaces. Lemma 3.8 (Hausdorff-Young Inequality) . Suppose that q ≥ and q − + r − = 1 . Then we havea continuous inclusion ˆ ℓ r ( T nθ ) ⊆ L q ( T nθ ) with norm inequality, (3.3) k x k L q ≤ k x k ˆ ℓ r ∀ x ∈ ˆ ℓ r ( T nθ ) . Proof.
The proof uses complex interpolation theory essentially in the same way as in the proof ofthe classical Hausdorff-Young inequality (see, e.g., [2]). For background on interpolation theorywe refer to the short survey of Connes [15, Appendix IV.B] and the references therein. The mainreference for complex interpolation theory there is the article of Calder´on [11] (see also [2, 32]).As mentioned above L ( T nθ ) and ˆ ℓ ( T nθ ) agree as Banach spaces and the inclusion of ℓ ( T nθ ) into C ( T nθ ) gives rise to a contraction into L ∞ ( T nθ ). In particular, we have the inequalities, k x k L = k x k ˆ ℓ ∀ x ∈ ˆ ℓ ( T nθ ) , (3.4) k x k L ∞ ≤ k x k ˆ ℓ ∀ x ∈ ˆ ℓ ( T nθ ) . (3.5)Suppose that q ∈ (2 , ∞ ). Then L q ( T nθ ) is a complex interpolation space for the pair of Banachspaces ( L ( T nθ ) , L ∞ ( T nθ )). Namely, in the notation of [15, Appendix IV.B] we have L q ( T nθ ) =[ L ( T nθ ) , L ∞ ( T nθ )] θ , where θ is such that q = (1 − θ ) , i.e., θ = 1 − q − (see [19, Section 4]and [33]). Note also that, as r ∈ (1 , ℓ r ( T nθ ) is an exact interpolation space for thepair ( ℓ ( T nθ ) , ℓ ( T nθ )). Namely, ℓ r ( T nθ ) = [ ℓ ( T nθ ) , ℓ ( T nθ )] θ , since we have(1 − θ ) 12 + θ = 12 (1 + θ ) = 12 (2 − q − ) = 1 − q − = r − . As the Fourier transform induces isometric isomorphisms between the spaces ℓ p ( Z n ) and ˆ ℓ p ( T nθ ),we see that ˆ ℓ r ( T nθ ) = [ˆ ℓ ( T nθ ) , ˆ ℓ ( T nθ )] θ . Combining all this with the inequalities (3.4)–(3.5) andusing complex interpolation theory (see [15, Theorem IV.B.1]) then shows we have a continuousinclusion ˆ ℓ r ( T nθ ) ⊆ L q ( T nθ ) with the norm inequality (3.3). The proof is complete. (cid:3) Lemma 3.9.
Suppose that ≤ p < and s > n (2 p − − . Then we have a continuous inclusion W s ( T nθ ) ⊂ ˆ ℓ p ( T nθ ) . roof. Let u ∈ P u k U k be in W s ( T nθ ). By H¨older’s inequality we have X k ∈ Z n | u k | p = X k ∈ Z n (cid:0) | k | (cid:1) − ps (cid:2) (cid:0) | k | (cid:1) s | u | k (cid:3) p ≤ (cid:20) X k ∈ Z n (cid:0) | k | (cid:1) − rps (cid:21) r (cid:20) X k ∈ Z n (cid:0) | k | (cid:1) s | u | k (cid:21) p (3.6) ≤ (cid:20) X k ∈ Z n (cid:0) | k | (cid:1) − psr (cid:21) r k u k pW s . Here r is such that r − + (2 p − ) − = 1, i.e., r − = 1 − p . By assumption s > n (2 p − −
1) = n (1 − p ) p − = n ( rp ) − , and hence psr > n and P (1+ | k | ) − psr < ∞ . Combining this with (3.6)shows that P k ∈ Z n | u k | p < ∞ , i.e., u ∈ ˆ ℓ p ( Z n ), and there is a constant C nps > u such that k u k ˆ ℓ p = (cid:20) X k ∈ Z n | u k | p (cid:21) p ≤ C nps k u k W s . This proves the result. (cid:3)
Remark . For p = 1 and s > n/
2, we get a continuous inclusion of W s ( T nθ ) into ˆ ℓ ( T nθ ).Combining it with the continuity of the inclusion of ˆ ℓ ( T nθ ) into C ( T nθ ) mentioned above, we geta continuous inclusion W s ( T nθ ) into C ( T nθ ).We are now in a position to prove the following result. Proposition 3.11.
Suppose that p − + q − = .(1) If g ∈ ℓ p ( Z n ) , then g ( − i ∇ ) maps continuously L ( T nθ ) to L q ( T n ) .(2) If g ∈ ℓ p, ∞ ( Z n ) , then g ( − i ∇ ) maps continuously W s ( T nθ ) to L q ( T n ) for every s > .Proof. Let g ∈ ℓ p ( Z n ). Lemma 3.7 ensures us that g ( − i ∇ ) maps continuously L ( T nθ ) to ˆ ℓ r ( T nθ ),where r − = + p − . Note that 1 − r − = − p − = q − , and so by Lemma 3.8 we havea continuous inclusion of ˆ ℓ r ( T nθ ) into L q ( T nθ ). It then follows that g ( − i ∇ ) maps continuously L ( T nθ ) to L q ( T n ). This proves the first part.To prove the 2nd part let g ∈ ℓ p, ∞ ( Z n ) and s >
0. In addition, let t ∈ [1 ,
2) be such that s > n (2 t − − W s ( T nθ ) into ˆ ℓ t ( T nθ ). Set p = [ p − − ( t − − )] − , i.e., p − = p − − ( t − − ). Note that p > p since t <
2. We alsoobserve that p − + t − = p − + = 1 − ( − p − ) = 1 − q − . The fact p > p implies that g ∈ ℓ p ( Z n ), and so Lemma 3.7 ensures us that g ( − i ∇ ) maps continuously ˆ ℓ t ( T nθ ) into ˆ ℓ r ( T nθ ) with r − = p − + t − = 1 − q − . Furthermore, by Lemma 3.8 we have a continuous inclusion of ˆ ℓ r ( T nθ )into L q ( T n ). It then follows that g ( − i ∇ ) maps continuously W s ( T nθ ) to L q ( T nθ ). This proves the2nd part and completes the proof. (cid:3) By combining Proposition 3.1 and Proposition 3.11 we arrive at the following result.
Proposition 3.12.
Let p ∈ [2 , ∞ ) .(1) If x ∈ L p ( T nθ ) and g ∈ ℓ p ( Z n ) , then the operator λ ( x ) g ( − i ∇ ) is bounded on L ( T nθ ) .(2) If x ∈ L p ( T nθ ) and g ∈ ℓ p, ∞ ( Z n ) , then the domain of λ ( x ) g ( − i ∇ ) contains ∪ s> W s ( T nθ ) .In particular, λ ( x ) g ( − i ∇ ) is densely defined.Remark . The inclusion ∪ s> W s ( T nθ ) ⊂ L ( T nθ ) is strict. For instance, if u = P u k U k with u k = (1 + | k | ) − n [log(1 + | k | )] − , then u ∈ L ( T nθ ), but u W s ( T nθ ) for any s > Cwikel Estimates on NC Tori
In this section, we establish Cwikel type estimates on NC tori for Schatten classes and theirweak versions. .1. Schatten classes and weak Schatten classes.
