Semiclassical Density of States for the Quantum Asymmetric Top
aa r X i v : . [ m a t h - ph ] J un Semiclassical Density of States for the QuantumAsymmetric Top
Alfonso Agnew and Alain Bourget ∗ Department of MathematicsCalifornia State University, Fullerton
Abstract
In the quantization of a rotating rigid body, a top, one is concernedwith the Hamiltonian operator L α = α L x + α L y + α L z , where α <α < α . An explicit formula is known for the eigenvalues of L α in thecase of the spherical top ( α = α = α ) and symmetrical top ( α = α = α ) [LL]. However, for the asymmetrical top, no such explicit expressionexists, and the study of the spectrum is much more complex. In thispaper, we compute the semiclassical density of states for the eigenvaluesof the family of operators L α = α L x + α L y + α L z for any α < α < α . Let S ⊆ R be the 2-sphere and let − ∆ S be the constant curvature spheri-cal Laplacian on S . It is well known that the spectrum of − ∆ S consists ofeigenvalues λ given by λ k = k ( k + 1) , k = 0 , , , .... Moreover, the eigenspace corresponding to λ k is of dimension 2 k + 1 and abasis of eigenfunctions is obtained by taking the standard spherical harmonicsof degree k , i.e. Y mk ( θ, φ ) = P mk (cos θ ) e imφ , | m | ≤ k, where P mk is the associated Legendre function of the first kind. For a moredetailed treatment of the spectral theory of ∆ S , we refer the reader to [Fo].From the fact that the eigenvalues λ k = k ( k + 1) of − ∆ S are of multiplicity2 k + 1, it is easy to see that spectrum of − ∆ S has clustering. A nice way toillustrate this fact is to observe that for any Schwartz function ϕ on R , k + 1 k X j = − k ϕ (cid:18) √ λ k k (cid:19) = φ (1) + O (cid:18) k (cid:19) , (1.1) ∗ Mailing address : Department of Mathematics, California State University (Fullerton),McCarthy Hall 154, Fullerton CA 92834 (US). k → ∞ (see [M]). Expressions like those appearingon the RHS of (1.1) are often referred to as a density of states (DOS) (seee.g.[T1]). Together with the mean level spacings and the pairs correlation, theDOS represents a useful quantity to measure the spread of the spectrum.In this paper we are interested in computing the DOS for √− L α , associatedwith the quantum asymmetric top with Hamiltonian L α , where L α is given by L α := ( α L x + α L y + α L z ) , and where L x = − i ( y∂ z − z∂ y ) ,L y = − i ( z∂ x − x∂ z ) ,L z = − i ( x∂ y − y∂ x ) . Here, we assume that α = ( α , α , α ) ∈ Λ , whereΛ := (cid:26) α ∈ R : 0 < α < α < α (cid:27) is the positive Weyl chamber. It is well known that − ∆ S and − L α are com-muting, self-adjoint, elliptic operators on L ( S ) and therefore possess a Hilbertbasis of joint eigenfunctions – the aforementioned spherical harmonics Y km [BT].Moreover, it is easy to verify that their principal symbols are linearly indepen-dent in T ∗ ( S ). For these reasons, we say that ∆ S and L α form a quantumintegrable system on S .An explicit formula is known for the eigenvalues of L α in the case of thespherical top ( α = α = α ) and symmetrical top ( α = α = α ) [LL].Although no such explicit formula exists for the eigenvalues of the asymmetri-cal top ( α = α = α ), the spectrum was recently characterized in terms ofparameters associated with the Lam´e equation (cf. proposition 2.2 in [T2]).For such a system, it is customary to compute the DOS of their joint spec-trum (see e.g. [Ch, Co]). Here, we are simply concerned with the density ofstates measures associated to the operators √− L α . In the following, we denoteby E k the eigenspace of ∆ S consisting of spherical harmonics of degree k , i.e. E k = Span { Y km : m = − k, − k + 1 , ..., k } , and by P k the projection onto E k . Wedefine the DOS measure associated to the operators L α by dρ DS ( x ; k, α ) := 12 k + 1 X λ ∈ σ ( √− P k L α ) δ (cid:18) x − λk (cid:19) (1.2)where σ ( √− P k L α ) denotes the spectrum of √− P k L α . Clearly, σ ( √− P k L α )consists of the eigenvalues p λ km , m ≤ | k | , of √− L α associated to the sphericalharmonics of degree k . Our purpose here is to compute the density of states forthe measure dρ DS ( x, k ; α ) in the semi-classical regime k → ∞ . For any given α ∈ Λ , let g be the function defined on the rectangle [0 , π ] × [0 , π/
2] by g ( ξ, θ ; α ) = ( α − α ) (cid:0) β cos ξ + ( β −
1) sin θ (cid:1) sin θ + α . where β = α − α α − α . Finally, let g + ( ξ, θ ; α ) = max { , g ( ξ, θ ; α ) } . Theorem 1.1.
