Szegö Limit Theorem on the Heisenberg Group
aa r X i v : . [ m a t h . F A ] M a r SZEG ¨O LIMIT THEOREM ON THE HEISENBERG GROUP
SHYAM SWARUP MONDAL AND JITENDRIYA SWAIN
Abstract.
Let H = V, H = B + V and H = L + V be the operators on the Heisenberggroup H n , where V is the operator of multiplication growing like | g | κ , < κ < B is abounded linear operator and L is the sublaplacian on H n . In this paper we prove Szeg¨olimit theorem for the operators H , H and H on L ( H n ) . Introduction
The classical Szeg¨o limit theorem describes the asymptotic distribution of eigenvalues ofthe operator P n T f P n , where T f is the multiplication operator on L ((0 , π )), associated witha positive function f ∈ C α [0 , π ] , α > { P n } of L [0 , π ]onto a linear subspace spanned by the functions { e imθ : 0 ≤ m ≤ n ; 0 ≤ θ < π } . For such atriple ( f, T f , { P n } ) , Szeg¨o proved that(1.1) lim n →∞ n + 1 log det P n T f P n = 12 π Z π log f ( θ ) dθ. The orthogonal projections P n coincide with the spectral projections π λ of the self-adjointoperator − d dx on L [0 , π ], with a periodic boundary condition, corresponding to the interval[0 , λ ) with n < λ < n +1 . The result in equation (1.1) is the well known as Szeg¨o limit theorem.We refer to [4, 16] for details and related results. More specifically, for a bounded real-valuedintegrable function f , Szeg¨o limit theorem is generalized to continuous functions F (instead ofthe logarithm in (1.1)) defined on [inf f, sup f ], containing the eigenvalues { λ ni } ni =1 of P n T f P n in Sect. 5.3 of [4]. The result for such F is,lim n →∞ n n X i =1 F ( λ ni ) = 12 π Z π F ( f ( θ )) dθ. Notice that the left hand side here can be seen to be the limit of tr ( F ( P n T f P n )) /tr ( P n )where tr ( X ) denotes the trace of the operator X , e imθ is an eigenfunction of ∆ = − d dx , witha periodic boundary condition and the asymptotic of the functional ρ λ ( F ) = tr ( π λ F ( π λ T f π λ ) π λ ) = X k F ( µ k ( λ )) Date : March 5, 2019.2010
Mathematics Subject Classification.
Primary 42C15; Secondary 47B38.
Key words and phrases.
Balian-Low Theorem; time-frequency analysis; Weyl transform; Zak transform. is precisely the sum of Dirac measures located at the eigenvalues µ k ( λ ) of the operator π λ T f π λ .Similar results were obtained for various classes of differential and pseudo-differential operatorsin [5], [6], [7], [15], [19] and [20].In [20], Zelditch considered a Schr¨odinger operator on R n of the form H = − ∆ + V ,where V is a smooth positive function that grows like V | x | κ , κ > A associated with a symbol a ( x, ξ ) relative toBeals-Fefferman weights ϕ ( x, ξ ) = 1 , ϕ ( x, ξ ) = (1 + | ξ | + V ( x )) / and proved the followingSzeg¨o type theorem: For any continuous function f ,lim λ →∞ tr f ( π λ Aπ λ )rank ( π λ ) = lim λ →∞ R H ( x,ξ ) ≤ λ f ( a ( x, ξ )) dxdξ Vol( H ( x, ξ ) ≤ λ )where H ( x, ξ ) = | ξ | + V ( x ), assuming one of the limits exists. Such asymptotic spectralformulae expressing the relation between functions of pseudo-differential operators and theirsymbols is an important and interesting problem in mathematical analysis.In [15], the authors consider the operator of the form H = ∆ + V on the lattice, where thediscrete Laplacian operator (∆ u )( k ) = P | k − j | =1 u ( j ) + 2 nu ( k ) and V is multiplication by apositive sequence V ( n ) = (cid:26) , n = 0 | n | κ , κ > H n . Un-like the symbols on R n or Z n the symbols on the Heisenberg group are operator valued.Establishing Szeg¨o limit theorem is therefore bit more complicated.To establish a Szeg¨o type theorem, we need to consider the ratios of distribution functionsassociated to different measures and their asymptotic behavior. The asymptotic limit ofsuch ratios is computed using a Tauberian theorem where some transform of these measures isconsidered and the limit taken for such transforms. For example, Zelditch [20] used the Laplacetransform (via Karamata’s Tauberian theorem ([18], p-192)) whereas Robert [11] suggestedthe use of Stieltjes transform (via Keldysh Tauberian theorem) in [8]. We use the Tauberiantheorem of Grishin-Poedintseva (see theorem 8 of [3]) and a theorem of Laptev-Safarov (see[9] and [10]), for estimating the errors, to prove our main theorems (Theorem 1.1, 1.2 and1.3).There is extensive work on the Szeg¨o’s theorem associated with orthogonal polynomials in L ( T , dµ ) with µ some probability measure on T , we refer to the monumental work of BarrySimon [1] for the details. ZEG ¨O LIMIT THEOREM ON THE HEISENBERG GROUP 3
In this article we prove Szeg¨o limit theorem for the operators H = V, H = B + V and H = L + V on the Heisenberg group H n , where V is the operator of multiplication growinglike | g | κ , κ > B is a bounded linear operator and L is the sublaplacian on H n . Throughoutthis article we assume V ( g ) = | g | κ , where | g | = ( | x | + | t | ) , g = ( x, t ) ∈ H n , (1.2)defining the homogenous norm on H n . Since the resolvent operators ( H j − iI ) − are compactfor j = 0 , , so the operators H , H and H have discrete spectrum. The eigenfunctionsof H j form a complete orthogonal basis for L ( H n ) . Then our main theorems are the following. We consider the operators H = V and M b ,the operators of multiplication by V and b respectably on L ( H n ), where b is a bounded realvalued integrable function on the Heisenberg group and obtain the following theorem: Theorem 1.1.
Let H = V and V ( g ) = | g | κ , κ ≥ . Let π r be the orthogonal projection of L ( H n ) onto the space of eigenfunctions of H with eigenvalue ≤ r . Let b be a bounded realvalued integrable function on H n and M b be the operator of multiplication by b on L ( H n ) .Then for any f ∈ C ( R ) we have lim r →∞ tr f ( π r M b π r ) tr ( π r ) = Z H n f ( b ( g )) dg. Further, we consider the operator H = B + V , where B is a bounded linear operatoron L ( H n ) and replace the multiplication operator M b by a 0-th order self-adjoint pseudo-differential operator on L ( H n ) and obtain the following variation of Theorem 1.1: Theorem 1.2.
Consider the operator H = B + V on the Heisenberg group H n , where B isa bounded operator on H n and V ( g ) = | g | κ , < κ < . Let π r be the orthogonal projectionof L ( H n ) onto the space of eigenfunctions of H with eigenvalue ≤ r ; let A be a 0-th orderself-adjoint pseudo-differential operator on L ( H n ) with symbol a ( g, λ ) , where g ∈ H n , λ ∈ R ∗ and let f ∈ C ( R ) . Then lim r →∞ tr f ( π r Aπ r ) tr ( π r ) = lim r →∞ R R ∗ R V ( g ) ≤ r tr f ( a ( g, λ )) dg dµ ( λ ) r R r − r dµ ( λ ) R V ( g ) ≤ r dg . (Assuming one limit exists) Finally we consider the Schr¨odinger operator H = L + V on the Heisenberg group withthe assumptions of Theorem 1.2 and obtain the following theorem on the Heisenberg group: SHYAM SWARUP MONDAL AND JITENDRIYA SWAIN
Theorem 1.3.
Consider the Schr¨odinger operator of the form H = L + V on the Heisen-berg group H n , where L is the sublaplacian associated with the Heisenberg group and V ( g ) = | g | κ , < κ < . Let π r be the orthogonal projection of L ( H n ) onto the space of eigenfunctionsof H with eigenvalue ≤ r ; let A be a 0-th order self-adjoint pseudo-differential operator on L ( H n ) with symbol a ( g, λ ) where g ∈ H n , λ ∈ R ∗ and let f ∈ C ( R ) . Then lim r →∞ tr f ( π r Aπ r ) tr ( π r ) = lim r →∞ R G r f ( a g,λ ( ξ, x )) dξ dx dg dλ R G r dξ dx dg dλ (Assuming one limit exists)where G r = { ( g, λ, ξ, x ) ∈ H n × R ∗ × R n × R n : | λ | ( | ξ | + | x | ) + V ( g ) ≤ r , V ( g ) ≤ r , | λ | >r − } and a ( g, λ ) = Op W ( a g,λ ) . Also we show that the above theorems are valid under a compact perturbation of thepseudo-differential operator A in Corollary 5.6.We organize the paper as follows. In section 2, we provide necessary background for theSzeg¨o limit theorem on the Heisenberg group, estimate the difference of the symbol of thecomposition and composition of the corresponding symbols of two pseudo-differential operatoron the Heisenberg group. Finally we prove the Szeg¨o limit theorem for the operators H , H and H in section 3, 4 and 5 respectively.2. Notations and Background
The main aim of this section is to discuss the symbol classes on the Heisenberg group viathe left invariant vector fields and their correspondence with the symbol classes on R n . Hermite Operator.