We briefly review the main definitions andproperties of Schatten classes and weak Schatten classes on Hilbert space. We refer to [26, 65] forfurther details.In what follows we let H be a (separable) Hilbert space with inner product h·|·i . We alsodenote by K the (closed) ideal of compact operators on H . Given any operator T ∈ K we let µ = ( µ j ( T )) j ≥ be its sequence of singular values, i.e., µ j ( T ) is the ( j + 1)-the eigenvalue countedwith multiplicity of the absolute value | T | = √ T ∗ T . By the min-max principle [26, 53] we have µ j ( T ) = min (cid:8) k T | E ⊥ k ; dim E = j (cid:9) , (4.1) = min {k T − R k ; rk( R ) ≤ j } , j ≥ . (4.2)This implies the following properties of singular values (see, e.g., [26, 65]), µ j ( T ) = µ j ( T ∗ ) = µ j ( | T | ) , (4.3) µ j + k ( S + T ) ≤ µ j ( S ) + µ k ( T ) , (4.4) µ j ( AT B ) ≤ k A k µ j ( T ) k B k , A, B ∈ L ( H ) . (4.5)In addition, we have the monotonicity principle,(4.6) 0 ≤ T ≤ S = ⇒ µ j ( T ) ≤ µ j ( S ) ∀ j ≥ . In what follows, we denote by L the trace-class with norm, k T k L := Tr | T | = X j ≥ µ j ( T ) , T ∈ L . Recall that for p ∈ (0 , ∞ ) the Schatten class L p consist of operators T ∈ K such that | T | p istrace-class. It is equipped with the quasi-norm, k T k L p := (cid:0) Tr | T | p (cid:1) p = (cid:18) X j ≥ µ j ( T ) p (cid:19) p , T ∈ L p . We obtain a quasi-Banach ideal. For p ≥ L p -quasi-norm is actually a norm, and so in thiscase L p is a Banach ideal.For p ∈ (0 , ∞ ), the weak Schatten class L p, ∞ is defined by L p, ∞ := n T ∈ K ; µ j ( T ) = O (cid:0) j − p (cid:1)o . This is a two-sided ideal. We equip it with the quasi-norm,(4.7) k T k L p, ∞ := sup j ≥ ( j + 1) p µ j ( T ) , T ∈ L p, ∞ . Note that(4.8) T ∈ L p, ∞ ⇐⇒ | T | p ∈ L , ∞ and k T k L p, ∞ = (cid:0) k| T | p k L , ∞ (cid:1) p . In addition, for p >
1, the quasi-norm k · k p, ∞ is equivalent to the norm, k T k ′ L p, ∞ := sup N ≥ N − p X j We shall distinguish between the following cases: • Case I: λ ( x ) g ( − i ∇ ) ∈ L p with p ≥ λ ( x ) g ( − i ∇ ) ∈ L p, ∞ with p > • Case II: λ ( x ) g ( − i ∇ ) ∈ L p and λ ( x ) g ( − i ∇ ) ∈ L p, ∞ with 0 < p < • Case III: λ ( x ) g ( − i ∇ ) ∈ L , ∞ .The first case is dealt with in [46]. Namely, we have the following result. Proposition 4.2 ([46, Theorem 3.1]) . The following holds.(1) If x ∈ L p ( T nθ ) and g ∈ ℓ p ( Z n ) with p ≥ , then λ ( x ) g ( − i ∇ ) ∈ L p , and we have (4.9) k λ ( x ) g ( − i ∇ ) k L p ≤ c p k x k L p k g k ℓ p . (2) If x ∈ L p ( T nθ ) and g ∈ ℓ p, ∞ ( Z n ) with p > , then λ ( x ) g ( − i ∇ ) ∈ L p, ∞ , and we have (4.10) k λ ( x ) g ( − i ∇ ) k L p, ∞ ≤ c p k x k L p k g k ℓ p, ∞ . Remark . Theorem 3.1 of [46] is a special case of [38, Theorem 3.4]. More generally, [38,Theorem 3.4] provides us with Cwikel-type estimates for any interpolation space between L and K . Remark . In the estimates (4.9)–(4.10) the best constants c p are ≤ 130 (see [38, 46]). Remark . The analogue of (4.9) on R n is known as the Kato-Seiler-Simon inequality [62].Combining Proposition 4.2 with Proposition 2.2 we immediately obtain the following statement. Proposition 4.6. Let s ∈ (2 , ∞ ) and set s = n (1 / − p − ) . The following holds.(1) If x ∈ W s ( T nθ ) and g ∈ ℓ p ( Z n ) , then λ ( x ) g ( − i ∇ ) ∈ L p , and we have k λ ( x ) g ( − i ∇ ) k L p ≤ c p k x k W s k g k ℓ p . (2) If x ∈ W s ( T nθ ) and g ∈ ℓ p, ∞ ( Z n ) , then λ ( x ) g ( − i ∇ ) ∈ L p, ∞ , and we have k λ ( x ) g ( − i ∇ ) k L p, ∞ ≤ c p k x k W s k g k ℓ p, ∞ . Case II is dealt with by the following result, the proof of which is postponed to next section. Theorem 4.7. Suppose that < p < . The following holds.(1) Let x ∈ L ( T nθ ) and g ∈ ℓ p ( Z n ) . Then λ ( x ) g ( − i ∇ ) ∈ L p , and we have k λ ( x ) g ( − i ∇ ) k L p ≤ k x k L k g k ℓ p . (2) Let x ∈ L ( T nθ ) and g ∈ ℓ p, ∞ ( Z n ) . Then λ ( x ) g ( − i ∇ ) ∈ L p, ∞ , and we have k λ ( x ) g ( − i ∇ ) k L p, ∞ ≤ c p k x k L k g k ℓ p, ∞ . Finally, for Case III we shall prove the following result. Theorem 4.8. For any x ∈ L p ( T θ ) with p > and g ∈ ℓ , ∞ ( Z n ) , the operator λ ( x ) g ( − i ∇ ) is in L , ∞ , and we have k λ ( x ) g ( − i ∇ ) k L , ∞ ≤ c p k x k L p k g k ℓ , ∞ . Proof. The proof uses real interpolation theory. Once again, for background on interpolationtheory we refer to the survey of Connes [15, Appendix IV.B] and the references therein. The mainreference for real interpolation theory there is the article of Lions-Peetre [42] (see also [2, 32]).Let x ∈ L p ( T nθ ), p > 2. By Proposition 4.2, for every g ∈ ℓ p, ∞ ( Z n ), the operator λ ( x ) g ( − i ∇ )is in L p, ∞ . Thus, we have a linear operator Φ x : ℓ p, ∞ ( Z n ) → L p, ∞ given byΦ x ( g ) = λ ( x ) g ( − i ∇ ) , g ∈ ℓ p, ∞ . Furthermore, the Cwikel-type estimate (4.10) gives(4.11) k Φ x ( g ) k L p, ∞ ≤ c p k x k L p k g k ℓ p, ∞ ∀ g ∈ ℓ p, ∞ . In addition, as L p ( T nθ ) ⊂ L ( T nθ ), it follows from Theorem 4.7 that, given any q ∈ (1 , g ∈ ℓ q, ∞ ( Z n ), then Φ x ( g ) ∈ L q, ∞ , and we have(4.12) (cid:13)(cid:13) Φ x ( g ) (cid:13)(cid:13) L q, ∞ ≤ c q k x k L k g k ℓ q, ∞ ≤ c q k x k L p k g k ℓ q, ∞ . s 2 ∈ ( q, p ), the spaces ℓ , ∞ ( Z n ) and L , ∞ are real interpolation spaces for the pairs ofBanach spaces ( ℓ q, ∞ ( Z n ) , ℓ p, ∞ ( Z n )) and ( L q, ∞ , L p, ∞ ) with the exact same exponents. Namely,in the notation of [15, Appendix IV.B], up to the same norm equivalences, we have ℓ , ∞ ( Z n ) =[ ℓ q, ∞ ( Z n ) , ℓ p, ∞ ( Z n )] θ, ∞ and L , ∞ = [ L q, ∞ , L p, ∞ ] θ, ∞ , with θ ∈ (0 , 1) such that 1 / − θ ) q − + θp − (see [15, § IV.2. α & Appendix IV.B] and [19, Section 4]). Thus, combining theestimates (4.11)–(4.12) with real interpolation theory (see [15, Theorem IV.B.2]) shows that Φ x induces a continuous linear map from ℓ , ∞ ( Z n ) to L , ∞ , and, for all g ∈ ℓ , ∞ ( Z n ), we have (cid:13)(cid:13) λ ( x ) g ( − i ∇ ) (cid:13)(cid:13) L , ∞ = (cid:13)(cid:13) Φ x ( g ) (cid:13)(cid:13) L , ∞ ≤ ( c q k x k L p ) − θ ( c p k x k L p ) θ k g k ℓ , ∞ ≤ c pq k x k L p k g k ℓ , ∞ . This gives the result. (cid:3) Remark . Theorem 4.8 does not hold for x ∈ L ( T nθ ) (see Remark 6.6). Corollary 4.10. For any x ∈ W s ( T θ ) with s > and g ∈ ℓ , ∞ ( Z n ) , the operator λ ( x ) g ( − i ∇ ) isin L , ∞ , and we have k λ ( x ) g ( − i ∇ ) k L , ∞ ≤ c s k x k W s k g k ℓ , ∞ . Proof. Let x ∈ W s ( T θ ) with s > g ∈ ℓ , ∞ ( Z n ). Let p > s ≥ n (1 / − p − ).By Proposition 2.2 this ensures us that W s ( T nθ ) embeds continuously into L p ( T nθ ). Combiningthis with Theorem 4.8 then gives the result. (cid:3) Proof of Theorem 4.7 In this section, we prove Theorem 4.7. The proof attempts to follow the approach to the proofof Cwikel-type estimates for noncommutative Euclidean spaces in [38]. However, some significantsimplification occurs thanks to Proposition 5.3 below.Recall that, given non-increasing sequences of non-negative numbers a = ( a j ) j ≥ and b =( b j ) j ≥ , the Hardy-Littlewood-P´olya majorization a ≺ b means that N X j =0 a j ≤ N X j =0 b j ∀ N ≥ ∞ X j =0 a j = ∞ X j =0 b j < ∞ . Given compact operators S and T on Hilbert space we shall write S ≺ T when µ ( S ) ≺ µ ( T ). Lemma 5.1 ([38, Proposition 2.7]) . Suppose that < p < , and let S, T ∈ L be such that | S | ≺ | T | .