Let g + be defined as above. Then, we have thatw- lim k →∞ dρ DS ( x, k ; α ) = 1 π Z π Z π/ F ( x ; θ, ξ, α ) cos θ dξdθ where F is a convex combination of delta functions given by F ( x ; θ, ξ, α ) = 14 δ (cid:18) x − p g + ( ξ, θ ; α ) (cid:19) + 34 δ (cid:18) x − p g + ( ξ, θ ; α ) (cid:19) . The weak limit is taken with respect to C c ( R + ) . The proof of Theorem 1.1 is given in the third section of the paper. In thesecond section, we show how one can separate the variables for the eigenvalueproblem − L α ψ = λψ and its connection to the Lam´e equation. In particular,we will show how the spectrum of the operators − L α can be explicitly computedthrough the Lam´e equation. As we mentioned earlier, − ∆ S and − L α are commuting, self-adjoint, ellipticoperators on L ( S ), hence they possess a Hilbert basis of joint eigenfunctionsthat form a class of spherical harmonics. Rather than working with the stan-dard spherical harmonics Y mk , we introduce a more suitable class of sphericalharmonics for our purpose, the so-called Lam´e harmonics [BT, WW].In terms of the Euclidean coordinates ( x, y, z ) ∈ R , the Lam´e harmonics ofdegree k are written as ψ ( x, y, z ) = x γ y γ z γ ( k −| γ | ) Y j =0 (cid:18) x θ j − α + y θ j − α + z θ j − α (cid:19) (2.1)where γ i ∈ { , } and | γ | = γ + γ + γ ; the value of | γ | is chosen so that k − γ is even. The values of the parameters θ j are determined by the condition∆ R ψ = 0. A simple computation shows that the θ j ’s must satisfy Niven’sequation X j =0 γ j θ i − α j + X j = i θ i − θ j = 0 , ( i = 1 , ...,
12 ( k − | γ | )) . Based on Whittaker-Watson [WW] terminology, we say that ψ is of the first,second, third or fourth species if | γ | = 0, | γ | = 1, | γ | = 2 or | γ | = 3 respectively.Note that there is no Lam´e harmonics of the second and fourth species for k even,whereas for k odd, there is none of the first and third species. We will see lateron that there exists respectively k/ k + 1) /
2, 3 k/ k − / k , there exist 2 k + 1 linearly independentLam´e harmonics, hence they form a basis for the space of spherical harmonics. In order to describe the Lam´e harmonics in greater detail, it is useful to introducea different system of coordinates on S , namely the sphero-conal coordinates[Sp, Vo]. We denote these by ( u , u ). They are defined for any given positivereal constants α < α < α by the zeros of the rational function R ( u ) = x u − α + y u − α + z u − α where ( x, y, z ) ∈ R . From the graph of R ( u ), it is easy to see that α < u <α < u < α .Figure 1: The graph of R ( u ) for fixed values of x, y, z and α i . The α i correspond tothe vertical asymptotes. The intersections with the u -axis are the two roots of R ( u )corresponding to the values of u i . The equation R ( u ) = 0 is invariant under rescaling ( x, y, z ) ( tx, ty, tz ),so the coordinates ( u , u ) are indeed coordinates on S under the assumption x + y + z = 1. They take their name from