Let H k denote the Hermite polynomial on R , defined by H k ( x ) = ( − k d k dx k ( e − x ) e x , k = 0 , , , · · · , and h k denote the normalized Hermite functions on R defined by h k ( x ) = (2 k √ πk !) − H k ( x ) e − x , k = 0 , , , · · · , The Hermite functions { h k } are the eigenfunctions of the Hermite operator H = − d dx + x with eigenvalues 2 k + 1 , k = 0 , , , · · · . These functions form an orthonormal basis for L ( R ).The higher dimensional Hermite functions denoted by Φ α are then obtained by taking tensorproduct of one dimensional Hermite functions. Thus for any multi-index α ∈ N n and x ∈ R n ,we define Φ α ( x ) = Q nj =1 h α j ( x j ) . The family { Φ α } is then an orthonormal basis for L ( R n ).They are eigenfunctions of the Hermite operator H = − ∆ + | x | corresponding to eigenvalues(2 | α | + n ), where | α | = P nj =1 α j . ZEG ¨O LIMIT THEOREM ON THE HEISENBERG GROUP 5
Pseudo-Differential Operator.
A linear differential operator p ( x, D ) = X | α |≤ m a α ( x ) ∂ α with variable coefficient a α can be expressed as p ( x, D ) f ( x ) = (2 π ) − n Z R n e ix · ξ p ( x, ξ ) ˆ f ( ξ ) dξ, ∀ x ∈ R n where the Fourier transform of f is defined byˆ f ( ξ ) = (2 π ) − n Z R n f ( x ) e − ix · ξ dx, ∀ ξ ∈ R n . When p ( x, ξ ) = a ( x, ξ ) is a measurable function on R n × R n , (not necessarily a polynomial)then the corresponding operator T a associated with the function a ( x, ξ ) is given by T a f ( x ) = a ( x, D ) f ( x ) = (2 π ) − n Z R n e ix · ξ a ( x, ξ ) ˆ f ( ξ ) dξ, ∀ x ∈ R n . The operator T a is called pseudo-differential operator and the function a ( x, ξ ) is called thesymbol of the pseudo-differential operator T a . Let m ∈ R , ≤ δ < ρ ≤ . Then the symbolclass S mρ,δ consists of those functions a ( x, ξ ) ∈ C ∞ ( R n × R n ) satisfying | ∂ αx ∂ βξ a ( x, ξ ) | ≤ C α,β (1 + | ξ | ) m − δ | α | + ρ | β | (2.1)for all multi-indices α, β . We take ρ = 1 and δ = 0 through out the paper and denote thesymbol class by S m .The Weyl quantization Op W for a “reasonable” symbol a in R n × R n is given by Op W ( a ) f ( u ) = (2 π ) − n Z R n Z R n e i ( u − v ) · ξ a (cid:18) ξ, u + v (cid:19) f ( v ) dv dξ, ∀ u ∈ R n , for all Schwartz class functions f on R n .2.3. Heisenberg Group.
One of the simple and natural example of non-abelian, non-compactgroup is the famous Heisenberg group H n , which plays an important role in several branches ofmathematics. The Heisenberg group H n is a Nilpotent Lie group whose underlying manifoldis R n +1 and the group operation is defined by( x, y, t )( x ′ , y ′ , t ′ ) = ( x + x ′ , y + y ′ , t + t ′ + 12 ( xy ′ − x ′ y )) , where ( x, y, t ) and ( x ′ , y ′ , t ′ ) are in R n × R n × R . Moreover, H n is a unimodular Lie group onwhich the Haar measure is the usual Lebesgue measure dx dy dt. The canonical basis for theLie algebra h n of H n is given by left-invariant vector fields: X j = ∂ x j − y j ∂ t , Y j = ∂ y j + x j ∂ t , j = 1 , , . . . n, and T = ∂ t , (2.2)satisfying the commutator relation [ X j , Y j ] = T, j = 1 , , . . . n. SHYAM SWARUP MONDAL AND JITENDRIYA SWAIN
The sublaplacian on the Heisenberg group is given by L = n X j =1 ( X j + Y j ) = n X j =1 (cid:18)(cid:18) ∂ x j − y j ∂ t (cid:19) + (cid:18) ∂ y j + x j ∂ t (cid:19) (cid:19) . By Stone-von Neumann theorem, the only infinite dimensional unitary irreducible representa-tions (up to unitary equivalence) are given by π λ , λ in R ∗ , where π λ is defined by π λ ( x, y, t ) f ( u ) = e iλ ( t + xy ) e i √ λyu f ( u + p | λ | x ) , ∀ u ∈ R n , ∀ f ∈ L ( R n ) and ( x, y, t ) ∈ H n . We use the convention √ λ := sgn( λ ) p | λ | = ( √ λ, λ > − p | λ | , λ < . For each λ ∈ R ∗ , the group Fourier transform of f ∈ L ( H n ) is a bounded linear map on L ( R n ) defined by ˆ f ( λ ) ≡ π λ ( f ) = Z H n f ( x, y, t ) π ∗ λ ( x, y, t ) dx dy dt. We denote B ( L ( R n )) to be the set of all bounded operators on L ( R n ). If f ∈ L ( H n ), thenˆ f ( λ ) is a Hilbert-Schmidt operator on L ( R n ) and satisfies the Plancherel formula Z R ∗ k ˆ f ( λ ) k B dµ ( λ ) = k f k L ( H n ) , where k . k B stands for the norm in the Hilbert space B of all Hilbert-Schmidt operators on L ( R n ) and dµ ( λ ) = c n | λ | n dλ where c n is a constant. Theorem 2.1.
For all Schwartz class functions on H n , the following inversion formula holds: f ( g ) = Z R ∗ tr ( π λ ( g ) ˆ f ( λ )) dµ ( λ ) , ∀ g ∈ H n . For a detailed study on the Heisenberg group we refer to Thangavelu [17].2.4.
Quantization and Pseudo-differential operator on the Heisenberg group.
Sincethe Fourier transform on H n is operator valued, the symbol of a pseudo-differential operatoron the Heisenberg group is also operator valued. We define the symbol class S mρ,δ ( H n ) for theHeisenberg group as in [12]. The symbol class S mρ,δ ( H n ) is defined by the set of all symbols σ for which the following quantities are finite: k σ k S mρ,δ ,a,b,c := sup g ∈ H n ,λ ∈ R ∗ k σ ( g, λ ) k S mρ,δ a,b,c , a, b, c ∈ N where(2.3) k σ ( g, λ ) k S mρ,δ a,b,c := sup [ α ] ≤ a [ β ] ≤ b [ γ ] ≤ c k π λ ( I − L ) ρ [ α ] − m − δ [ β ]+ γ X βg ∆ ′ α σ ( g, λ ) π λ ( I − L ) − γ k op ZEG ¨O LIMIT THEOREM ON THE HEISENBERG GROUP 7 with α = ( α , α , α ) = ( α , α . . . α n , α , α . . . α n , α ) ∈ N n × N n × N and [ α ] = | α | + | α | + 2 α . For each α we write X α = X α Y α T α , where X α = X α X α . . . X α n n and Y α = Y α Y α · · · Y α n n and ∆ ′ α := ∆ α x ∆ α y ∆ α t , where ∆ α x = ∆ α x ∆ α x · · · ∆ α n x n , ∆ α y = ∆ α y ∆ α y . . . ∆ α n y n , where the difference operators are defined (see [12] or [14]) as∆ x j ˆ f ( π λ ) = π λ ( x j f ) , ∆ y j ˆ f ( π λ ) = π λ ( y j f ) and ∆ t ˆ f ( π λ ) = π λ ( tf ) . The operator corresponding to the symbol a is denoted by Op ( a ) and is defined by Op ( a ) f ( g ) = ( T a f )( g ) = Z R ∗ tr ( π ∗ λ ( g ) a ( g, λ ) ˆ f ( λ )) dµ ( λ ) , (2.4)for all Schwartz class functions f on H n . The relation between the Weyl quantization andthe representations of the Heisenberg group enables to work with the symbols in S mρ,δ by σ ( g, λ ) = Op W ( a g,λ ) which can be seen in the following theorem. Theorem 2.2. [14]
Let ρ , δ be real numbers such that ≥ ρ ≥ δ ≥ and ( ρ, δ ) = (0 , . (1) if σ = σ ( g, λ ) is in S mρ,δ ( H n ) , then there exist a smooth function a = a ( g, λ, ξ, u ) = a g,λ ( ξ, u ) on H n × R ∗ × R n × R n such that : σ ( g, λ ) = Op W ( a g,λ ) . (2.5) It satisfies for any α, β ∈ N n , ¯ β ∈ N n +10 and ¯ α ∈ N , | ∂ αξ ∂ βu ˜ ∂ ¯ αλ,ξ,u X ¯ βg a g,λ ( ξ, u ) | ≤ C α,β, ¯ α, ¯ β | λ | ρ | α | + | β | (1 + | λ | (1 + | ξ | + | u | )) m − ρ ¯ α + δ [¯ β ] − ρ ( | α | + | β | )2 , (2.6) where the operator ˜ ∂ λ,ξ,u was defined in [14] . (2) Conversely, if a = { a ( g, λ, ξ, u ) = a g,λ ( ξ, u ) } is a smooth function on H n × R ∗ × R n × R n satisfying (2.6) for every α, β ∈ N n and ¯ α ∈ N , then there exist a unique symbol σ ∈ S mρ,δ ( H n ) such that (2.5) holds. In order to prove the Szeg¨o limit theorem we restrict ourselves to a special symbol class onthe Heisenberg group S mρ,δ, Φ ( H n ), where Φ( g, λ ) = | λ | + V ( g ), S mρ,δ, Φ ( H n ) : = n σ ( g, λ ) : | ∂ αξ ∂ βu ˜ ∂ ¯ αλ,ξ,u X ¯ βg a g,λ ( ξ, u ) | (2.7) ≤ C α,β, ¯ α, ¯ β | λ | ρ | α | + | β | (1 + Φ( g, λ )(1 + | ξ | + | u | )) m − ρ ¯ α + δ [¯ β ] − ρ ( | α | + | β | )2 o SHYAM SWARUP MONDAL AND JITENDRIYA SWAIN and σ ( g, λ ) , a g,λ are related by (2.5). Thus symbol class S mρ,δ, Φ ( H n ) is defined by the set of allsymbols σ for which the following quantities are finite: k σ k S mρ,δ, Φ ,a,b,c := sup g ∈ H n ,λ ∈ R ∗ k σ ( g, λ ) k S mρ,δ, Φ a,b,c , a, b, c ∈ N where(2.8) k σ ( g, λ ) k S mρ,δ, Φ a,b,c := sup [ α ] ≤ a [ β ] ≤ b [ γ ] ≤ c k ( π λ ( I −L )+ V ( g )) ρ [ α ] − m − δ [ β ]+ γ X βg ∆ ′ α σ ( g, λ )( π λ ( I −L )+ V ( g )) − γ k op . We take ρ = 1 and δ = 0 throughout the article and denote the symbol classes S m , ( H n )and S m , , Φ ( H n ) by S m ( H n ) and S m Φ ( H n ) respectively. Note that if σ ( g, λ ) ∈ S m Φ ( H ) then k σ k S mρ,δ ,a,b,c < ∞ for all a, b, c ∈ N . For example, for each λ ∈ R ∗ , the operators V ( g ) I, λI, | λ | H + V ( g ) I ∈ S ( H n ) . Composition of symbols.