(i) If S ∈ L p , then T ∈ L p , and k T k L p ≤ k S k L p .(ii) If S ∈ L p, ∞ , then T ∈ L p, ∞ , and k T k L p, ∞ ≤ c p k S k L p, ∞ . Recall that a sequence of bounded operators ( T j ) j ≥ on Hilbert space is called right-disjoint (resp., left-disjoint ) when T j T ∗ l = 0 (resp., T ∗ j T l = 0) when j = l . When all the operators T j areselfadjoint notions of left-disjointness and right-disjointness are both equivalent to the condition T j T l = 0 when j = 0. In that case we simply that say that we have a disjoint sequence. Lemma 5.2 ([38, Lemma 2.9]) . Let ( T j ) j ≥ be a sequence in L such that P k T j k L < ∞ .Assume further that the sequence ( T j ) j ≥ is left-disjoint or right-disjoint. Then, we have µ (cid:18) M j ≥ T j (cid:19) ≺ µ (cid:18) X j ≥ T j (cid:19) . The operator-theoretic underpinning of Theorem 4.7 is the following majorization estimate. Proposition 5.3. Let x ∈ L ( T nθ ) and g ∈ ℓ ( Z n ) . Then, we have (5.1) k x k L µ (cid:0) g ( − i ∇ ) (cid:1) ≺ µ (cid:0) λ ( x ) g ( − i ∇ ) (cid:1) . roof. The fact that g ∈ ℓ ( Z n ) ensures us that g ( − i ∇ ) ∈ L . Furthermore, as x ∈ L ( T nθ ) weknow by Proposition 4.2 that λ ( x ) g ( − i ∇ ) is in L as well.Bearing this in mind, given any k ∈ Z n , we denote by Π k the orthogonal projection onto C U k .Thus, for all u = P ˆ u k U k in L ( T nθ ), we haveΠ k u = (cid:10) U k | u (cid:11) U k = ˆ u k U k . Note that (Π k ) k ∈ Z n is a disjoint family of rank 1 projections.For k ∈ Z n , we also set T k := λ ( x ) g ( − i ∇ )Π k . Each operator T k has rank ≤ 1, and so this is a Hilbert-Schmidt operator. Moreover, if k = l , then T k T ∗ l = ( λ ( x ) g ( − i ∇ ))Π k Π l ( λ ( x ) g ( − i ∇ )) ∗ = 0. Thus, the sequence ( T k ) k ∈ Z n is right-disjoint. Wealso note that, as U k is in the domain of λ ( x ), for all u = P ˆ u k U k in L ( T nθ ), we have T k u = λ ( x ) g ( − i ∇ )Π k u = ˆ u k λ ( x ) g ( −∇ ) U k = ˆ u k g ( k ) λ ( x ) U k . Claim. For all k ∈ Z n , we have | T k | = k x k L | g ( k ) | Π k . Proof of the claim. Let k ∈ Z n , and set A = λ ( x ) g ( − i ∇ ). Note that A is a bounded operator byProposition 3.12. We have T ∗ k T k = ( A Π k ) ∗ ( A Π k ) = Π k A ∗ A Π k . We observe that Π k A ∗ AU k = (cid:10) U k | A ∗ AU k (cid:11) U k = (cid:10) AU k | AU k (cid:11) U k . As AU k = λ ( x ) g ( −∇ ) U k = g ( k ) λ ( x ) U k , we get (cid:10) AU k | AU k (cid:11) = | g ( k ) | (cid:10) λ ( x ) U k | λ ( x ) U k (cid:11) = | g ( k ) | τ (cid:2) xU k ( xU k ) ∗ (cid:3) = | g ( k ) | τ [ xx ∗ ] = | g ( k ) | k x k L . Thus, Π k A ∗ AU k = (cid:10) AU k | AU k (cid:11) U k = | g ( k ) | k x k L U k . It then follows that, for all u = P ˆ u k U k , we have T ∗ k T k = Π k A ∗ A Π k u = ˆ u k Π k A ∗ AU k = ˆ u k | g ( k ) | k x k L U k = | g ( k ) | k x k L Π k u. That is, T ∗ k T k = | g ( k ) | k x k L Π k . Thus, | T k | = p T ∗ k T k = | g ( k ) |k x k L p Π k = | g ( k ) |k x k L Π k . This proves the claim. (cid:3) Combining the claim above with the fact that g ∈ ℓ ( Z n ) gives X k ∈ Z n k T k k L = X k ∈ Z n k| T k |k L = X k ∈ Z n | g ( k ) | k x k L k Π k k L = k x k L k g k ℓ < ∞ . All this allows us to apply Lemma 5.2 to get(5.2) µ (cid:18) M k ∈ Z n T k (cid:19) ≺ µ (cid:18) X k ∈ Z n T k (cid:19) = µ (cid:0) λ ( x ) g ( − i ∇ ) (cid:1) . Set S = L k ∈ Z n T k . The claim above implies that | S | = L k ∈ Z n | T k | = L k ∈ Z n k x k L | g ( k ) | Π k . Therefore, we have µ ( S ) = µ (cid:18) M k ∈ Z n k x k L | g ( k ) | Π k (cid:19) = k x k L µ (cid:0) | g ( − i ∇ ) | (cid:1) = k x k L µ (cid:0) g ( − i ∇ ) (cid:1) . Combining this with (5.2) then gives the majorization k x k L µ ( g ( − i ∇ )) ≺ µ ( λ ( x ) g ( − i ∇ )). Theproof is complete. (cid:3) We are now in a position to prove Theorem 4.7. roof of Theorem 4.7. Let x ∈ L ( T nθ ) and g ∈ ℓ p, ∞ , p < 2. Recall that g ( − i ∇ ) ∈ L p, ∞ and k g ( − i ∇ ) k L p, ∞ = k g k ℓ p, ∞ . Note also that g ∈ ℓ ( Z n ), since p < 2. Thus, combining the majoriza-tion (5.1) with Lemma 5.1 shows that λ ( x ) g ( − i ∇ ) ∈ L p, ∞ , and we have (cid:13)(cid:13) λ ( x ) g ( − i ∇ ) (cid:13)(cid:13) L p, ∞ ≤ c p k x k L k g ( − i ∇ ) k L p, ∞ ≤ c p k x k L k g k ℓ p, ∞ . If g ∈ ℓ p ( Z n ), then g ( − i ∇ ) ∈ L p and k g ( − i ∇ ) k L p = k g k ℓ p . Therefore, in the same way asabove, we deduce that λ ( x ) g ( − i ∇ ) ∈ L p , and we have (cid:13)(cid:13) λ ( x ) g ( − i ∇ ) (cid:13)(cid:13) L p ≤ k x k L k g ( − i ∇ ) k L p ≤ k x k L k g k ℓ p . The proof of Theorem 4.7 is complete. (cid:3) Specific Cwikel Estimates It is worth specializing the Cwikel estimates of Section 4 to operators of the forms λ ( x )∆ − n/ p and ∆ − n/ p λ ( x )∆ − n/ p , since these estimates are used in the derivation of the CLR inequalitieson NC tori in Section 8. In the terminology of [72] these Cwikel-type estimates are called specific Cwikel estimates.If p > 0, then ∆ − n/ p = λ ( x ) g p ( − i ∇ ), where g p ( k ) = | k | − n/p for k = 0 and g p (0) = 0. Inparticular, g p ∈ ℓ p, ∞ ( Z n ). Thus, by specializing Proposition 4.2 (resp., Theorem 4.8, Theorem 4.7)to g = g p with p > p = 2, p < 2) we arrive at the following statement. Theorem 6.1. The following holds.(1) If p > and x ∈ L p ( T nθ ) , then λ ( x )∆ − n/ p ∈ L p, ∞ , and we have (cid:13)(cid:13) λ ( x )∆ − n p (cid:13)(cid:13) L p, ∞ ≤ c np k x k L p . (2) If x ∈ L p ( T nθ ) , p > , then λ ( x )∆ − n/ ∈ L , ∞ , and we have (cid:13)(cid:13) λ ( x )∆ − n (cid:13)(cid:13) L , ∞ ≤ c np k x k L p . (3) If < p < and x ∈ L ( T nθ ) , then λ ( x )∆ − n/ p ∈ L p, ∞ , and we have (cid:13)(cid:13) λ ( x )∆ − n p (cid:13)(cid:13) L p, ∞ ≤ c np k x k L . Let us now turn to the Cwikel operators ∆ − n/ p λ ( x )∆ − n/ p . Suppose that p − + 2 q − = 1. Asmentioned in Section 2, if x ∈ L p ( T nθ ), then λ ( x ) makes sense as a bounded operator from L q ( T nθ )to its anti-linear dual L q ( T nθ ) ∗ . Moreover, by Lemma 3.4, if s > n/ p , or if s = n/ p and p > λ ( x ) induces a bounded operator λ ( x ) : W s ( T nθ ) → W − s ( T nθ ). This allows us to makes senseof the composition ∆ − s/ λ ( x )∆ − s/ as a bounded operator on L ( T nθ ) ( cf . Proposition 3.5).Let y ∈ L p ( T nθ ). As (2 p ) − + q − = ( p − + 2 q − ) = , we know from