the fact that they can be obtainedby the intersection of the unit sphere with confocal cones.The relations between the sphero-conal and Euclidean coordinates are givenby x = ( u − α )( u − α )( α − α )( α − α ) ,y = ( u − α )( u − α )( α − α )( α − α ) ,z = ( u − α )( u − α )( α − α )( α − α ) . In particular, ( u , u ) form an orthogonal system of coordinates on S . Thiscan easily be seen by considering the vectors ~r i = ( ∂ u i x, ∂ u i y, ∂ u i z ) for which ~r · ~r = x ( u − α )( u − α ) + y ( u − α )( u − α ) + z ( u − α )( u − α )= R ( u ) − R ( u ) u − u = 0 . The great advantage of sphero-conal coordinates over other coordinate systemson S is that they allow us to simultaneously separate variables in both ofthe spectral problems for − ∆ S and − L α (see [Sp]). For example, in thesecoordinates, the Laplace equation − ∆ S ψ = k ( k + 1) ψ takes the form4 u − u X i =1 ( − i (cid:20)p A ( u i ) ∂∂u i (cid:18)p A ( u i ) ∂ψ∂u i (cid:19)(cid:21) = k ( k + 1) ψ (2.2)where A ( u i ) = ( u i − α )( u i − α )( u i − α ). One can then separate the variablesand write ψ ( u , u ) = ψ ( u ) ψ ( u ). Denoting the separation constant by − λ ,it follows directly from (2.2) that both ψ and ψ are solutions of the same Lam´eequation A ( x ) ψ ′′ i ( x ) + 12 A ′ ( x ) ψ ′ i ( x ) = 14 ( k ( k + 1) x − λ ) ψ i ( x ) ( i = 1 , . (2.3)From the general theory of Lam´e equation [WW], it is well known that thesolutions of (2.3) are given by the Lam´e functions ψ ( x ) = ψ ( x ) = | x − α | γ / | x − α | γ / | x − α | γ / φ ( x ) (2.4)where φ is a polynomial of degree ( k − | γ | ) / γ chosen as above. Conse-quently, the joint eigenfunctions of − ∆ S and − L α are given by ψ ( u , u ) = Y j =1 | u j − α | γ / | u j − α | γ / | u j − α | γ / φ ( u j ) (2.5)Note that, up to a constant depending only on the α ’s and the solutions θ j ’s ofthe Niven’s equations, (2.5) are the Lam´e harmonics (2.1) expressed in sphero-conal coordinates.Based on these observations, we can now compute the eigenvalues of − L α .Let E be such an eigenvalue; we will show that E = λ , the separation constantobtained previously. First, we use the fact that − ( L x + L y + L z ) ψ = − ∆ S ψ = k ( k + 1) ψ to deduce that( α − α ) L x ψ + ( α − α ) L z ψ = ( α k ( k + 1) − E ) ψ. In terms of sphero-conal coordinates, we can rewrite last equation as4 u − u (cid:20) ( α + u ) p A ( u ) ∂∂u (cid:18)p A ( u ) ∂ψ∂u (cid:19) − ( α + u ) p A ( u ) ∂∂u (cid:18)p A ( u ) ∂ψ∂u (cid:19)(cid:21) = ( α k ( k + 1) − E ) ψ. Upon separating the variables, ψ ( u , u ) = ψ ( u ) ψ ( u ), we obtain A ( u i ) ψ ′′ i ( u i ) + 12 A ′ ( u i ) ψ ′ i ( u i ) = 14 ( µu i − E ) ψ i ( u i ) ( i = 1 , . (2.6)By comparison of (2.6) with (2.3), we conclude that µ = k ( k + 1) and E = λ as desired. All that remains to prove is that we get all the possible eigenvaluesof − L α in this way. This is a consequence of the following result due to Stieltjesand Sz¨ego (see [Sz], § Theorem 2.1.