In this section we discuss the composition formula for twopseudo-differential operators in different symbol classes in our setup and estimate the differenceof the symbol of the composition with the composition of the corresponding symbols of thepseudo-differential operators on H n . Theorem 2.3.
Let a ∈ S µ Φ ( H n ) and b ∈ S ν Φ ( H n ) . Then the composition Op ( a ) ◦ Op ( b ) is a pseudo-differential operator with symbol a H n b ∈ S µ + ν Φ ( H n ) . Moreover a H n b has thefollowing asymptotic formula: a H n b = b a + 12 √ λ X ≤ j ≤ n X j b T j a + Y j b T ′ j a (2.9) + 18 | λ | X ≤ j,k ≤ n X j X k b T j T k a + Y j Y k b T ′ j T ′ k a + X j Y k b T j T ′ k a + Y j X k b T ′ j T k a + R , where R depends only on X α for α ≥ and a b = N X s =0 s ! (cid:16) i (cid:17) s a (cid:16) ←− ∂∂ξ −→ ∂∂u − −→ ∂∂ξ ←− ∂∂u (cid:17) s b + S N (2.10) where S N ∈ S | µ | + | ν |− N with T j a = − ∂ u j a and T ′ j a = ∂ ξ j a. Proof.
The proof follows along the similar lines as in Proposition 3.5, page 57 of [2], withappropriate modifications. (cid:3)
Corollary 2.4.
Let m ≤ , a ( g, λ ) ∈ S m Φ ( H n ) and b ( g, λ ) = ( | λ | H + V ( g )+ r I ) − ∈ S − ( H n ) .Then a H n b ( g, λ ) ≈ b ( g, λ ) ◦ a ( g, λ ) + R ( g, λ ) , where k R ( g, λ ) k B ( L ( R n )) → as r → ∞ . ZEG ¨O LIMIT THEOREM ON THE HEISENBERG GROUP 9
Proof.
Using (2.7) and (2.10) we have | a b ( ξ, u ) − b ( ξ, u ) a ( ξ, u ) | ≤ ∞ X j =1 s s ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a ( ξ, u ) (cid:16) ←− ∂∂ξ −→ ∂∂u − −→ ∂∂ξ ←− ∂∂u (cid:17) s b ( ξ, u ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ r ∞ X s =1 s s ! 1 r s/ . Therefore k a b − ab k = O ( r − ). Again using the above estimate repeatedly in (2.9) and usingthe fact that n << r , we have k a H n b − ba k B ( L ( R n )) ≤ r ∞ X s =0 (cid:18) nr (cid:19) s → r → ∞ . (cid:3) Lemma 2.5.
Let m ≤ and a ∈ S m Φ ( H n ) . Then for sufficiently large N ∈ N , g, g ∈ H n , h ∈ L ( H n ) and for B ∈ B ( L ( R n )) we have (1) | tr ( π ∗ λ ( g ) a ( g, λ ) B ) | ≤ M | tr (cid:0) π λ ( I − L ) − N (cid:1) |k a ( g, λ ) kk B k . (2) (cid:12)(cid:12)(cid:12)(cid:12)Z R ∗ tr ( π ∗ λ ( g ) a ( g, λ )ˆ h ( λ )) dµ ( λ ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C k h k . Proof.
For any N ∈ N , we make use of the identity π λ ( g ) = T N ( λ − N I ) π λ ( g )(2.11)to obtain the required bound, where the vector field T = ∂∂t . For h ∈ L ( H n ), consider | tr (cid:0) π ∗ λ ( g ) a ( g, λ ) B (cid:1) | = | tr (cid:0) T N ( λ − N I ) π ∗ λ ( g ) a ( g, λ ) B (cid:1) | = | tr (cid:0) π λ ( I − L ) − N π λ ( I − L ) N T N ( λ − N I ) π ∗ λ ( g ) a ( g, λ ) B (cid:1) |≤ | tr (cid:0) π λ ( I − L ) − N (cid:1) |k π λ ( I − L ) N T N ( λ − N I ) π ∗ λ ( g ) k a ( g, λ ) kk B k . Since ( λ − N I ) π ∗ λ ( g ) as a pseudo-differential operator of order − N on H n , the operator normof π λ ( I −L ) N T N ( λ − N I ) π ∗ λ ( g ) is uniformly bounded by M (say) (see (2.8)). Using the abovefact we have | tr (cid:0) π ∗ λ ( g ) a ( g, λ ) π λ ( g ) (cid:1) | ≤ M | tr (cid:0) π λ ( I − L ) − N (cid:1) |k a ( g, λ ) kk B k . This proves part(a).
For part (b), choose sufficiently large N such that the sum X α | α | + n ) N and the integral Z R ∗ dµ ( λ )(1 + | λ | ) N converges. Since π λ ( I − L ) = 1 + | λ | H , we have tr (1 + | λ | H ) − N = X α h (1 + | λ | H ) − N Φ α , Φ α i = X α | λ | (2 | α | + n )) N ≤ X α | λ | ) N (2 | α | + n )) N = 1(1 + | λ | ) N X α | α | + n )) N . Thus by part ( a ) we have Z R ∗ (cid:12)(cid:12)(cid:12) tr (cid:16) π ∗ λ ( g ) a ( g, λ ) b h ( λ ) (cid:17)(cid:12)(cid:12)(cid:12) dµ ( λ ) ≤ C Z R ∗ k b h ( λ ) k tr (cid:0) π λ ( I − L ) − N (cid:1) dµ ( λ ) ≤ C X α | α | + n ) N Z R ∗ k b h ( λ ) k B (1 + | λ | ) − N dµ ( λ ) ≤ C k h k (cid:18)Z R ∗ (1 + | λ | ) − N dµ ( λ ) (cid:19) < ∞ . (cid:3) Corollary 2.6.
Under the assumptions of Corollary 2.4 one has (1) k Op ( a H n b ) − Op ( b ◦ a ) k B ( L ( H n )) → as r → ∞ . (2) For ℓ ∈ N , k [ Op ( b )] ℓ − Op ( b ℓ ) k B ( L ( H n )) → as r → ∞ . Proof.