Proposition 3.1 that λ ( y ) makes sense as a bounded operator from L q ( T nθ ) to L ( T nθ ). By duality we get a boundedoperator λ ( y ) ∗ : L ( T nθ ) → L q ( T nθ ) ∗ such that(6.1) h λ ( y ) ∗ u, v i = h u | λ ( y ) v i , u ∈ L ( T nθ ) , v ∈ L q ( T nθ ) . In addition, if s > n/ p , or if s = n/ p and p > 1, then we have a continuous embedding of W s ( T nθ ) into L q ( T nθ ), and so the operator λ ( x )∆ − s/ is bounded on L ( T nθ ). Lemma 6.2. Let x ∈ L p ( T nθ ) , p ≥ , be of the form x = yz with y, z ∈ L p ( T nθ ) . In addition,assume that either s > n/ p , or s = n/ p and p > . Then λ ( x ) = λ ( y ∗ ) ∗ λ ( z ) , ∆ − s λ ( x )∆ − s = (cid:2) λ ( y ∗ )∆ − s (cid:3) ∗ λ ( z )∆ − s . Proof. Let u, v ∈ L q ( T nθ ), p + 2 q − = 1. In view of (3.1) we have h λ ( x ) u, v i = τ (cid:2) v ∗ xu (cid:3) = τ (cid:2) v ∗ yzu (cid:3) . As zu = λ ( z ) u ∈ L ( T nθ ) and v ∗ y = ( λ ( y ∗ ) v ) ∗ ∈ L ( T nθ ), we get(6.2) h λ ( x ) u, v i = τ (cid:2) ( λ ( y ∗ ) v ) ∗ λ ( z ) u (cid:3) = (cid:10) λ ( z ) u | λ ( y ∗ ) v (cid:11) . ombining this with (6.1) gives h λ ( x ) u, v i = (cid:10) λ ( y ∗ ) ∗ λ ( z ) u, v (cid:11) ∀ u, v ∈ L q ( T nθ ) . That is, λ ( x ) = λ ( y ∗ ) ∗ λ ( z ).Suppose that, either s > n/ p , or s = n/ p and p > 1. Using (3.2) and (6.2) shows that, for all u, v ∈ L ( T nθ ), we have (cid:10) ∆ − s λ ( x )∆ − s u | v (cid:11) = (cid:10) λ ( x )∆ − s u, ∆ − s v (cid:11) = (cid:10) λ ( z )∆ − s u | λ ( y ∗ )∆ − s v (cid:11) = (cid:10)(cid:2) λ ( y ∗ )∆ − s (cid:3) ∗ λ ( z )∆ − s u | v (cid:11) . It then follows that ∆ − s/ λ ( x )∆ − s/ = [ λ ( y ∗ )∆ − s/ ] ∗ λ ( z )∆ − s/ . The proof is complete. (cid:3) Lemma 6.3. Any x ∈ L p ( T nθ ) , p ≥ , can be written in the form x = yz with y, z ∈ L p ( T nθ ) suchthat k y k L p = k z k L p = ( k x k L p ) / .Proof. Let x = u | x | be the polar decomposition of x . As mentioned in Section 2 the phase u isa partial isometry in L ∞ ( T nθ ), and hence k u k L ∞ ≤ 1. Thus, if we set y = u | x | / and z = | x | / ,then x = yz and z ∈ L p ( T nθ ) with k z k L p = ( k x k L p ) / . In addition, H¨older’s inequality (seeProposition 2.1) ensures that y = u | x | / ∈ L p ( T nθ ), and we have k y k L p ≤ k u k L ∞ k| x | / k L p ≤ ( k x k L p ) / . As H¨older’s inequality also gives k x k L p = k yz k L p ≤ k y k L p k z k L p ≤ k y k L p (cid:0) k x k L p (cid:1) / , we deduce that k y k L p = ( k x k L p ) / . The proof is complete. (cid:3) We are now in a position to establish Cwikel estimates for the operators ∆ − n/ p λ ( x )∆ − n/ p . Theorem 6.4. The following holds.(1) If x ∈ L p ( T nθ ) , p > , then ∆ − n/ p λ ( x )∆ − n/ p ∈ L p, ∞ , and we have (cid:13)(cid:13) ∆ − n p λ ( x )∆ − n p k L p, ∞ ≤ c np k x k L p . (2) If x ∈ L p ( T nθ ) , p > , then ∆ − n/ λ ( x )∆ − n/ ∈ L , ∞ , and we have (6.3) (cid:13)(cid:13) ∆ − n λ ( x )∆ − n k L , ∞ ≤ c np k x k L p . (3) If x ∈ L ( T nθ ) and p < , then ∆ − n/ p λ ( x )∆ − n/ p ∈ L p, ∞ , and we have (cid:13)(cid:13) ∆ − n p λ ( x )∆ − n p k L p, ∞ ≤ c np k x k L . Proof. Let x ∈ L p ( T n ), p > 1. By Lemma 6.3 we may write x = yz , with y, z ∈ L p ( T nθ ) such that k y k L p = k z k L p = ( k x k L p ) / . By Lemma 6.2 we have∆ − n p λ ( x )∆ − n p = (cid:2) λ ( y ∗ )∆ − n p (cid:3) ∗ λ ( z )∆ − n p . It follows from Theorem 6.1.(i) that [ λ ( y ∗ )∆ − n p ] ∗ and λ ( z )∆ − n p are both in the weak Schattenclass L p, ∞ , and we have (cid:13)(cid:13) λ ( z )∆ − n p (cid:13)(cid:13) L p, ∞ ≤ c np k z k L p , (cid:13)(cid:13)(cid:2) λ ( y ∗ )∆ − n p (cid:3) ∗ (cid:13)(cid:13) L p, ∞ = (cid:13)(cid:13) λ ( y ∗ )∆ − n p (cid:13)(cid:13) L p, ∞ ≤ c np k y ∗ k L p = c np k y k L p . H¨older’s inequality for weak Schatten classes ( cf . Proposition 4.1) and the fact that k y k L p = k z k L p = ( k x k L p ) / then imply that ∆ − n p λ ( x )∆ − n p ∈ L p, ∞ , and we have (cid:13)(cid:13) ∆ − n p λ ( x )∆ − n p k L p , ∞ ≤ (cid:13)(cid:13)(cid:2) λ ( y ∗ )∆ − n p (cid:3) ∗ (cid:13)(cid:13) L p, ∞ (cid:13)(cid:13) λ ( z )∆ − n p (cid:13)(cid:13) L p, ∞ ≤ c np k x k L p . This proves (i). Parts (ii) and (iii) are proved similarly. The proof is complete. (cid:3) Remark . The Cwikel estimates provided by Theorem 6.1 and Theorem 6.4 hold verbatim ifwe replace ∆ by 1 + ∆. emark . The estimate (6.3) does not hold for x ∈ L ( T nθ ) (see [43, Lemma 5.7]). Remark . For the ordinary torus T n , i.e., θ = 0, by a recent result of Sukochev-Zanin [72]the estimate (6.3) still holds if x is in Zygmund’s class L log L ( T n ) (see also [67]). It would beinteresting to have an analogue of this result for θ = 0.7. Borderline Birman-Schwinger Principle In this section, we establish a “borderline” version of the abstract Birman-Schwinger principlefor the number of negative values of relatively form-compact perturbations of non-negative semi-bounded operators on Hilbert space.To a large extent we follow the original approach of Birman-Solomyak [9], which we recast inthe framework of [66]. However, our ultimate result (Theorem 7.10) seems to be new, at leastat the level of generality it is stated. In particular, it can be applied to Schr¨odinger operators∆ g + V , and more generally fractional Schr¨odinger operators ∆ αg + V , in the following setups: closedRiemannian manifolds, compact manifolds with boundary with suitable boundary condition, oreven hyperbolic manifolds with infinite volume.Throughout this section we let H be a (separable) Hilbert space.7.1. Glazman’s lemma. The proof of the abstract Birman-Schwinger principle by Birman-Solomyak [9] is based on the use of Glazman’s lemma, which is an elementary version of themin-max principle. We shall now briefly recall this result and fix some notation along the way.Let A be a bounded from below selfadjoint operator on H . We denote by Q A its quadraticform. Thus, Q A has domain dom( A + µ ) / with µ > Q A ) containsdom( A ) as a dense subspace. Moreover, on dom( A ) we have Q A ( ξ, η ) := h Aξ | η i ∀ ξ, η ∈ dom( A ) . Denote by Sp ess ( A ) the essential spectrum of A , i.e., the complement of the discrete spectrum.Thus, the part of the spectrum of A below inf Sp ess ( A ) consists of isolated eigenvalues with finitemultiplicity. Assuming there is any, we list them as a non-decreasing sequence,(7.1) λ ( A ) ≤ λ ( A ) ≤ λ ( A ) ≤ · · · , where each eigenvalue is repeated according to multiplicity. This sequence may be finite or infinite.We introduce the counting function, N ( A ; λ ) := (cid:8) j ; λ j ( A ) < λ (cid:9) , λ < inf Sp ess ( A ) . (7.2)If inf Sp ess ( A ) ≥ 0, we also set N − ( A ) = N ( A ; 0).In what follows, given any λ < inf Sp ess ( A ), we denote by F ( A ; λ ) the collection of subspaces F of dom( Q A ) such that(7.3) Q A ( ξ, ξ ) < λ k ξ k ∀ ξ ∈ F \ . Lemma 7.1 (Glazman’s Lemma; see [9, Theorem 10.2.3]) . For all λ < inf Sp ess ( A ) , we have (7.4) N ( A ; λ ) = max { dim F ; F ∈ F ( A ; λ ) } . The maximum is reached by F = L µ<λ ker( A − µ ) .Remark . The equality (7.4) continues to hold for λ ≥ inf Sp ess ( A ) provided we define N ( A ; λ )as the rank of the spectral projection E ( λ ) := ( −∞ ,λ ) ( A ), in which case both sides of (7.4) areinfinite (see [9]). Remark . Given λ < inf Sp ess ( A ), let F +0 ( A ; λ ) be the collection of subspaces E of dom( Q A )such that Q A ( ξ, ξ ) ≥ λ k ξ k for all ξ ∈ E . Then we also have(7.5) N ( A ; λ ) = min { dim E ⊥ ; E ∈ F +0 ( A ; λ ) } . f T is a selfadjoint compact operator on H we may be interested in counting its positiveeigenvalues. We arrange them as a non-increasing sequence, λ ( T ) ≥ λ ( T ) ≥ λ ( T ) ≥ · · · , where each eigenvalue is repeated according to multiplicity. Once again the sequence may be finiteor infinite. Note that, if T ≥ 0, then λ j ( T ) is just the ( j + 1)-th singular value µ j ( T ).We define N + ( T ; µ ) := (cid:8) j ; λ j ( T ) > µ (cid:9) , µ > . Equivalently, N + ( T ; µ ) = N ( − T ; − µ ). In particular,(7.6) N + ( T ; µ ) = (cid:8) j ; µ j ( T ) > µ (cid:9) if T ≥ . In what follows, for µ > F + ( T ; µ ) the collection of subspaces F ⊂ H such that(7.7) Q T ( ξ, ξ ) > µ k ξ k ∀ ξ ∈ F \ . As F + ( T ; µ ) = F ( − T ; − µ ), we obtain the following version of Glazman’s lemma for compactoperators. Lemma 7.4 (Glazman’s Lemma, compact version) . Let T be a selfadjoint compact operator on H . Then, for all µ > , we have (7.8) N + ( T ; µ ) = max { dim F ; F ∈ F + ( A ; µ ) } . The Abstract Birman-Schwinger principle. From now on we let H be a (densely de-fined) selfadjoint operator on H with non-negative spectrum containing 0. Its quadratic form Q H has domain dom( Q H ) = dom( H + 1) . We denote by H + the Hilbert space obtained by endowingdom( Q H ) with the Hilbert space norm, k ξ k + = (cid:0) Q H ( ξ, ξ ) + k ξ k (cid:1) = (cid:13)(cid:13) (1 + H ) / ξ (cid:13)(cid:13) , ξ ∈ dom( Q H ) . We also let H − be the Hilbert space of continuous anti-linear functionals on H + . Note that wehave a continuous embedding ι : H ֒ → H − with dense range given by h ι ( ξ ) , η i = h ξ | η i , ξ ∈ H , η ∈ H + . The operator ( H + 1) / : H + → H is a unitary isomorphism. By selfadjointness it extendsto a unitary isomorphism ( H + 1) / : H → H − such that(7.9) (cid:10) ( H + 1) / ξ, η (cid:11) = (cid:10) ξ (cid:12)(cid:12) ( H + 1) / η (cid:11) , ξ ∈ H , η ∈ H + , where h· , ·i : H − × H + → C is the natural duality pairing. In particular, we have bounded inverses( H + 1) − / : H → H + and ( H + 1) − / : H − → H such that(7.10) (cid:10) ( H + 1) − / ξ | η (cid:11) = (cid:10) ξ, ( H + 1) − / η (cid:11) , ξ ∈ H − , η ∈ H . More generally, for any λ < 0, we have bounded operators ( H − λ ) / : H + → H and ( H − λ ) / : H → H − with bounded inverses ( H − λ ) − / : H → H + and ( H − λ ) − / : H − → H .Similarly, the operator H extends to a bounded operator H : H + → H − such that(7.11) h Hξ, η i = Q H ( ξ, η ) ξ, η ∈ H + . More generally, for any λ < 0, the operator H − λ = ( H − λ ) / ( H − λ ) / extends to a boundedoperator from H + to H − with bounded inverse ( H − λ ) − : H − −→ H + .In what follows, we let V : H + → H − be a bounded operator. We denote by Q V the corre-sponding quadratic form with domain H + and given by Q V ( ξ, η ) := h V ξ, η i , ξ, η ∈ H + . We assume that Q V is symmetric and H -form compact . The latter condition means that theoperator V : H + → H − is compact, or equivalently, ( H + 1) − / V ( H + 1) − / is a compactoperator on H .Our main focus is the operator H V := H + V . It makes sense as a bounded operator H V : H + → H − . Furthermore, as the symmetric quadratic form Q V is H -form compact, it is H -formbounded with zero H -bound (see [66, § he restriction of H V to dom( H V ) := H − V ( H ) is a bounded from below selfadjoint operator on H whose quadratic form is precisely Q H + Q V . Lemma 7.5 (see [66, Theorem 7.8.4]) . The following holds.(i) For all λ Sp( H ) ∪ Sp( H V ) , the operator ( H V − λ ) − − ( H − λ ) − is compact.(ii) The operators H and H V have the same essential spectrum.(iii) If H has compact resolvent, then so does H V . As H V is bounded from below, in the same way as in (7.1) we may list its eigenvalues belowthe essential spectrum as a non-decreasing sequence λ ( H V ) ≤ λ ( H V ) ≤ · · · . We then definethe counting function N ( H V ; λ ) as in (7.2). As H V has the same essential spectrum as H andSp( H ) ⊂ [0 , ∞ ), it follows that the essential spectrum of H V is contained in [0 , ∞ ). In particular,inf Sp ess ( H ) ≥ 0, and hence we may define N − ( H V ) := N ( H V ; 0) as above.Given operators V j : H + → H − , j = 1 , 2, we shall write V ≤ V when Q V ≤ Q V , i.e., Q V and Q V are both symmetric and Q V ( ξ, ξ ) ≤ Q V ( ξ, ξ ) for all ξ ∈ H + . In this case Q H V ( ξ, ξ ) ≤ Q H V ( ξ, ξ ) for all ξ ∈ H + . By using Glazman’s lemma we then get the following monotonicityprinciple,(7.12) V ≤ V = ⇒ N ( H V ; λ ) ≤ N ( H V ; λ ) ∀ λ < inf Sp ess ( H ) . If λ < inf Sp ess ( H ), it then follows that, in the notation of Lemma 7.1, we have F ( H V ; λ ) ⊂ F ( H V ; λ ). Therefore, by using (7.4) we get the The Birman-Schwinger principle was establishedby Birman [3] and Schwinger [59] for Schr¨odinger operators ∆ + V on R n , n ≥ 3. It is the mainimpetus for using Cwikel estimates to establish the Cwikel-Lieb-Rozenblum inequality (see [17,64]). In our setting it relates the counting function N ( H V ; λ ) to the eigenvalues of the Birman-Schwinger operators, K V ( λ ) = − ( H − λ ) − V ( H − λ ) − , λ < . The compactness of V ensures us that K V ( λ ) is a compact operator on H .Note also that K V ( λ ) is related to the quadratic form Q V by h K V ( λ ) ξ | η i = − (cid:10) V ( H − λ ) − / ξ, ( H − λ ) − / η (cid:11) = − Q V (cid:0) ( H − λ ) − / ξ, ( H − λ ) − / η (cid:1) , ξ, η ∈ H . (7.13)Thus, the fact that Q V is symmetric ensures us that K V ( λ ) is a selfadjoint compact operator. Wethen define the counting function N + ( K V ( λ ); µ ), µ > 0, as in (7.6). If in addition V ≤ 0, then K V ( λ ) ≥ 0, and so in this case the eigenvalues of K V ( λ ) agree with its singular values. Proposition 7.6 (Abstract Birman-Schwinger Principle [9, Lemma 1.4]) . For all λ < , we have (7.14) N ( H V ; λ ) = N + ( K V ( λ ); 1) . Proof. Given λ < 0, let F ( H V ; λ ) be the collection of subspaces F ⊂ H + satisfying (7.3) relativelyto Q H V . We also denote by F + ( K V ( λ ); 1) the collection of subspaces F + ⊂ H satisfying (7.7)relatively to Q K V ( λ ) for µ = 1.Let ξ ∈ H + \ η = ( H − λ ) / ξ ∈ H , i.e., ξ = ( H − λ ) − / η . By using (7.11) we get Q H V ( ξ, ξ ) < λ h ξ | ξ i ⇐⇒ Q H ( ξ, ξ ) − λ h ξ | ξ i < − Q V ( ξ, ξ ) , ⇐⇒ h ( H − λ ) ξ, ξ i < − h V ξ, ξ i . In the same way as in (7.9) we have h ( H − λ ) ξ, ξ i = (cid:10) ( H − λ ) η, ( H − λ ) − η (cid:11) = h η | η i . Moreover, by using (7.13) we get − h V ξ, ξ i = − (cid:10) V ( H − λ ) − η, ( H − λ ) − η (cid:11) = h K V ( λ ) η | η i = Q K V ( λ ) ( η, η ) . Thus,(7.15) Q H V ( ξ, ξ ) < λ h ξ | ξ i ⇐⇒ Q K V ( λ ) ( η, η ) > h η | η i . s ( H − λ ) / is a linear isomorphism from H + onto H , it follows from (7.15) that it inducesa one-to-one correspondence between the spaces in F ( H V ; λ ) and those in F + ( K V ( λ ); 1). Thiscorrespondence preserves the dimension. Thus, by using (7.4) and (7.8) we get N ( H V ; λ ) = max { dim F ; F ∈ F ( H V ; λ ) } , = max { dim F + ; F + ∈ F + ( K V ( λ ); 1) } = N + ( K V ( λ ); 1) . The proof is complete. (cid:3) Remark . The idea of the above proof of the abstract Birman-Schwinger principle is due toBirman-Solomyak [9]. In fact, we merely recasted the proof of [9] into the framework of [66, § Corollary 7.8. Assume V ≤ and ( H + 1) − / V ( H + 1) − / ∈ L q, ∞ for some q ∈ (0 , ∞ ) . Then,for all λ < , the operator K V ( λ ) is in the weak Schatten class L q, ∞ , and we have (7.16) N ( H V ; λ ) ≤ k K V ( λ ) k q L q, ∞ . Proof. We recall the proof for reader’s convenience. Let λ < A = ( H + 1) / ( H − λ ) − / .The fact that ( H + 1) − / V ( H + 1) − / is in L q, ∞ ensures us that K V ( λ ) ∈ L q, ∞ , since A is abounded operator on H , and we have(7.17) K V ( λ ) = − A ∗ ( H + 1) − / V ( H + 1) − / A. Bearing this in mind, set N = N ( H V ; λ ). We may assume N ≥ 1, since otherwise the inequal-ity (7.16) is trivially satisfied. The Birman-Schwinger principle (7.14) asserts that N + ( K V ( λ ); 1) = N . As V ≥ 0, we know from (7.6) that N + ( K V ( λ ); 1) is the number of singular values of K V ( λ ) that are > 1. Thus, K V ( λ ) has exactly N singular values > 1. In particular, we have µ N − ( K V ( λ )) ≥ 1. Therefore, in view of the definition (4.7) of the quasi-norm of L q, ∞ , we get(7.18) N ≤ N µ N − (cid:0) K V ( λ ) (cid:1) q ≤ (cid:18) sup j ≥ ( j + 1) q µ j (cid:0) K V ( λ ) (cid:1)(cid:19) q = (cid:13)(cid:13) K V ( λ ) (cid:13)(cid:13) q L q, ∞ . The proof is complete. (cid:3) Borderline Birman-Schwinger principle. The proofs of the standard CLR inequality forSchr¨odinger operators on R n , n ≥ 3, uses the fact that for the Laplacian on R n with n ≥ − λ ) − has a weak limit as λ → − . This allowsus to take λ = 0 in (7.16) and get an upper bound for N − ( H V ) (see [65, Theorem 7.9.11]).However, on closed manifolds and NC tori, 0 is in the discrete spectrum of the Laplacian, andso the resolvent has a pole singularity at λ = 0. This prevents us from letting λ → − in (7.16).This stresses out the need for a “borderline” version of the Birman-Schwinger (7.16) when 0 is thediscrete spectrum.Assume that 0 lies in the discrete spectrum of H , i.e., 0 is an isolated eigenvalue of H . Thus,the essential spectrum of H is contained in some interval [ a, ∞ ) with a > 0. As H and H V havesame essential spectrum by Lemma 7.5, it follows that H V has at most finitely many non-positiveeigenvalues.We denote by H − the partial inverse of H . That is, H − vanishes on ker H and inverts H on (ker H ) ⊥ = ran H . This is a selfadjoint bounded operator with non-negative spectrum.We then define H − / to be ( H − ) / . Equivalently, H − = f ( H ) and H − / = g ( H ), where f ( t ) = [ ǫ, ∞ ) ( t ) t − and g ( t ) = [ ǫ, ∞ ) ( t ) t − / , with ǫ > H ) ∩ (0 , ǫ ] = ∅ .We shall need the following abstract version of Lemma 6.2. Lemma 7.9. Suppose that V ≤ . Then there is a compact operator W : H + → H such that V = − W ∗ W , where W ∗ : H → H − is the adjoint map.Proof. Set K = K V ( − V ≤ 0, we know that K is a positive compact operator. Set W = K / ( H + 1) / . Then W is a compact operator from H + to H . By duality we get a linearoperator W ∗ : H → H − such that(7.19) h W ∗ ξ, η i = h ξ | W η i , ξ ∈ H , η ∈ H + . n particular, for ξ, η ∈ H , we have h W ∗ W ξ, η i = h W ξ | W η i = (cid:10) K ( H + 1) ξ | K ( H + 1) η (cid:11) = (cid:10) K ( H + 1) ξ | ( H + 1) η (cid:11) = (cid:10) ( H + 1) K ( H + 1) ξ, η (cid:11) = − h V ξ, η i . This shows that V = − W ∗ W . The proof is complete. (cid:3) In what follows we denote by Π the orthogonal projection on ker H . This is a selfadjointfinite rank operator whose range is contained in H + , and hence we get a bounded operatorΠ : H → H + . By duality we obtain a bounded operator Π : H − → H such that h Π ξ | η i = h ξ, Π η i , ξ ∈ H − , η ∈ H . The composition Π V Π then is a selfadjoint finite-rank operator on H . As above, we denote by N − (Π V Π ) its number of negative eigenvalues counted with multiplicity. Proposition 7.10 (Borderline Birman-Schwinger Principle) . Assume that is in the discretespectrum of H . Suppose that V ≤ and H − / V H − ∈ L p, ∞ for some p ∈ (0 , ∞ ) . Then (7.20) 0 ≤ N − ( H V ) − N − (Π V Π ) ≤ (cid:13)(cid:13) H − V H − (cid:13)(cid:13) p L p, ∞ . In particular, if V (ker H ) ⊂ ran H , then (7.21) N − ( H V ) ≤ (cid:13)(cid:13) H − V H − (cid:13)(cid:13) p L p, ∞ . Proof. Set A = H / ( H + 1) − / . Then ( H + 1) − / = H − / A + Π , and so in the same way asin (7.17) we have ( H + 1) − / V ( H + 1) − / = A ∗ (cid:2) H − V H − (cid:3) A + R, where R is a finite rank operator. This implies that ( H + 1) − / V ( H + 1) − / is in the weakSchatten class L p, ∞ . In particular, this is a compact operator. Likewise, the Birman-Schwingeroperators K V ( λ ), λ < 0, are in L p, ∞ as well.Bearing this in mind, set N = N − ( H V ) and N = N − (Π V Π ). Let us first show that that N ≤ N . We have N = dim F , where F = L λ< ker(Π V Π − λ ). For all ξ ∈ H , we have h Π V Π ξ | ξ i = h V Π ξ, Π ξ i = Q V (Π ξ, Π ξ ) = Q H V (Π ξ, Π ξ ) . Furthermore, if ξ ∈ ker(Π V Π − λ ) with λ < 0, then ξ = λ − Π V Π ξ ∈ ker H . It follows that F ⊂ ker H , and hence Π ξ = ξ for all ξ ∈ F . We deduce that, for all ξ ∈ F \ 0, we have Q H V ( ξ, ξ ) = Q H V (Π ξ, Π ξ ) = h Π V Π ξ | ξ i ≤ µ k ξ k < , where µ = λ N − (Π V Π ) is the smallest negative eigenvalue of Π V Π . In the notation ofLemma 7.1, this means that F ∈ F ( H V ; 0). Thus, by using (7.4) we get N = dim F ≤ N ( H V ; 0) = N. It remains to show the 2nd inequality in (7.20). We may assume N − N ≥ 1, since otherwisethe inequality is trivially satisfied. Note that N = N ( H V ; λ ) as soon as λ < N = N + ( K V ( λ ); 1). As K V ( λ ) ≥ 0, inthe same way as in the proof of Corollary 7.8 this ensures that K V ( λ ) has exactly N singularvalues > 1, and hence µ N − ( K V ( λ )) ≥ W : H + → H such that V = − W ∗ W . Set T ( λ ) = W ( H − λ ) − . This is a compact operator on H . Moreover, by using (7.10) and (7.19)we see that, for all ξ, η ∈ H , we have (cid:10) ( H − λ ) − W ∗ ξ | η (cid:11) = (cid:10) W ∗ ξ, ( H − λ ) − η (cid:11) = (cid:10) ξ | W ( H − λ ) − η (cid:11) = h ξ | T ( λ ) η i . This shows that T ( λ ) ∗ = ( H − λ ) − / W ∗ . Thus, K V ( λ ) = ( H − λ ) − W ∗ W ( H − λ ) − = T ( λ ) ∗ T ( λ ) = | T ( λ ) | . et ˜ K V ( λ ) = T ( λ ) T ( λ ) ∗ = W ( H − λ ) − W ∗ . By using (4.3) we get(7.22) µ j ( K V ( λ )) = µ j ( T ( λ )) = µ j ( T ( λ ) ∗ ) = µ j (cid:0) ˜ K V ( λ ) (cid:1) , j ≥ . Observe that ˜ K V ( λ ) = T ( λ )(1 − Π ) T ( λ ) ∗ + T ( λ )Π T ( λ ) ∗ . By using (4.4) we get µ N − (cid:0) ˜ K V ( λ ) (cid:1) ≤ µ N − N − (cid:0) T ( λ )(1 − Π ) T ( λ ) ∗ (cid:1) + µ N (cid:0) T ( λ )Π T ( λ ) ∗ (cid:1) . As rk T ( λ )Π T ( λ ) ∗ ≤ rk Π = N , it follows from (4.2) that µ N ( T ( λ )Π T ( λ ) ∗ ) = 0. Thus,(7.23) µ N − (cid:0) ˜ K V ( λ ) (cid:1) ≤ µ N − N − (cid:0) T ( λ )(1 − Π ) T ( λ ) ∗ (cid:1) . We also have T ( λ )(1 − Π ) T ( λ ) ∗ = W ( H − λ ) − / (1 − Π )( H − λ ) − / W ∗ , = W ( H + 1) − / · (1 − Π )( H + 1)( H − λ ) − · ( H + 1) − / W ∗ , = T ( − − Π )( H + 1)( H − λ ) − T ( − ∗ . As (1 − Π )( H + 1)( H − λ ) − ≤ ( H + 1) H − , it follows that T ( λ )(1 − Π ) T ( λ ) ∗ ≤ T ( − H + 1) H − T ( − ∗ = W H − W ∗ . Therefore, by using (7.23) and the monotonicity principle (4.6) we get(7.24) µ N − (cid:0) ˜ K V ( λ ) (cid:1) ≤ µ N − N − (cid:0) T ( λ )(1 − Π ) T ( λ ) ∗ (cid:1) ≤ µ N − N − (cid:0) W H − W ∗ (cid:1) . As in (7.22) we have µ j ( W H − W ∗ ) = µ j ( H − / V H − / ) for all j ≥ 0. Combining thiswith (7.24) and the fact that µ N − ( ˜ K V ( λ )) ≥ ≤ µ N − (cid:0) ˜ K V ( λ ) (cid:1) ≤ µ N − N − (cid:0) H − V H − (cid:1) . In the same way as in (7.18) this implies that N − N ≤ ( N − N ) µ N − N − (cid:0) H − V H − (cid:1) p ≤ (cid:13)(cid:13) H − V H − (cid:13)(cid:13) p L p, ∞ . This gives the 2nd inequality in (7.20). The proof is complete. (cid:3) Remark . A weaker version of (7.20) is used in the work of Birman and Solomyak, in particularfor dealing with Schr¨odinger operators on bounded domains under Neuman’s boundary condition(see, e.g., [5, 7, 67]). In our general setup it amounts to the following inequality(7.26) N − ( H V ) − dim ker H ≤ (cid:13)(cid:13) H − V H − (cid:13)(cid:13) p L p, ∞ . This inequality immediately follows from (7.20) since N − (Π V Π ) ≤ dim ker H . However, we geta direct proof by applying Galzman’s lemma in the version given by (7.5) to the restriction of Q H V to H + ∩ (ker H ) ⊥ and by using the usual Birman-Schwinger principle (7.16) on (ker H ) ⊥ .In fact, in all the instances where it is used by Birman and Solomyak the inequality (7.26) isrelevant, since in those situations ker H = 1 and Π V Π has a negative eigenvalue is V ≤ V = 0. However, in general we might have N − ( H V ) − dim ker H < V (ker H ) ⊂ ran H ),and so in such a case the inequality (7.26) is not sharp (see Example 7.12 below). Example . Let ( M n , g ) be a closed Riemannian manifold, and take H to be the (positive)Laplacian ∆ g on H = L ( M ; g ) with domain the Sobolev space W ( M ). In this case, ker ∆ g isspanned by the characteristic functions of the connected components of M , and so dim ker ∆ g =dim H ( M ), where H ( M ) is the zero-th degree de Rham cohomology space of M . Let V ∈ L ∞ ( M ). Then, ∆ − / g V ∆ − / g ∈ L n/ , ∞ , and so from (7.26) we get(7.27) N − (∆ g + V ) − dim H ( M ) ≤ (cid:13)(cid:13) ∆ − g V ∆ − g (cid:13)(cid:13) n L n , ∞ . Suppose now that M has at least two connected components, and let M be such a component.If V = − α M with α > 0, then Π V Π = − α Π , where Π is the orthogonal projection onto M , and hence N − (Π V Π ) = 1. Furthermore, if α is small enough, then N − (∆ g − α M ) = N − (∆ g | M − α ) + N − (∆ g | M \ M ) = N − (∆ g | M ; α ) = 1 < dim H ( M ) . This proof was pointed out to us by Grigori Rozenblum. hus, in this case the l.h.s. of (7.27) is < CLR Inequalities on NC Tori In this section, we combine the result of the previous two sections to establish CLR inequalitiesfor fractional Schr¨odinger operators ∆ n/ p + λ ( V ) on T nθ .In the notation of the previous section we let H = L ( T nθ ) and let H be the fractional Laplacian∆ n/ p , p > 0, with domain W n/ p ( T nθ ). Note that ∆ n/ p has compact resolvent and its nullspaceis C · 1. In particular, 0 is an isolated eigenvalue. Up to equivalent norms, the Hilbert spaces H + and H − are the Sobolev spaces W n/ p ( T nθ ) and W − n/ p ( T nθ ), respectivelySuppose that, either s > n/ q and q ≥ 1, or s = n/ q and q ≥ 1. In this case, if V ∈ L q ( T nθ ),then we know by Lemma 3.4 that λ ( V ) induces a bounded operator λ ( V ) : W s ( T nθ ) → W − s ( T nθ ).Thus, it defines a quadratic form on W s ( T nθ ) by Q λ ( V ) ( u, v ) = h λ ( V ) u, v i = τ (cid:2) v ∗ V u (cid:3) , u, v ∈ W s ( T nθ ) . In particular, if V ∗ = V , then Q λ ( V ) ( u, v ) = τ (cid:2) u ∗ V v (cid:3) = Q λ ( V ) ( v, u ) . That is, Q λ ( V ) is symmetric. If in addition V ≥ 0, then Q λ ( V ) ( u, u ) = τ (cid:2) u ∗ V u ] = τ (cid:2) ( V / u ) ∗ V / u (cid:3) ≥ . Thus, Q λ ( V ) ≥ 0. In particular, in the notation of Section 7 we have(8.1) V ≤ V = ⇒ λ ( V ) ≤ λ ( V ) . Bearing this in mind, Theorem 6.4 implies that the operator λ ( V ) : W n/ p ( T nθ ) → W − n/ p ( T nθ )is compact under any of the following conditions:(i) p > V ∈ L p ( T nθ ).(ii) p = 1 and V ∈ L q ( T nθ ), q > p < V ∈ L ( T nθ ).It follows that, if V is selfadjoint and any of the above conditions (i)–(iii) holds, then the pair(∆ n/ p , λ ( V )) fits into the framework of the previous section. Thus, we may define the fractionalSchr¨odinger operator ∆ n p + λ ( V ) as a form sum. We obtain a bounded from below selfadjointoperator on L ( T n ) with compact resolvent. In particular, it has no essential spectrum. As abovewe denote by N − (∆ n/ p + λ ( V )) the number of negative eigenvalues counted with multiplicity.In what follows, given any selfadjoint element V ∈ L p ( T nθ ), we let V ± = ( | V | ± V ) be thepositive and negative parts of V . Note that V + and V − are positive elements of L p ( T n ).We are now in a position to establish CLR inequalities on NC tori in the following form. Theorem 8.1 (CLR Inequalities on NC Tori) . Let n ≥ . The following holds.