Let ρ , ρ , ρ be any three real positive numbers and let a , a , a be any three real distinct numbers. There exist exactly m +1 distinct real numbers ν for which the generalized Lam´e equation A ( x ) y ′′ ( x ) + X j =0 ρ j Y i = j ( x − a i ) y ′ ( x ) = ( m ( m + 1 + | ρ | ) x − ν ) y ( x ) (2.7) has a polynomial solution y of degree m . Moreover, the m + 1 polynomial solu-tions obtained in this way are linearly independent. Replacing the expression of the Lam´e function ψ i given in (2.4) into (2.3),one can easily verify that the polynomial φ of degree ( k − | γ | ) / A ( x ) φ ′′ ( x ) + X j =0 (cid:18) γ j + 12 (cid:19) Y l = j ( x − α l ) φ ′ ( x )= 14 (cid:18) ( k − | γ | )( k + | γ | + 1) x − λ + D ( α, γ ) (cid:19) φ ( x ) , (2.8)where D ( α, γ ) = ( α + α ) γ + ( α + α ) γ + ( α + α ) γ + 2 γ γ α + 2 γ γ α +2 γ γ α . The values taken by ν = λ − D ( α, γ ) in terms of the different valuesof γ are given in the table below.species γ , γ , γ ν γ = γ = γ = 0 λγ = 1 , γ = γ = 0 λ − α − α γ = 1 , γ = γ = 0 λ − α − α γ = 1 , γ = γ = 0 λ − α − α γ = 0 , γ = γ = 1 λ − α − α − α γ = 0 , γ = γ = 1 λ − α − α − α γ = 0 , γ = γ = 1 λ − α − α − α γ = γ = γ = 1 λ − α + α + α )Table 1: The values taken by ν By Stieltjes’ result with ρ i = γ i + 1 /
2, we deduce that there are exactly( k − | γ | ) / ν for which (2.8) has a polynomial solution φ ofdegree ( k − | γ | ) /
2. In particular, the number of Lam´e harmonics of degree k and of specie 1 is k/ k + 1) /
2, of specie 3 is 3 k/ k − /
2. It follows that for any k ∈ N , there exist 2 k + 1 linearlyindependent Lam´e harmonics, so they form a Hilbert basis of L ( S ).Furthermore, for each k ∈ N , we also obtain 2 k + 1 values of ν (multiplicityincluded) to which correspond by Table 1, 2 k +1 values of λ . In other words, theeigenvalues of the linearly independent Lam´e harmonics of degree k are exactlygiven by the 2 k + 1 values of λ . Therefore, we have shown the first part of thefollowing theorem. Theorem 2.2.
Let α = ( α , α , α ) ∈ Λ , then the spectrum of the operator − L α is given by all numbers λ appearing on the RHS of the Lam´e equation (2.3) . Moreover, the λ ’s corresponding to the Lam´e harmonics of degree k liewithin the interval ( α ( k − k + 1) , α k ( k + 4) + 4 | α | ) . The second part is an immediate consequence of a result due to Van Vleck[Va] where he proves that all numbers ν corresponding to the polynomial solu-tions of degree m of the generalized Lam´e equation (2.7) lie inside the interval( α m ( m + 1 + | ρ | ) , α m ( m + 1 + | ρ | )). It follows from this and (2.8) that theeigenvalues λ lie inside the intervalmin γ { α ( k −| γ | )( k + | γ | +1)+ D ( α, γ ) } ≤ λ ≤ max γ { α ( k −| γ | )( k + | γ | +1)+ D ( α, γ ) } . Since γ i ∈ { , } , it is then easy to see thatmin γ { α ( k − | γ | )( k + | γ | + 1) + D ( α, γ ) } ≥ α ( k − k + 1)and max γ { α ( k − | γ | )( k + | γ | + 1) + D ( α, γ ) } ≤ α k ( k + 4) + 4 | α | from which the conclusion of the theorem follows. Based on the different species of the eigenvalues, we partition the spectrum of − L α into four disjoint subsets σ k , ..., σ k defined by σ ki := { λ : λ is an eigenvalue of a Lam´e harmonics of degree k and of species i } . For each k ∈ N , we denote the eigenvalues of √− L α corresponding to the 2 k + 1Lam´e harmonics of degree k by q λ k − k ( α ) < q λ k − k +1 ( α ) < · · · < q λ kk ( α ) . Based on the definition of the σ i , we can decompose dρ DS ( ϕ ; k, α ) into fourdisjoints sums, i.e. dρ DS ( ϕ ; k, α ) = 12 k + 1 k X j = − k ϕ q λ kj ( α ) k + O (cid:18) k (cid:19) = 12 k + 1 X i =1 X λ ∈ σ ki ϕ √ λk ! + O (cid:18) k (cid:19) . (3.1)As we mentioned before, when k is even, only the Lam´e harmonics of thefirst and third species will contribute to the sum above, whereas only the secondand fourth species will contribute when k is odd. Therefore, we can write k X j = − k ϕ q λ kj k = P λ ∈ σ k ϕ (cid:16) √ λk (cid:17) + P λ ∈ σ k ϕ (cid:16) √ λk (cid:17) , k even P λ ∈ σ k ϕ (cid:16) √ λk (cid:17) + P λ ∈ σ k ϕ (cid:16) √ λk (cid:17) , k odd . The key observation here is that the eigenvalues can be obtained by simplyregarding the polynomial solution of the generalized Lam´e equation (2.8). Moreprecisely, we introduce the sets Z i , i = 1 , , ,
4, defined by Z ki := { ν | There exist λ ∈ σ ki and γ ∈ { , } such that ν = λ − D ( α, γ ) } . Consequently, the four sums above can now be taken over the sets Z ki insteadof σ ki . That is, X λ ∈ σ ki ϕ √ λk ! = X ν ∈ Z ki ϕ p ν + D ( α, γ ) k ! (3.2)Moreover, since ϕ is compactly supported, we can approximate uniformly ϕ by smooth functions. Without loss of generality, we may therefore assume that ϕ satisfies ϕ p ν + D ( α, γ ) k ! = ϕ (cid:18) √ νk (cid:19) + O (cid:18) k (cid:19) since D ( α, γ ) = O (1). The equation (3.2) easily implies that1 | σ ki | X λ ∈ σ ki ϕ √ λk ! = 1 | Z ki | X ν ∈ Z ki ϕ (cid:18) √ νk (cid:19) + O (cid:18) k (cid:19) . (3.3)The asymptotic of the sums in RHS of (3.3) are obtained through the fol-lowing lemma. Lemma 3.1.
Let ν , ..., ν m denote the m + 1 real numbers for which the Lam´eequation A ( x ) y ′′ ( x ) + X j =0 ρ j Y i = j ( x − α i ) y ′ ( x ) = ( m ( m + 1 + | ρ | ) x − ν ) y ( x ) admits a polynomial solution y of degree m . For any ϕ ∈ C c ( R + ) , we have that m + 1 m X j =0 ϕ (cid:18) √ ν j m (cid:19) = 1 π Z π Z π/ ϕ (cid:16)p g + ( ξ, θ ; α ) (cid:17) cos θ dθ dξ + O (cid:18) m (cid:19) where g + ( ξ, θ ; α ) = max { , ( α − α ) (cid:0) β sin θ cos ξ + ( β −
1) sin θ (cid:1) + α } , and β = α − α α − α . The proof of Lemma 3.1 is rather long and technical, so we prefer to postponeit until the end of the present section. With this lemma in hand, we can nowcomplete the proof of Theorem 1.1. As a consequence of Lemma 3.1, we obtainfor k even,12 k + 1 k X j = − k ϕ q λ kj ( α ) k = 12 k + 1 X λ ∈ σ k ϕ √ λk ! + X λ ∈ σ k ϕ √ λk ! = 12 k + 1 X ν ∈ Z k ϕ (cid:18) √ νk (cid:19) + X ν ∈ Z k ϕ (cid:18) √ νk (cid:19) + O (cid:18) k (cid:19)