Take h ∈ L ∩ L ( H n ). Since a H n b − ba ∈ S ( H n ) , applying part (1) of Lemma 2.5and Corollary 2.4 we get | Op ( a H n b ) − Op ( b ◦ a ) h ( g ) | = (cid:12)(cid:12)(cid:12)(cid:12)Z R ∗ tr (cid:16) π ∗ λ ( g )[ a H n b ( g, λ ) − ( b ◦ a )( g, λ )]ˆ h ( λ ) (cid:17) dµ ( λ ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ Cr Z R ∗ k ˆ h ( λ ) k B | λ | ) N − dµ ( λ ) . An application of Cauchy-Schwarz inequality proves part (1) by choosing sufficiently large N .To prove (2) we use mathematical induction on ℓ . For ℓ = 1, the hypothesis is trivially true.Assume that (2) holds for [ Op ( b )] ℓ − . Using the composition formula (2.9) and Corollary 2.4we have ( b ℓ − H n b )( g, λ ) ≈ b ℓ − ( g, λ ) ◦ b ( g, λ ) + R ℓ ( g, λ ) , where k R ℓ ( g, λ ) k B ( L ( R n )) → r → ∞ . This proves (2). (cid:3)
Further, using the asymptotic formula (2.9), we prove the that the ℓ -fold composition of Op ( a ) ∈ S φ ( H n ) with itself is asymptotically equal to Op ( a ℓ ) in the following lemma. ZEG ¨O LIMIT THEOREM ON THE HEISENBERG GROUP 11
Lemma 2.7.
Let A be the pseudo-differential operator on L ( H n ) with symbol a ( g, λ ) ∈ S ( H n ) . Then for any ℓ ∈ N , the symbol a ℓ ( g, λ ) of the operator A ℓ can be written as a ℓ ( g, λ ) = ( a ( g, λ )) ℓ + E l ( g, λ ) with E ℓ ( g, λ ) ∈ B ( L ( R n )) . Proof.
We prove the lemma using mathematical induction on ℓ. For ℓ = 1 our hypothe-sis holds trivially. Assume that our hypothesis is true for the symbol a ℓ − . That means a ℓ − ( g, λ ) = ( a ( g, λ )) ℓ − + E ℓ − ( g, λ ) with E ℓ − ( g, λ ) ∈ B ( L ( R n )) . But by (2.9) and (2.10)we have a ℓ ≈ ( a ℓ − a ) + T ≈ a ℓ + S + S , where S = √ λ P ≤ j ≤ n X j b T j a + Y j b T ′ j a + | λ | P ≤ j,k ≤ n X j X k b T j T k a + Y j Y k b T ′ j T ′ k a + X j Y k a T j T ′ k a + Y j X k b T ′ j T k a + R and S ( ξ, u ) = ∞ X s =1 s s ! a ℓ − ( ξ, u ) (cid:16) ←− ∂∂ξ −→ ∂∂u − −→ ∂∂ξ ←− ∂∂u (cid:17) s a ( ξ, u ) . Since (cid:12)(cid:12)(cid:12)(cid:12) ∂ s ∂ξ s ∂ s ∂u s X αg a ℓ − ( ξ, u ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C s ,s | λ | s s ((1 + Φ( g, λ ))(1 + | ξ | + | u | )) − s s , and (cid:13)(cid:13)(cid:13)(cid:13) ∂ s ∂ξ s ∂ s ∂u s T ′ j β T γk a (cid:13)(cid:13)(cid:13)(cid:13) ≤ C s ,s | λ | | β | + | γ | + s s (1 + Φ( g, λ )(1 + | ξ | + | u | )) − | β | + | γ | + s s , we have k S k ≤ C ∞ X s =0 | λ | s k (1 + | λ | H ) k s < ∞ (2.12)and k X j a ℓ − T j a k ≤ C n | λ | k (1 + | λ | H ) k ∞ X s =0 (cid:18) λ k (1 + | λ | H ) k (cid:19) s . Again k X j X k a ℓ − T j T k a k ≤ nC j | λ |k (1 + | λ | H ) k ∞ X s =0 (cid:18) λ k (1 + | λ | H ) k (cid:19) s . Therefore k S k ≤ C " ∞ X s =0 (cid:18) λ k (1 + | λ | H ) k (cid:19) s n k (1 + | λ | H ) k + n k (1 + | λ | H ) k + n k (1 + | λ | H ) k + · · · (cid:19) ≤ C n k (1 + | λ | H ) k ∞ X s =0 (cid:18) λ k (1 + | λ | H ) k (cid:19) s ∞ X l =0 (cid:18) k (1 + | λ | H ) k (cid:19) l . Consequently T is a bounded operator and k a l − a l k B ( L ( R n )) < ∞ . (cid:3) Szeg¨o limit theorem for H We need to observe the following facts before proving Theorem 1.1.
Lemma 3.1.
Let M b be the multiplication operator defined in Theorem 1.1, then trf ( π r M b π r ) = tr ( π r f ( M b ) π r ) for any f ∈ C ( R ) . Proof.
Notice that k ( I − π r ) M b π r k B = tr ( π r M b ( I − π r ) M b π r ) = tr ( π r M b π r ) − tr ( π r M b π r ) . The operator π r M b π r is an operators on L ( H n ) with kernel K ( g, g ) = X k ,k ≤ r h b e k , e k i e k ( g ) e k ( g ), for any orthonormal basis { e k } of L ( H n ).Therefore tr ( π r M b π r ) = R H n K ( g, g ) dg = X k ≤ r h b e k , e k i . Further, tr ( π r M b π r ) = R H n K ( g, g ) dg = X k ≤ r h b e k , e k i , where the operator π r M b π r M b π r is an integral opera-tor with kernel K ( g, g ) = X k ,k ,k ≤ r h be k , e k ih be k , e k i e k ( g ) e k ( g ). So tr ( π r M b π r ) = tr ( π r M b π r ) . So k ( I − π r ) M b π r k B = 0.Observe that for n ∈ N , π r M nb π r = π r M b ( π r + ( I − π r )) M b · · · M b π r = ( π r M b π r ) n +terms with a factor of ( I − π r ) M b π r . By Cauchy-Schwarz inequality, tr (terms with a factor of( I − π r ) M b ) is dominated by some constant (depending on b ) times k ( I − π r ) M b π r k B .Therefore | tr ( π r M nb π r ) − tr ( π r M b π r ) n | = 0. Thus tr f ( π r M b π r ) = tr ( π r f ( M b ) π r )for f ( x ) = x n , ∀ n ∈ N and this result can be extended to continuous functions as an applicationof the Weierstrass approximation theorem and spectral theorem. (cid:3) Lemma 3.2.
For r > define I r : L ( H n ) → L ( H n ) by I r ( φ )( g ) = Z r − r tr ( π ∗ λ ( g ) ˆ φ ( λ )) dµ ( λ ) .Then lim r →∞ tr ( π r f ( M b ) π r ) tr ( π r ) = lim r →∞ tr ( π r I r f ( M b ) π r I r ) tr ( π r I r ) . (3.1) Proof.
We know that if X is a positive trace class operator and Y is a bounded operator on L ( H n ) then | tr ( XY X ) | ≤ k Y k| tr ( X ) . Using this inequality we get (cid:12)(cid:12) tr ( π r ) − tr ( π r I r ) (cid:12)(cid:12) = (cid:12)(cid:12) tr ( π r ( I − I r )) (cid:12)(cid:12) ≤ k I − I r k| tr ( π r ) | . But for ψ ∈ L ( H n ), an application of Plancherelformula gives k ( I − I r ) ψ k = Z | λ | >r k ˆ ψ ( λ ) k B dµ ( λ ) → r → ∞ . Therefore, (cid:12)(cid:12)(cid:12)(cid:12) tr ( π r I r ) tr ( π r ) − (cid:12)(cid:12)(cid:12)(cid:12) ≤ k I − I r k → r → ∞ . (3.2)We add a suitable constant to make the operator M b positive and any f ∈ C ( R ) can bewritten as the difference of two positive functions namely the positive and the negative part ZEG ¨O LIMIT THEOREM ON THE HEISENBERG GROUP 13 of f . So without loss of generality we take f ( M b ) as a positive operator. Further, (cid:12)(cid:12) tr ( π r f ( M b ) π r ) − tr ( π r I r f ( M b ) π r I r ) (cid:12)(cid:12) = (cid:12)(cid:12) tr ( π r f ( M b ) π r ( I − I r )) − tr ( π r ( I − I r ) f ( M b ) π r I r ) (cid:12)(cid:12) ≤ (cid:12)(cid:12) tr ( π r f ( M b ) π r ) (cid:12)(cid:12) k I − I r k + (cid:12)(cid:12) tr ( π r f ( M b ) π r I r ) (cid:12)(cid:12) k I − I r k . Therefore, (cid:12)(cid:12)(cid:12)(cid:12) tr ( π r I r f ( M b ) π r I r ) tr ( π r f ( M b ) π r ) − (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:18) (cid:12)(cid:12)(cid:12)(cid:12) tr ( π r f ( M b ) π r I r ) tr ( π r f ( M b ) π r ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) k I − I r k → , (3.3)as r → ∞ . Combining (3.2) and (3.3) we get (3.1). (cid:3) Proof of theorem 1.1:
The operator π r I r f ( M b ) π r I r is an integral operator with kernel K r ( g, g ) = Z r − r tr ( π ∗ λ ( g ) I r × r f ( b ( g )) π λ ( g )) dµ ( λ ) . Therefore tr ( π r I r f ( M b ) π r I r ) = Z H n K r ( g, g ) dg = r Z r − r dµ ( λ ) Z H n f ( b ( g )) dg and tr ( π r I r ) = X i ≤ r h [ I r φ i , ˆ φ i i = X i ≤ r h ˆ φ i X [ − r ,r ] , ˆ φ i i = X i ≤ r Z r − r tr ( ˆ φ i ∗ ( λ ) ˆ φ i ( λ )) dµ ( λ ) = r Z r − r dµ ( λ ) , for any orthonormal basis { φ i } ∞ i =1 of L ( H n ) . Thuslim r →∞ tr ( π r f ( M b ) π r ) tr ( π r ) = lim r →∞ tr ( π r I r f ( M b ) π r I r ) tr ( π r I r ) = Z H n f ( b ( g )) dg. Szeg¨o limit theorem for H Consider the operators H and V as defined in Theorem 1.2. Since the operators ( H + r I ) − and ( V + r I ) − are compact for r >
0, we choose a suitable m ∈ N such that( H + r I ) − m and ( V + r I ) − m are trace class operators on L ( H n ). We observe the followingfacts before proving Theorem 1.2. Lemma 4.1.