(i) If p > and V = V ∗ ∈ L p ( T nθ ) , then (8.2) N − (cid:0) ∆ n p + λ ( V ) (cid:1) − ≤ c np τ (cid:2) | V − | p (cid:3) . (ii) If V = V ∗ ∈ L p ( T nθ ) , p > , then (8.3) N − (cid:0) ∆ n + λ ( V ) (cid:1) − ≤ c np τ (cid:2) | V − | p (cid:3) p . (iii) If p < and V = V ∗ ∈ L ( T nθ ) , then (8.4) N − (cid:0) ∆ n p + λ ( V ) (cid:1) − ≤ c np τ (cid:2) | V − | (cid:3) p . Proof. Suppose that q = max( p, 1) if p = 1, or q > p = 1. The proof amounts to show that if V = V ∗ ∈ L q ( T nθ ), then(8.5) N − (cid:0) ∆ n p + λ ( V ) (cid:1) − ≤ c npq τ (cid:2) | V − | q (cid:3) pq . Note also that if V − = 0, then V = V + ≥ Q λ ( V ) ≥ 0, and hence ∆ n/ p + λ ( V ) ≥ 0. Thus,in this case N − (∆ n/ p + λ ( V )) = 0, and so the inequality (8.5) is trivially satisfied. Therefore, wemay assume V − = 0. s V = V + − V − ≥ − V − , by (8.1) we have λ ( V ) ≥ − λ ( V − ), and so the monotonicity princi-ple (7.12) implies that(8.6) N − (cid:0) ∆ n p + λ ( V ) (cid:1) ≤ N − (cid:0) ∆ n p − λ ( V − ) (cid:1) . Let Π be the orthogonal projection onto ker ∆ n/ p = C · 1. It is a rank 1 projection given byΠ u = τ ( u )1 , u ∈ L ( T nθ ) . The formula continues to make sense for any u ∈ L ( T nθ ). We have − Π λ ( V − )Π − τ (1)Π V − = − τ ( V − ) < . Thus, − τ ( V − ) is a negative eigenvalue of − Π λ ( V − )Π . As − Π λ ( V − )Π has rank 1 this is theunique negative eigenvalue, and hence N − ( − Π λ ( V − )Π ) = 1. Combining this with (8.6) gives(8.7) N − (cid:0) ∆ n p + λ ( V ) (cid:1) − ≤ N − (cid:0) ∆ n p − λ ( V − ) (cid:1) − N − ( − Π λ ( V − )Π ) . Moreover, Theorem 6.1 ensures us that ∆ − n/ p λ ( V )∆ − n/ p ∈ L p, ∞ with norm estimate,(8.8) (cid:13)(cid:13) ∆ − n p λ ( V )∆ − n p (cid:13)(cid:13) L p, ∞ ≤ c npq k V − k L q = c npq τ (cid:2) | V − | q (cid:3) q . This allows us to apply the borderline Birman-Schwinger principle (Theorem 7.10) to get N − (cid:0) ∆ n p − λ ( V − ) (cid:1) − N − ( − Π λ ( V − )Π ) ≤ (cid:13)(cid:13) ∆ − n p λ ( V )∆ − n p (cid:13)(cid:13) p L p, ∞ . Combining this with (8.7) and (8.8) gives the inequality (8.5). The proof is complete. (cid:3) Remark . Suppose that either q = max( p, 1) and p = 1, or q > p . The CLR inequali-ties (8.2)–(8.4) are consistent with Lieb’s version of the CLR inequality for Schr¨odinger operatorson closed manifolds (see [39, 40]). Note that we cannot expect estimates of the form,(8.9) N − (cid:0) ∆ n p + λ ( V ) (cid:1) ≤ c npq τ (cid:2) | V − | q (cid:3) pq . To see this take V = − ǫ with ǫ > 0. In that case the right-hand side of (8.9) is c npq τ [ ǫ q ] p/q = c npq ǫ p .In addition (∆ n/ p − ǫ )1 = − ǫ 1, and hence − ǫ is always a negative eigenvalue of ∆ n/ p − ǫ . Thus, N − (∆ n/ p − ǫ ) ≥ 1. Therefore, if we had the inequality (8.9), then we would have1 ≤ N − (cid:0) ∆ n p − ǫ (cid:1) ≤ c npq ǫ p ∀ ǫ > , which does not hold as soon as ǫ is small enough. Remark . For the ordinary torus T n , i.e., θ = 0, the Cwikel estimates of [72] allow us to extendthe critical CLR inequality (8.3) to all potentials V in Zygmund’s class L log L ( T n ).The most interesting case of the above CLR inequalities is for the usual Schr¨odinger operator∆ + λ ( V ), i.e., p = n/ 2. In this case Theorem 8.1 specializes to the following statement. Theorem 8.4. The following holds.(i) If n ≥ and V = V ∗ ∈ L n/ ( T nθ ) , then (8.10) N − (cid:0) ∆ + λ ( V ) (cid:1) − ≤ c n τ (cid:2) | V − | n (cid:3) . (ii) If n = 2 and V = V ∗ ∈ L p ( T θ ) , p > , then (8.11) N − (cid:0) ∆ + λ ( V ) (cid:1) − ≤ c np τ (cid:2) | V − | p (cid:3) p . Remark . The 2-dimensional case (8.11) shows a stark contrast with the Euclidean space case,since on R the Birman-Schwinger principle and the CLR inequality for Schr¨odinger operators∆ + V do not hold. In particular, recent results of Hoang et al. [29] show that if V ∈ L ( R ) \ R V ( x ) dx ≤ 0, then ∆ + V always has at least one negative eigenvalue (see also [63]). Remark . The case p = n/ p = n we get the hyper-relativistic Schr¨odinger operator |∇| + V (see, e.g., [18]). lthough the inequality (8.9) does not hold, it holds semiclassically . Namely, we have thefollowing result. Corollary 8.7 (Semiclassical CLR Inequality) . Assume that, either q = max( p, with p = 1 , or q > p = 1 . Let V = V ∗ ∈ L q ( T nθ ) . Then, as h → + we have (8.12) N − (cid:0) h np ∆ n p + λ ( V ) (cid:1) ≤ c npq h − n τ (cid:2) | V − | q (cid:3) pq + O(1) . Proof. As h n/p ∆ n/ p + λ ( V ) = h n/p (∆ + λ ( h − n/p V )), the operators h n/p ∆ n/ p + λ ( V ) and ∆ + λ ( h − n/p V ) have the same number of negative eigenvalues counted with multiplicity. Therefore,the CLR inequality (8.5) for h − n/p V gives N − (cid:0) h np ∆ n p + λ ( V ) (cid:1) = N − (cid:0) ∆ n p + λ (cid:0) h − np V (cid:1)(cid:1) ≤ c npq τ (cid:2)(cid:12)(cid:12) h − np V − (cid:12)(cid:12) q (cid:3) pq + 1 ≤ c npq h − n τ (cid:2) | V − | q (cid:3) pq + O(1) . The proof is complete. (cid:3) The semiclassical CLR inequality on R n , n ≥ 3, is the main tool to extend the semi-classicalWeyl’s law for Schr¨odinger operators h ∆ + V with smooth potentials V to Schr¨odinger operatorswith potentials in L n/ ( R n ) (see [64, Proposition 5.2]). This result can be traced back to the workof Birman and his collaborators in the late 60s and early 70s (see [5, 7]).Likewise, we can extend the semiclassical Weyl’s law for fractional Schr¨odinger operators h np ∆ n p + V for smooth potentials to potentials in a suitable L q -class (see [67] and the refer-ences therein). Similarly, the semiclassical CLR inequality (8.12) for NC tori would enable usto extend the semiclassical Weyl law on NC tori for smooth potentials to non-smooth potentials.However, to date the semiclassical Weyl law for smooth potentials has still not been established.We believe it can be established in a similar fashion as in the Euclidean case by setting up asemi-classical pseudodifferential calculus on NC tori. However, this falls out of the scope of thispaper. Therefore, we only state the semiclassical Weyl’s law on NC tori as a conjecture. Conjecture 8.8 (Semiclassical Weyl Law on NC Tori) . Let n ≥ . Suppose that, either p = 1 and q = max( p, , or p = 1 < q . If V = V ∗ ∈ L q ( T nθ ) , then as h → + we have (8.13) N − (cid:0) h np ∆ n p + λ ( V ) (cid:1) = c n h − n τ (cid:2) | V − | p (cid:3) + o (cid:0) h − n (cid:1) , where c n = | B n | . In particular, if n ≥ and V = V ∗ ∈ L n/ ( T nθ ) , or if n = 2 and V = V ∗ ∈ L q ( T n ) with q > ,then N − (cid:0) h ∆ + λ ( V ) (cid:1) = c n h − n τ (cid:2) | V − | n (cid:3) + o (cid:0) h − n (cid:1) . Remark . In the critical case p = n/ L log L ( T nθ ). However, this would requireextending the Cwikel estimate (6.3) and the CLR inequality (8.3) to this class of potentials. References [1] Bellissard, J.; van Elst, A.; Schulz-Baldes, H.: The noncommutative geometry of the quantum Hall effect . J.Math. Phys. (1994), 5373–5451.[2] Bennett, C; Sharpley, R.: Interpolation of operators . Pure and Applied Mathematics 129, Academic Press,Inc., Boston, MA, 1988.[3] Birman, M.Sh.: The spectrum of singular boundary problems . Amer. Math. Soc. Transl. 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