0= 14 k/ X ν ∈ Z k ϕ (cid:18) √ νk/ (cid:19) + 34 k/ X ν ∈ Z k ϕ (cid:18) √ ν k/ (cid:19) + O (cid:18) k (cid:19) (3.4)By Lemma 3.1, the first sum in the brackets of (3.4) is equal to1 π Z π Z π/ ϕ (cid:18) g + ( ξ, θ ; α ) (cid:19) cos θ dθ dξ + O (cid:18) k (cid:19) , (3.5)and the second sum in brackets of (3.4) is equal to1 π Z π Z π/ ϕ (cid:18) g + ( ξ, θ ; α ) (cid:19) cos θ dθ dξ + O (cid:18) k (cid:19) , (3.6)Combining equations (3.5) and (3.6), we deduce that12 k + 1 k X j = − k ϕ q λ kj ( α ) k = 1 π Z π Z π/ F ( ϕ ; ξ, θ ; α ) cos θ dθ dξ + O (cid:18) k (cid:19) (3.7)where the function F is defined by F ( ϕ ; ξ, θ ; α ) = 14 ϕ (cid:18) g ( ξ, θ ; α ) (cid:19) + 34 ϕ (cid:18) g ( ξ, θ ; α ) (cid:19) . The conclusion of Theorem 1.1 for k even then follows from (3.1) and (3.7).Similarly, for k odd, we have that12 k + 1 k X j = − k ϕ q λ kj ( α ) k = 12 k + 1 X λ ∈ σ k ϕ √ λk ! + X λ ∈ σ k ϕ √ λk ! = 12 k + 1 X ν ∈ Z k ϕ (cid:18) √ νk (cid:19) + X ν ∈ Z k ϕ (cid:18) √ νk (cid:19) + O (cid:18) k (cid:19) = 34 k/ X ν ∈ Z k ϕ (cid:18) √ ν k/ (cid:19) + 14 k − / X ν ∈ Z k ϕ (cid:18) √ νk/ (cid:19) + O (cid:18) k (cid:19) . (3.8)1As for the case k even, we apply Lemma 3.1 to conclude that (3.7) holdswhen k is a positive odd integer.To complete the proof of Theorem 1.1, it remains to prove Lemma 3.1. According to Theorem 2.1 with a = − a = 0 and a = β >
0, there exist m + 1 real values ˜ ν , ..., ˜ ν m for which the generalized Lam´e equation x ( x − β )( x + 1) Y ′′ ( x ) + (cid:2) ρ x ( x − β ) + ρ ( x + 1)( x − β )+ ρ x ( x + 1)] Y ′ ( x ) = ( m ( m + 1 + | ρ | ) x − ˜ ν ) Y ( x ) , (3.9)admits a polynomial solution Y of degree m . First, we show that for any ϕ ∈ C c ( R + )1 m + 1 m X j =0 ϕ (cid:18) ˜ ν j m (cid:19) = 1 π Z π Z π/ ϕ ( h ( ξ, θ ; α )) cos θ dθ dξ + O (cid:18) m (cid:19) (3.10)where h ( ξ, θ ; α ) = β sin θ cos ξ + ( β −
1) sin θ .The starting point in proving (3.10) consists of establishing a three-termrecurrence relation satisfied by the Lam´e polynomials Y . In particular, thiswill allow us to obtain the eigenvalues of − L α as the those of some tridiagonalmatrix.More precisely, we consider a Lam´e polynomial of degree m of the form Y ( x ) = m X j =0 a j x j . If we replace the expression for Y ( x ) into the Lam´e equation (3.9), we obtainthe following three-term recurrence relation: A j ( ρ, β ) a j + B j ( ρ, β ) a j +1 + C j ( ρ, β ) a j − = ˜ νa j ( j = 0 , ..., m ) (3.11)where a − = 0, a m +1 = 0, and A j ( ρ, β ) = ( β − j ( j − ρ ) − ρ j + β ρ j,B j ( ρ, β ) = ( j + 1)( j + ρ ) β C j ( ρ, β ) = µ − ( j − j − | ρ | ) . (3.12)Note that, as a result of the above, A = B m = C . These relations are moreconveniently expressed in matrix form. Indeed, if we introduce the tridiagonalmatrix A = ( a ij ), i, j = 0 , ..., m , given by a ij = B i ( ρ,β ) µ if i = j − A i ( ρ,β ) µ if i = j C i ( ρ,β ) µ if i = j + 1 , (3.13)2then the three-term recurrence relation (3.11) implies that AX = ˜ νµ X, where X = ( a , a , ..., a m ) T . Throughout the rest of the proof, we denote by ˜ ν µ , ..., ˜ ν m µ the m + 1 eigenvalues of A . Note that the components of the eigen-vectors X are exactly the coefficients of the Lam´e polynomials Y .We will divide the rest of the proof into several lemmas. The first one consistsof computing the trace of the powers A n for any n ∈ N . Lemma 3.2.