Consider the self-adjoint operators V and H as defined in Theorem 1.2. Then(a) (cid:12)(cid:12)(cid:12)(cid:12) tr (( H + r I ) − m ) tr (( V + r I ) − m ) − (cid:12)(cid:12)(cid:12)(cid:12) → r → ∞ . (b) If B is any bounded operator on L ( H n ) , then (cid:12)(cid:12)(cid:12)(cid:12) tr ( B ( H + r I ) − m ) tr ( B ( V + r I ) − m ) − (cid:12)(cid:12)(cid:12)(cid:12) → r → ∞ . Proof.
Without loss of generality we prove the result the positive operator B by adding asuitable constant c > B + cI positive.(a) Since B and ( V + r I ) − are bounded and positive operators, we have( H + r I ) = ( V + r I ) (( V + r I ) − ( B )( V + r I ) − + 1)( V + r I ) . (4.1)Therefore ( H + r I ) − m = ( V + r I ) − m + ( V + r I ) − m ((1 + K r ) − m − V + r I ) − m , (4.2)where K r = ( V + r I ) − ( B )( V + r I ) − . Here K r is a positive operator and k ( I + K r ) − k ≤ , for any r > . Thus (cid:12)(cid:12) tr (cid:0) ( H + r I ) − m ) − tr (( V + r I ) − m (cid:1)(cid:12)(cid:12) = (cid:12)(cid:12) tr (cid:0) ( V + r I ) − m ((1 + K r ) − m − V + r I ) − m (cid:1)(cid:12)(cid:12) ≤ tr (cid:0) ( V + r I ) − m (cid:1)(cid:13)(cid:13) ((1 + K r ) − m − (cid:13)(cid:13) ≤ m (cid:13)(cid:13) K r (cid:13)(cid:13) tr (cid:0) ( V + r I ) − m (cid:1) ≤ m (cid:13)(cid:13) B (cid:13)(cid:13)(cid:13)(cid:13) ( V + r I ) − (cid:13)(cid:13) tr (cid:0) ( V + r I ) − m (cid:1) . Therefore, (cid:12)(cid:12)(cid:12)(cid:12) tr (( H + r I ) − m ) tr (( V + r I ) − m ) − (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:13)(cid:13) B (cid:13)(cid:13)(cid:13)(cid:13) ( V + r I ) − (cid:13)(cid:13) → r → ∞ . (b) Using (4.2) we have (cid:12)(cid:12) tr (cid:0) B ( H + r I ) − m ) − tr ( B ( V + r I ) − m (cid:1)(cid:12)(cid:12) = (cid:12)(cid:12) tr (cid:0) B ( V + r I ) − m ((1 + K r ) − m − V + r I ) − m (cid:1)(cid:12)(cid:12) = (cid:12)(cid:12) tr (cid:0) ( V + r I ) − m B ( V + r I ) − m ((1 + K r ) − m − (cid:1)(cid:12)(cid:12) = (cid:12)(cid:12) tr (cid:0) W r ((1 + K r ) − m − (cid:1)(cid:12)(cid:12) = (cid:12)(cid:12) tr (cid:0) W r ((1 + K r ) − m − W r (cid:1)(cid:12)(cid:12) ≤ m (cid:13)(cid:13) B (cid:13)(cid:13)(cid:13)(cid:13) ( V + r I ) − (cid:13)(cid:13) tr (cid:0) B ( V + r I ) − m (cid:1) , where W r = ( V + r I ) − m B ( V + r I ) − m is a positive and trace class operator on L ( H n ). (cid:3) Therefore, (cid:12)(cid:12)(cid:12)(cid:12) tr ( B ( H + r I ) − m ) tr ( B ( V + r I ) − m ) − (cid:12)(cid:12)(cid:12)(cid:12) ≤ k B k (cid:13)(cid:13) ( V + r I ) − (cid:13)(cid:13) → r → ∞ . Corollary 4.2.
Consider the self-adjoint operators V and H as given in Theorem 1.2. Let φ H ( r ) = tr ( π r ) and φ V ( r ) = V ol ( { g ∈ H n : V ( g ) ≤ r } ) , then we have the followingasymptotic : (1) φ V is multiplicatively continuous. ZEG ¨O LIMIT THEOREM ON THE HEISENBERG GROUP 15 (2) φ H ( r ) ≈ Cr n +1) κ as r → ∞ . (3) sup µ ≤ r [ tr ( µ + r ) − tr ( µ )] ≤ tr ( π r ) (cid:18) n +1) κ rr + O (cid:16) r (cid:17) (cid:19) , as r → ∞ . Proof. (1) The function φ V is given by φ V ( r ) = V ol ( { g ∈ H n : V ( g ) ≤ r } ) = V ol ( { g ∈ H n : | g | ≤ r k } ) ≈ r n +1) k lim r →∞ τ → φ V ( τ r ) φ V ( r ) = lim r →∞ τ → ( τ r ) n +1) κ r n +1) κ = lim τ → τ n +1) κ = 1Therefore φ V is multiplicatively continuous function. Further from part (a) of Lemma 4.1 Z ∞ r m ( u + r ) m dφ H ( u ) / Z ∞ r m ( u + r ) m dφ V ( u ) → , as r → ∞ . Now, applying Theorem 8 of Grishin-Poedintseva [3], part (3) follows directly from part (2):sup µ ≤ r [ tr ( π µ + r ) − tr ( π µ )] ≤ tr ( π r ) n + 1) κ rr + O (cid:18) r (cid:19) ! . (cid:3) Lemma 4.3.
Let V and H defined as in Theorem 1.2, then lim r →∞ tr ( B ( H + r I ) − m ) tr (( H + r I ) − m ) = lim r →∞ tr ( B ( V + r I ) − m ) tr (( V + r I ) − m ) . the above equality valid in the sense that if one of limits exist then the other also does and thelimits are same. Proof.
For each r >
0, we have (cid:18) tr ( B ( H + r I ) − m ) tr ( B ( V + r I ) − m ) (cid:19)(cid:18) tr (( H + r I ) − m ) tr (( V + r I ) − m ) (cid:19) = (cid:18) tr ( B ( H + r I ) − m ) tr (( H + r I ) − m ) (cid:19)(cid:18) tr (( B ( V + r I ) − m ) tr (( V + r I ) − m ) (cid:19) . (4.3)Since the left hand side has limit 1 (by part (b) of Lemma 4.1), the right hand side limit in(4.3) exists and equal to 1. Therefore if the numerator or the denominator in the fraction inthe right hand side has a limit in (4.3), then the other also has a limit and they both agree.Therefore, lim r →∞ tr ( B ( H + r I ) − m ) tr (( H + r I ) − m ) = lim r →∞ tr ( B ( V + r I ) − m ) tr (( V + r I ) − m ) . (cid:3) Lemma 4.4.
Define π V,r : L ( H n ) → L ( H n ) by π V,r ( h )( g ) = h ( g ) X { g ∈ H n : V ( g ) ≤ r ( g ) for all h ∈ L ( H n ) . Then (cid:12)(cid:12)(cid:12)(cid:12) tr ( π r f ( A ) π r ) tr ( π r ) − tr f ( π V,r Aπ V,r ) tr ( π V,r ) (cid:12)(cid:12)(cid:12)(cid:12) → r → ∞ . Proof.
Without loss of generality add a suitable constant to make the function f is positive.Then f ( A ) is a positive operator on L ( H n ).Let φ H ( r ) = tr ( π r ) and φ V ( r ) = tr ( π π V,r ). Set φ H ,f ( r ) = tr ( π r f ( A ) π r ) = tr ( f ( A ) π r f ( A ) ) and φ V,f ( r ) = tr ( π V,r f ( A ) π V,r ) = tr ( f ( A ) π V,r f ( A ) ) . Using thespectral theorem for both H , V and Lemma 4.3 we getlim r →∞ R ∞ φ H ,f ( u )(1+ ur ) m +1 du R ∞ φ H ( u )(1+ ur ) m +1 du = lim r →∞ φ H ,f ( r ) φ H ( r ) = lim r →∞ φ V,f ( r ) φ V ( r ) = lim r →∞ R ∞ φ V,f ( u )(1+ ur ) m +1 du R ∞ φ V ( u )(1+ ur ) m +1 du . Therefore (cid:12)(cid:12)(cid:12)(cid:12) tr ( π r f ( A ) π r ) tr ( π r ) − tr f ( π V,r Aπ V,r ) tr ( π V,r ) (cid:12)(cid:12)(cid:12)(cid:12) → r → ∞ . (cid:3) Lemma 4.5.
Let I r be defined as in Lemma 3.2, then (a) (cid:12)(cid:12)(cid:12)(cid:12) tr ( π V,r I r ) tr ( π V,r ) − (cid:12)(cid:12)(cid:12)(cid:12) → r → ∞ . (b) (cid:12)(cid:12)(cid:12)(cid:12) tr ( π r π V,r I r π r ) tr ( π V,r I r ) − (cid:12)(cid:12)(cid:12)(cid:12) → r → ∞ . Proof.
The proof of the lemma follows similarly as in Lemma 3.2. (cid:3)
Lemma 4.6. (a) Let H , A be the operators defined in Theorem 1.2 and k ∈ (0 , , then theoperator [ H , A ] is bounded on L ( H n ) . (b) Under the assumption of Theorem 1.2, we have (cid:12)(cid:12)(cid:12)(cid:12) tr f ( π r Aπ r ) tr ( π r ) − tr ( π r f ( A ) π r ) tr ( π r ) (cid:12)(cid:12)(cid:12)(cid:12) → as r → ∞ . Proof.
Since [
B, A ] is bounded on L ( H n ), the boundedness of the operator [ V, A ] will implyboundedness of the operator [ H , A ] on L ( H n ). For h ∈ L ( H n ),[ A, V ] h ( g ) = ( AV − V A ) h ( g ) = Z H n K ( g, g ) h ( g ) dg , where K ( g, g ) = (cid:0) | g | k − | g | k (cid:1) R R ∗ tr (cid:0) π ∗ λ ( g ) a ( g, λ ) π λ ( g ) (cid:1) dµ ( λ ) . By Lemma 2.5 we have | tr (cid:0) π ∗ λ ( g ) a ( g, λ ) π λ ( g ) (cid:1) | ≤ M | tr (cid:0) π λ ( I −L ) − N (cid:1) |k (1+ | λ | H ) − N π λ ( g g − ) π ∗ λ ( g g − ) kk a ( g, λ ) k .Since (1 + | λ | H ) − N π λ ( g g − ) has order − N , (2.7) gives k (1 + | λ | H ) − N π λ ( g g − ) k ≤ c (1 + V ( g g − )) N . (4.4)Using the above inequality, the identity π λ ( I − L ) = 1 + | λ | H and the fact that κ ∈ (0 , | K ( g, g ) | ≤ c N (cid:12)(cid:12) gg − (cid:12)(cid:12) κ (1 + V ( g g − )) N , ZEG ¨O LIMIT THEOREM ON THE HEISENBERG GROUP 17 and k [ A, V ] h k = Z H n (cid:12)(cid:12)(cid:12)(cid:12) Z H n K ( g, g ) h ( g ) dg (cid:12)(cid:12)(cid:12)(cid:12) dg ≤ c N Z H n (cid:18) Z H n (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12) gg − (cid:12)(cid:12) k h ( g )(1 + V ( g g − )) N (cid:12)(cid:12)(cid:12)(cid:12) dg (cid:19) d g = k| h | ∗ K k where K ( g ) = | g | k (1 + | g | ) N . Since for a sufficiently large N , K ∈ L ( H n ), an application ofMinkowski’s inequality gives k [ A, V ] h k ≤ C k K k k h k . (b) Since A is bounded self-adjoint, the spectrum of A , σ ( A ) is a compact subset of R . Sinceany continuous function can be approximated in the supremum norm by smooth functions, itis enough to if we assume that f ∈ C ( σ ( A )) . By Theorem 1.6 of Laptev-Safarov [9], we get(setting A = H , B = A, χ = 0 , ψ = f, P λ = π r ) | tr ( π r f ( A ) π r − π r f ( π r Aπ r ) π r ) |≤ k f ′′ k ∞ N r ( r ) (cid:16) k π r A k + π r k π r − r [ A, H ] k (cid:17) . Dividing both sides by tr ( π r ) and setting r = r α , α ∈ (0 ,
1) and using the boundness of A, [ A, H ] we have | tr ( π r f ( A ) π r − π r f ( π r Aπ r ) π r ) | tr ( π r ) ≤ C N r α ( r ) tr ( π r ) → r → ∞ by (3) of Corollary 4.2, where N r ( r ) = sup µ ≤ r ( tr ( π µ − π µ − r )). (cid:3) Proof of theorem 1.2 : Using Lemma 4.4, Lemma 4.5 and Lemma 4.6 we havelim r →∞ tr f ( π r Aπ r ) tr ( π r ) = lim r →∞ tr ( π r f ( A ) π r ) tr ( π r )= lim r →∞ tr ( π V,r f ( A ) π V,r ) tr ( π V,r )= lim r →∞ tr ( π V,r f ( A ) π V,r ) tr ( π V,r I r )= lim r →∞ tr ( π V,r f ( A ) π V,r ) tr ( π r π V,r I r π r ) . Let ℓ ∈ N and ψ ∈ L ( H n ). Now applying Lemma 2.7 we have π V,r A ℓ π V,r ψ ( g ) = π V,r A ℓ ψ ( g ) X V ≤ r ( g )= π V,r Z R ∗ tr ( π ∗ λ ( g ) a ℓ ( g, λ )( ψ X V ≤ r ( g )ˆ)( λ )) dµ ( λ )= Z V ( g ) ≤ r Z R ∗ tr ( π ∗ λ ( g ) a ℓ ( g, λ ) ψ ( g ) π λ ( g )) dµ ( λ ) dg = Z H n (cid:18) X V ≤ r Z R ∗ tr ( π ∗ λ ( g ) a ℓ ( g, λ ) π λ ( g )) dµ ( λ ) (cid:19) ψ ( g ) dg = Z H n K ( g, g ) ψ ( g ) dg , where the kernel of the operator π V,r A ℓ π V,r is given by K ( g, g ) = X V ≤ r Z R ∗ tr ( π ∗ λ ( g ) a ℓ ( g, λ ) π λ ( g )) dµ ( λ ) . Thus, tr ( π V,r A ℓ π V,r ) = Z H n K ( g, g )) dg (4.5) = Z H n X V ≤ r Z R ∗ tr ( π ∗ λ ( g ) a ℓ ( g, λ ) π λ ( g )) dµ ( λ ) dg = Z V ≤ r Z R ∗ tr ( a ℓ ( g, λ )) dµ ( λ ) dg = Z V ≤ r Z R ∗ tr ( a ( g, λ ) ℓ + E ℓ ( g, λ )) dµ ( λ ) dg = Z V ≤ r Z R ∗ tr ( a ( g, λ ) ℓ ) dµ ( λ ) dg + Z V ≤ r Z R ∗ tr ( E ℓ ( g, λ )) dµ ( λ ) dg. Since R V ≤ r R R ∗ tr ( E ℓ ( g, λ ) dµ ( λ ) dg = R V ≤ r R R ∗ tr ( π ∗ λ ( g ) π λ ( g ) E ℓ ( g, λ )) dµ ( λ ) dg and E ℓ ( g, λ )is a bounded operator (see Lemma 2.7), part ( a ) of Lemma 2.5 and (4.4) gives Z V ≤ r Z R ∗ tr ( E ℓ ( g, λ )) dµ ( λ ) dg < ∞ . (4.6)Since π r I r = I r π r , π r π V,r I r π r ψ ( g ) = Z H n K ( g, g ) ψ ( g ) dg , where K ( g, g ) = X V ≤ r ( g ) Z r − r tr ( π ∗ λ ( g ) I r × r f ( b ( g )) π λ ( g )) dµ ( λ )Therefore tr ( π r π V,r I r π r ) = Z H n K ( g, g ) dg = Z V ≤ r Z r − r tr ( π ∗ λ ( g ) I r × r π λ ( g )) dµ ( λ ) dg = Z V ≤ r Z r − r tr ( I r × r ) dµ ( λ ) dg = r Z r − r dµ ( λ ) Z V ≤ r dg. Therefore tr ( π V,r A ℓ π V,r ) tr ( π r π V,r I r π r ) = R V ≤ r R R ∗ tr ( a ( g, λ ) ℓ ) dµ ( λ ) dgr R r − r dµ ( λ ) R V ≤ r dg + E ( r ) , where | E ( r ) | ≤ R V ≤ r R R ∗ tr ( E ℓ ( g, λ )) dµ ( λ ) dgr R r − r dµ ( λ ) → r → ∞ . This result can be extended to continuous functions as an application of theWeierstrass approximation theorem and spectral theorem. This proves our Theorem 1.2.