We have thatTr ( A n ) = [[ n/ X j =0 (cid:18) nj, j, n − j (cid:19) × m X i =1 (cid:18) − i m (cid:19) j (cid:18) i m (cid:19) n − j ( β − n − j β j + O (1) (3.14) for any positive integer n . Here, [[ n/ denotes the greatest integer less or equalto n/ .Proof of Lemma : We decompose A as a sum of three matrices, A = L + D + U ,where D = µ diag(0 , A , ..., A m ) and L = 1 µ · · · C · · · C · · · · · · C m , U = 1 µ B · · ·
00 0 B · · · · · · B m − · · · . The trace of A n is then given by the trace of ( L + D + U ) n . When we expandlast expression, the non-commutativity of the matrices L, D and U implies thatthe trace of A n is the sum of 3 n terms of the form M M · · · M n , where M i = L, D, or U . This is unmanageable in its full generality for arbitrary n. However, we are interested primarily in the asymptotic information containedin the trace, which allows us to make significant simplifications.First, we point out that our need to consider A n stems from the fact that wewill use polynomials to approximate the continuous function ϕ in Lemma 3.1.Thus, we need to extract asymptotic information about Tr( A n ) for fixed, butarbitrary n. In our case, we will ultimately be taking a limit m → ∞ for fixed n, and so in this limit, n/m → M M · · · M n are products ofmatrices, each being lower diagonal ( L ), diagonal ( D ), or upper diagonal ( U ).This allows us to make definite statements about the zero structure of the matrixproducts, i.e., the entries that are necessarily zero in the matrix product. Forexample, multiplication on the left or right by a diagonal matrix preserves thezero structure: LD and DL are both lower diagonal if L is. The analogousstatement holds for U D and
DU.
The effect of multiplying by L or U is onlyslightly less simple. In fact, as far as the effect on zero structure is concerned, L and U behave like quantum mechanical creation and annihilation operators(respectively). In detail, if we denote by R M (resp. L M ) the operation of right(resp. left) multiplication by a matrix M , then for any matrix B :(i) R U B corresponds to shifting all columns of B one place to the right:col i +1 ( R U B ) = col i ( B ), creating a zero column in the first column.(ii) R L B corresponds to shifting all columns of B one place to the left:col i − ( R U B ) = col i ( B ), creating a zero column in the last column.(iii) L U B corresponds to shifting all rows of B up one place: row i − ( R U B ) =row i ( B ), creating a zero row in the last row.(iv) L U B corresponds to shifting all rows of B down one place: row i +1 ( R U B ) =row i ( B ), creating a zero row in the first row.As a result, the diagonal of a term M M · · · M n in A n will have zero traceunless the number of factors j of L is the same as the number of factors of U. The remaining n − j factors must all be D. Thus, many of the 3 n terms do notcontribute to Tr( A n ) . The last issue concerns the lack of commutativity in the terms that do con-tribute to the trace. Some of these terms are of the form( LU ) j D n − j , j = 0 , . . . , [[ n/ . (3.15)Since LU and D are diagonal, the trace is particularly simple to compute in thecase of the canonical terms (3.15):Tr( M M · · · M n ) = Tr(( LU ) j D n − j )= m X i =1 (cid:18) − i m (cid:19) j (cid:18) i m (cid:19) n − j ( β − n − j β j + O (1) . Noncanonical terms will differ from