ZEG ¨O LIMIT THEOREM ON THE HEISENBERG GROUP 19 Szeg¨o limit theorem for H In this section we prove Szeg¨o limit theorem for H under certain assumptions on thesymbol a ( g, λ ) as in [20]. We assume thatlim E →∞ ¯ a ( E ) = a, (5.1)where ¯ a ( E ) = 1 S ( E ) Z G E a g,λ ( ξ, x ) dξ dx dg dλ and S ( E ) = Z G E dξ dx dg dλ, with G E = { ( g, λ, ξ, x ) ∈ H n × R ∗ × R n × R n : | λ | ( | ξ | + | x | ) + V ( g ) = E, V ( g ) ≤ E , | λ | ≥ E − } . Proposition.
Let G E = { ( g, λ, ξ, x ) ∈ H n × R ∗ × R n × R n : | λ | ( | ξ | + | x | ) + V ( g ) ≤ E, V ( g ) ≤ E , | λ | ≥ E − } . Then volume of G r = V ol ( G r ) ≈ Cr n +1 κ + n − as r → ∞ . Proof. : Evaluating the volume of G E using the homogeneous norm on H n by changing tospherical coordinate we have V ol ( G r ) = Z G r dξ dx dg dµ ( λ ) grows asymptotically similar tothat of Cr n +1 κ + n − . (cid:3) Lemma 5.1.
Let φ ( r ) = tr ( π r ) and ψ ( r ) = tr ( π r Aπ r ) . Then under the assumption (5.1)we have Φ( u ) = Z ∞ φ ( r )( r + u ) m +1 dr = Z ∞ Z G E dξ dx dg dµ ( λ )( E + u ) m +1 dE + E ( u ) and Ψ( u ) = Z ∞ ψ ( r )( r + u ) m +1 dr = Z ∞ Z G E a g,λ ( ξ, x ) dξ dx dg dµ ( λ )( E + u ) m +1 dE + E ( u ) , where E ( u ) = m Z H n × R ∗ tr ( E ( g, λ, u )) dg dµ ( λ ) and E ( u ) = m Z H n × R ∗ tr ( E ( g, λ, u )) dg dµ ( λ ) with | E i ( u ) | → as u → ∞ , i = 1 , . Proof.
The operator H has discrete spectrum of eigenvalues 0 ≤ c ≤ c · · · ∞ . Let { ψ j } ∞ j =1 be the complete set of eigenfunctions on corresponding to the eigenvalues { c j } on L ( H n ).Then φ ( r ) = tr ( π r ) = X c j ≤ r h ψ j , ψ j i and φ ′ ( r ) = ∞ X j =1 h ψ j , ψ j i δ ( r − c j ). NowΦ( u ) = Z ∞ φ ( r )( r + u ) m +1 dr = m ∞ X j =1 h ψ j , ψ j i Z ∞ δ ( r − c j )( r + u ) m dr = m ∞ X j =1 h ψ j , ψ j i c j + u ) m . Now applying Corollary 2.4 on the symbols of H we getΦ( u ) = m tr ( H + u ) − m ! = m Z H n × R ∗ tr (cid:0) | λ | H + V ( g ) + u (cid:1) − m + E ( g, λ, u ) ! dg dµ ( λ )= m Z H n × R ∗ Z R n × R n (cid:16)(cid:0) | λ | ( | ξ | + | x | ) + V ( g ) + u (cid:1) − m (cid:17) dg dµ ( λ ) + E ( u )= Z ∞ Z G E dξ dx dg dµ ( λ )( E + u ) m +1 dE + E ( u ) , where E ( u ) = m Z H n × R ∗ tr ( E ( g, λ, u )) dgdµ ( λ ). Similarly taking ψ ( r ) = tr ( π r Aπ r ), wehave Ψ( u ) = Z ∞ Z G E a g,λ ( ξ, x ) dξ dx dg dµ ( λ )( E + u ) m +1 dE + E ( u )where E ( u ) = m Z H n × R ∗ tr ( E ( g, λ, u )) dgdµ ( λ ). Observe that E ( g, λ, u ) is a symbol on H n . Therefore by Lemma 2.5 and (4.4) we have E ( u ) → u → ∞ . Similarly E ( u ) → u → ∞ . (cid:3) In order to prove the Szeg¨o limit theorem for the Schr¨odinger operator H , we need toestimate the asymptotic growth of the measures tr ( π r Aπ r ) and tr ( π r ). Therefore we applyKeldysh Tauberian Theorem to compare the measures and obtain the following Corollary. Corollary 5.2.
Consider the self-adjoint operator π r and v ( r ) as given in Theorem 1.3and Proposition 5.1 respectively. Let φ ( r ) = tr ( π r ) , ψ ( r ) = tr ( π r Aπ r ) then we have thefollowing asymptotic : (1) v ( r ) ≈ Cr n +1 κ + n − as r → ∞ . (2) v is multiplicatively continuous. (3) tr ( π r ) ≈ Cr n +1 κ + n − as r → ∞ . (4) sup µ ≤ r [ tr ( µ + r ) − tr ( µ )] ≤ tr ( π r ) ( n + 1) k rr + O (cid:18) r (cid:19) ! , as r → ∞ . (5) ψ is multiplicatively continuous. Proof.
Clearly (1) directly follows from Proposition 5.1. Nowlim r →∞ τ → v ( τ r ) v ( r ) = lim r →∞ τ → ( τ r ) n +1 κ + n − r n +1 κ + n − = lim τ → τ n +1 κ + n − = 1 ZEG ¨O LIMIT THEOREM ON THE HEISENBERG GROUP 21
Therefore v is multiplicatively continuous function. We choose sufficiently large m such thatthe operator ( H + uI ) − m is in trace class. Therefore by Lemma 5.1 and Theorem 8 of Grishin-Poedintseva [3], we get φ ( r ) /v ( r ) → r → ∞ . This proves (3). Using the asymptotic in(3), it is easy to check thatsup µ ≤ r [ tr ( µ + r ) − tr ( µ )] ≤ tr ( r ) ( n + 1) k rr + O (cid:18) r (cid:19) ! . To prove (5), notice that if ϕ and χ are two distribution functions satisfying lim r →∞ ϕ ( r ) χ ( r ) = 1then ϕ is multiplicatively continuous whenever χ is. Therefore ψ is also a multiplicativelycontinuous function. (cid:3) Theorem 5.3.
Under the assumption (5.1) we have lim r →∞ tr ( π r Aπ r ) tr ( π r ) = lim r →∞ R G r a g,λ ( ξ, x ) dξ dx dg dµ ( λ ) R G r dξ dx dg dµ ( λ ) . Proof.
The proof follows directly by Lemma 5.1, as all the requirements (by our the assumption(5.1) on the symbol a ( g, λ )) of Theorem 8 of Grishin-Poedintseva [3] are satisfied. (cid:3) Corollary 5.4.
Let P ( r ) be a polynomial in R . Then lim r →∞ tr ( π r P ( A ) π r ) tr ( π r ) = lim r →∞ R G r P ( a g,λ ( ξ, x )) dξ dx dg dµ ( λ ) R G r dξ dx dg dµ ( λ ) Proof.
The operator P ( A ) has symbol P ( a ( g, λ )) + E − ( g, λ ) where E − ( g, λ ) ∈ B ( L ( R n )) . Proceeding exactly as in (4.6) we get Z H n × R ∗ tr ( E − ( g, λ )) dg dµ ( λ ) < ∞ . Therefore we getthe corollary as r → ∞ . (cid:3) Lemma 5.5.
Let H , A be the operators defined in Theorem 1.3 and κ ∈ (0 , , then(a) the operator [ H , A ] is bounded on L ( H n ) . (b) Under the assumptions of Theorem 1.3, we have (cid:12)(cid:12)(cid:12)(cid:12) tr f ( π r Aπ r ) tr ( π r ) − tr ( π r f ( A ) π r ) tr ( π r ) (cid:12)(cid:12)(cid:12)(cid:12) → as r → ∞ . Proof.
For κ ∈ (0 ,
1) the operator [
V, A ] is bounded (see part ( a ) of Lemma 4.6). Theboundedness of the operator [ L , A ] will imply boundedness of the operator [ H , A ] on L ( H n ). Using the identity (2.11) we get A L h ( g ) = Z R ∗ tr (cid:16) π ∗ λ ( g ) a ( g, λ ) c L h ( λ ) (cid:17) dµ ( λ )= Z R ∗ tr (cid:18) π λ ( I − L ) N T N λ − N π ∗ λ ( g ) a ( g, λ )ˆ h ( λ ) | λ | Hπ λ ( I − L ) − N (cid:19) dµ ( λ ) ≤ C (1 + V ( g )) − N +2 Z R ∗ k ˆ h ( λ ) k B | λ | ) N − dµ ( λ ) . For sufficiently large N , an application of Cauchy-Schwarz inequality gives k A L h k ≤ M k h k . Similarly, L Ah ( g ) = Z R ∗ tr (cid:16) π ∗ λ ( g ) [ L Ah ( λ ) (cid:17) dµ ( λ )= Z R ∗ tr (cid:16) π ∗ λ ( g ) | λ | c Ah ( λ ) H (cid:17) dµ ( λ )= Z R ∗ Z H n Ah ( g ) tr (cid:18) π ∗ λ ( g ) π λ ( g ) | λ | H (cid:19) dµ ( λ ) dg = Z R ∗ Z H n (cid:18)Z R ∗ tr (cid:16) π ∗ λ ( g ) a ( g , λ ) b h ( λ ) (cid:17) dµ ( λ ) (cid:19) tr (cid:18) π ∗ λ ( g ) π λ ( g ) | λ | H (cid:19) dµ ( λ ) dg By Lemma 2.5 we get, Z R ∗ (cid:12)(cid:12)(cid:12) tr (cid:16) π ∗ λ ( g ) a ( g , λ ) b h ( λ ) (cid:17)(cid:12)(cid:12)(cid:12) dµ ( λ ) ≤ C k h k . Further, choose suf-ficiently large N we get Z R ∗ Z H n (cid:12)(cid:12)(cid:12)(cid:12) tr (cid:18) π ∗ λ ( g ) π λ ( g ) | λ | H (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) dµ ( λ ) dg ≤ C (1+ V ( g )) − N +4 Z R ∗ Z H n (1 + V ( g )) − N +2 (1 + | λ | ) N − dµ ( λ ) dg . Therefore k [ A, H ] h k ≤ M k h k , which completes part (a). Part (b) follows similarly as inthe proof of part (b) of Lemma 4.6. (cid:3) Corollary 5.6.
Theorem 1.1, 1.2 and 1.3 also holds under the compact perturbation of thepseudo-differential operator A .Proof. To prove the above result, enough to show lim r →∞ tr ( π r A n π r ) tr ( π r ) = lim r →∞ tr ( π r ( A + B ) n π r ) tr ( π r )for any compact operator B on L ( H n ). Notice that ( A + B ) n = A n + terms with factor A p B n − p or B p A n − p where p ∈ { , , · · · , n } . Since the class of compact operators form atwo sided ideal of the class of bounded operators, ( A + B ) n = A n + a compact operator.We are done if we can prove that for a compact operator T , lim r →∞ tr ( π r T π r ) tr ( π r ) = 0 . SinceT is a compact operator, for given ǫ > T k such that k T k − T k → k → ∞ . Then (cid:12)(cid:12)(cid:12)(cid:12) tr ( π r T π r ) − tr ( π r T k π r ) tr ( π r ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ k T − T k k → k → ∞ . Therefore for given ǫ > N ∈ N such that (cid:12)(cid:12)(cid:12)(cid:12) tr ( π r T π r ) − tr ( π r T k π r ) tr ( π r ) (cid:12)(cid:12)(cid:12)(cid:12) < ǫ k ≥ N . Further, (cid:12)(cid:12)(cid:12)(cid:12) tr ( π r T N π r ) tr ( π r ) (cid:12)(cid:12)(cid:12)(cid:12) → r → ∞ i.e, for given ǫ > , ∃ N ∈ N such that (cid:12)(cid:12)(cid:12)(cid:12) tr ( π r T N π r ) tr ( π r ) (cid:12)(cid:12)(cid:12)(cid:12) < ǫ , ∀ r > N . Thus (cid:12)(cid:12)(cid:12)(cid:12) tr ( π r T π r ) tr ( π r ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) tr ( π r T N π r ) tr ( π r ) (cid:12)(cid:12)(cid:12)(cid:12) + k T − T N k < ǫ ∀ r ≥ N . (cid:3) ZEG ¨O LIMIT THEOREM ON THE HEISENBERG GROUP 23
Remark 5.7. (1) The result in Theorem 1.1 is valid if the operator H is replaced by H ′ : B + V where B is a bounded operator on L ( H n ) for all κ > κ in (0,1) is required to show that the boundedness of the commutator[ V, A ] (see the proof of ( a ) in Lemma 4.6). However if we proceed along the similarlines of [20] then the boundedness of commutator [ V, A ] can be proved for κ > . Acknowledgments
The first author wishes to thank the Ministry of Human Resource Development, India forthe research fellowship and Indian Institute of Technology Guwahati, India for the supportprovided during the period of this work.6.
Appendix
We collect few definitions and theorems of Grishin-Poedintseva [3], that we use in our paperfor the reader’s convenience.
Definition 6.1.
Let φ be a positive function on the half line [0 , ∞ ) . Let S = { α : ∃ M, R with φ ( tr ) ≤ M t α φ ( r ) , for all t ≥ , r ≥ R } and G = { β : ∃ M, R with φ ( tr ) ≥ M t β φ ( r ) , for all t ≥ , r ≥ R } Then the numbers α ( φ ) := inf S and β ( φ ) := sup G are called the upper and lower Matu-shevskaya index of φ respectively. Theorem 6.2. ( [3] ,Theorem 2)Let m > − . Assume that ϕ is positive measurable function on [0 , ∞ ) that does not vanishidentically in any neighborhood of infinity. Let Φ( r ) = Z ∞ ϕ ( rt )(1 + t ) m +1 dt be finite. Then thefunctions ϕ and Φ have same growth at infinity if and only if β ( ϕ ) > − and α ( ϕ ) < m . Definition 6.3.
A function ϕ is said to be multiplicatively continuous at infinity if it satisfies lim r →∞ τ → ϕ ( τ r ) ϕ ( r ) = 1 . Theorem 6.4. ( [3] ,Theorem 8) Let ϕ and ψ be positive functions on [0 , ∞ ) satisfying thefollowing conditions: (1) the functions ϕ and ψ do not vanish identically in any neighborhood of infinity; (2) the function ϕ is multiplicatively continuous at infinity and β ( ϕ ) > − ; (3) the function ψ is increasing; (4) at least one of the inequalities α ( ϕ ) < m and α ( ψ ) < m holds, where m > − ; (5) the functions Φ( r ) = Z ∞ ϕ ( ru )(1 + u ) m +1 du and Ψ( r ) = Z ∞ ψ ( ru )(1 + u ) m +1 du are finite and lim r →∞ Ψ( r )Φ( r ) = 1 then lim r →∞ ψ ( r ) ϕ ( r ) = 1 . References B. Simon , Szeg¨o’s theorem and its descendants: Spectral theory for L perturbations of orthogonal poly-nomials, Princeton University Press, Princeton, N. J., 2011.2. H. Bahouri , C. Kammerer and
I. Gallagher , Phase-space analysis and pseudodifferential calculus onthe Heisenberg group,
Ast´erisque , (2010).3. A. F. Grishin and
I. V. Poedintseva , Towards the Keldysh Tauberian theorem,
J. Math. Sci. , (4),(2006), 2272–2287.4. U. Grenander and
G. Szeg¨o , Toeplitz forms and their applications, Chelsea Publishing Co., New York,1984.5.
V. Guillemin , Some classical theorems in spectral theory revisited, “Seminar on singularity of solutionsof differential equations,” Princeton University Press, Princeton, 1979, 219–259.6.
L. H¨ormander , The analysis of linear partial differential operators, Vol. IV, Springer-Verlag, Berlin,Heidelberg, New York, Tokyo, 1984.7.
A. J. E. M. Janssen and
S. Zelditch , Szeg¨o limit theorems for the harmonic oscillator,
Trans. Amer.Math. Soc. (2), (1983), 563–587.8.
M. V. Keldysh,
On a Tauberian theorem,
Trudy Mat. Inst. Steklov. , (1951), 77–86.9. A. Laptev, and
Yu. Safarov,
Szeg¨o type limit theorems,
J. Func. Anal. , (1996), 544–559.10.
A. Laptev and
Yu. Safarov,
Error estimate in the generalized Szeg¨o theorem,
Equations aux deriveesparielles , XV-1–XV-7, Saint-Jean-De-Monts, 1991.11.
D. Robert , Remarks on the paper of S. Zelditch: “Szeg¨o limit theorems in quantum mechanics,”
J. Funct.Anal. (1983), 304–308.12. M. Ruzhansky and
V. Fischer , A pseudo-differential calculus on the Heisenberg group,
C. R. Acad.Paris, Ser.I , (2014), 197–204.13. M. Ruzhansky and
V. Turunen , Quantization of pseudo-differential operators on the torus,
J. FourierAnal. Appl.
M. Ruzhansky and
V. Turunen , Quantisation of Nilpotent Lie Groups, vol. , Birkh¨auser, Springer,Progress in Mathematics (2016).15.
J. Swain and
M. Krishna , Szeg¨o limit theorem on the lattice,
J. Pseudo-Differ. Oper. Appl. ,https.//doi.org/10.1007/s11868-018-0266-8.16.
G. Szeg¨o , On certain Hermitian forms associated with the Fourier series of a positive function,
Comm.Sem. Math. Univ. Lund. , (1952), 228–238.17.
S. Thangavelu , Harmonic Analysis on the Heisenberg Group, 2nd edn. vol. , Springer, New York,1998.18.
D. Widder , The Laplace transform, Princeton Mathematical Series, v. 6, Princeton University Press,Princeton, N. J., 1941.19.
H. Widom , Eigenvalue distribution theorems for certain homogeneous space,
J. Funct. Anal. , (1979),139–147.20. S. Zelditch , Szeg¨o limit theorems in quantum mechanics,
J. Funct. Anal. (1), (1983), 67–80. Department of Mathematics, Indian Institute of Technology Guwahati, Guwahati 781039, In-dia.
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