canonical terms only at order O ( n/m ) = O (1 /m ) , and so for asymptotic purposes, we may assume that all terms have thecanonical form (3.15). To see this, note that multiplication of matrices of theform L, D, and U constitutes a shifting of their rows and columns. For termswith n factors, the number of shifts is at most n. Being products of matrices that4are (lower, upper) diagonal, the noncanonical terms will yield sums of productsof the form Γ p ∆ q , where Γ p , ∆ q ∈ { A l /µ, B l /µ, C l /µ | l = 0 , , . . . , m } and | p − q | = O ( n ) . As anexample, A p µ B q µ = β ( β − p q m + O (cid:18) m (cid:19) = β ( β − p ( p + O ( n )) m + O (cid:18) m (cid:19) = β ( β − p m + O ( n/m ) + O (cid:18) m (cid:19) = β ( β − p m + O (cid:18) m (cid:19) . Since there are exactly (cid:0) nj,j,n − j (cid:1) matrices M M · · · M n that contain j fac-tors of L , j factors of U and ( n − j ) factors of D , we finally deduce thatTr( A n ) = [[ n/ X j =0 (cid:18) nj, j, n − j (cid:19) Tr(( LU ) j D n − j ) + O (1)= [[ n/ X j =0 (cid:18) nj, j, n − j (cid:19) m X i =1 (cid:18) − i m (cid:19) j (cid:18) i m (cid:19) n − j ( β − n − j β j + O (1) . (3.16)This completes the proof of the Lemma.The next result deals with the inner sum P mi =1 (cid:16) − i m (cid:17) j (cid:16) i m (cid:17) n − j in (3.16).As the next lemma shows, this sum is asymptotically given by a Beta integral. Lemma 3.3.
We have that m m X i =0 (cid:18) − i m (cid:19) j (cid:18) i m (cid:19) n − j = 12 B ( j + 1 , n − j + 1 /
2) + O (cid:18) m (cid:19) (3.17) where B ( p, q ) is the standard Beta integral defined by B ( p, q ) = 2 Z π/ cos p − θ sin q − θ dθ. Proof of Lemma : This is obvious. The LHS of (3.17) is a Riemann sum forthe function (1 − x ) j ( x ) n − j on [0 , m m X i =0 (cid:18) − i m (cid:19) j (cid:18) i m (cid:19) n − j = Z (1 − x ) j ( x ) n − j dx + O (cid:18) m (cid:19) . x = sin θ andusing the trigonometric representation of the Beta integral.As a consequence of (3.16) and Lemma 3.2, it follows that1 m Tr( A n ) = 1 m m X i =0 (cid:18) ν i µ (cid:19) n = 12 [[ n/ X j =0 (cid:18) nj, j, n − j (cid:19) B ( j + 1 , n − j + 1 / β − n − j β j + O (cid:18) m (cid:19) . (3.18)In order to evaluate the sum inside the integral sign, we use the sinc functiondefined by sinc( x ) = ( x = 0 , sin xx for x = 0 . The key point here is to observe that sinc( πx ) = 0 when x is a non-zero integer,and that sinc(0) = 1. Using this function, we can then replace the sum in (3.18)by the more appropriate sum over multi-index γ = ( γ , γ , γ ) such that | γ | = n .More precisely, we have [[ n/ X j =0 (cid:18) nj, j, n − j (cid:19) B ( j + 1 , n − j + 1 /
2) ( β − n − j β j = X | γ | = n (cid:18) nγ (cid:19) ( β − γ β γ + γ B ( γ ) sinc( π ( γ − γ )) . (3.19)where B ( γ ) := B (cid:0) γ + γ + 1 , n − γ − γ + (cid:1) . Based on the representationof sinc( x ) as the integralsinc( π ( γ − γ )) = 12 π Z π − π e iξ ( γ − γ ) dξ, (3.20)the RHS of (3.19) can be written as12 π Z π − π X | γ | = n (cid:18) nγ (cid:19) ( β − γ β γ + γ B ( γ ) e iξ ( γ − γ ) dξ. (3.21)Replacing B ( γ ) by the expression B ( γ ) = 2 Z π/ (cos θ ) γ + γ +1 (sin θ ) n − γ − γ dθ, X | γ | = n (cid:18) nγ (cid:19) ( β − γ β γ + γ (cos θ ) γ + γ (sin θ ) n − γ − γ e iξ ( γ − γ ) = ( β cos ξ sin 2 θ + ( β −
1) sin θ ) n . (3.22)If we denote by h ( ξ, θ ) = ( β cos ξ sin 2 θ + ( β −