Spectral analysis of transfer operators associated to Farey fractions
aa r X i v : . [ m a t h - ph ] A ug Spectral analysis of transfer operators associated toFarey fractions
Claudio Bonanno ∗ Sandro Graffi † Stefano Isola ‡ November 4, 2018
Abstract
The spectrum of a one-parameter family of signed transfer operatorsassociated to the Farey map is studied in detail. We show that whenacting on a suitable Hilbert space of analytic functions they are self-adjoint and exhibit absolutely continuous spectrum and no non-zeropoint spectrum. Polynomial eigenfunctions when the parameter is anegative half-integer are also discussed.
Keywords:
Transfer operators, Farey fractions, spectral theory, periodfunctions, self-reciprocal functions
Riassunto:
Analisi spettrale di operatori di trasferimento associati allefrazioni di Farey.
Presentiamo uno studio dettagliato dello spettro di una famiglia ad unparametro di operatori di trasferimento segnati associati alla trasformazionedi Farey dell’intervallo unitario in s´e. Se fatti agire su un opportuno spazio diHilbert di funzioni analitiche essi risultano autoaggiunti e con spettro asso-lutamente continuo (ad eccezione dell’autovalore nullo). Diamo altres`ı una aclassificazione completa delle autofunzioni polinomiali quando il parametro`e un semintero negativo.
Scientific Chapter:
Mathematical Physics ∗ Dipartimento di Matematica Applicata, Universit`a di Pisa, via F. Buonarroti 1/c,I-56127 Pisa, Italy, email: < [email protected] > † Dipartimento di Matematica, Universit`a di Bologna, Piazza di Porta S. Donato 5,I-40127 Bologna, Italy, e-mail: < graffi@dm.unibo.it > ‡ Dipartimento di Matematica e Informatica, Universit`a di Camerino, via Madonnadelle Carceri, I-62032 Camerino, Italy. e-mail: < [email protected] > Preliminaires and statement of the main results
Let F : [0 , → [0 ,
1] be the
Farey map defined by F ( x ) = ( x − x if 0 ≤ x ≤ − xx if ≤ x ≤ x ∈ [0 , x = 1 a + 1 a + 1 a + · · · ≡ [ a , a , a , . . . ]then x = [ a , a , a , . . . ] F ( x ) = [ a − , a , a , . . . ] (1.2)with [0 , a , a , . . . ] ≡ [ a , a , . . . ]. Differently said, let F n be the ascendingsequence of irreducible fractions between 0 and 1 constructed inductively inthe following way: set first F = ( , ), then F n is obtained from F n − byinserting among each pair of neighbours a ′ b ′ and a ′′ b ′′ in F n − their Farey sum ab := a ′ + a ′′ b ′ + b ′′ . Thus F = (cid:0) , , (cid:1) F = (cid:0) , , , , (cid:1) F = (cid:0) , , , , , , , , (cid:1) and so on. The elements of F n are called Farey fractions . It is easy to verifythat the set of pre-images ∪ nk =0 F − k { } coincides with F n for all n ≥
1. Thisimplies that ∪ ∞ k =0 F − k { } = Q ∩ [0 , ab belongs to F n \ F n − if and only if itscontinued fraction expansion ab = [ a , a , . . . , a k ] with a k > P ki =1 a i = n .In this paper we shall study a family of signed generalized transfer operators P ± q associated to the map F , whose action on a function f ( x ) : [0 , → C is given by a weighted sum over the values of f on the set F − ( x ), namely f ( x ) ( P ± q f )( x ) = (cid:18) x + 1 (cid:19) q (cid:20) f (cid:18) xx + 1 (cid:19) ± f (cid:18) x + 1 (cid:19)(cid:21) (1.3)where q is a real or complex parameter. The operator P +1 is referred toas the Perron-Frobenius operator for the map F : its fixed function is thedensity of an absolutely continuous F -invariant measure. In this case one2asily checks that the function 1 /x has this property. However, since 1 /x does not belong to L ([0 , , dx ) the statistical properties of the map F haveto be described in the framework of infinite ergodic theory [Aa]. We referto [Bal] for a general review of transfer operator techniques in dynamicalsystems theory. Here, one motivation to study signed transfer operatorsarises from their appearing in dynamical zeta functions such as Selberg andRuelle’s (see [DEIK], Corollary 3.13, and also [BI]).Using the Farey fractions, the iterates P ± q n f of the above operators canbe expressed as suitable sums over the Stern-Brocot tree , the binary treewith root node 1 and whose n -th level L n is given by L n = ( F n \ F n − ) ∪ S ( F n \ F n − ), where S is the map S : x → /x and such that the elementsof S ( F n \ F n − ) are in reverse order. An important feature of this tree isthat each positive rational number appears as a vertex exactly once. Theleft part of the Stern-Brocot tree (starting from the node ) is called the Farey tree , with vertex-set Q ∩ (0 , Figure 1:
First four levels of the Stern-Brocot tree.
An easy generalisation of Proposition 5.9 in [DEIK] yields for all x ∈ R + and q ∈ C , ( P ± q n f )( x ) = X ab ∈ L n f (cid:16) n ( x,a/b ) ax + b (cid:17) ± f (cid:16) n ( x,a/b ) ax + b (cid:17) ( ax + b ) q (1.4)3here n ( x, a/b ) = µx + ν and n ( x, a/b ) = ( a − µ ) x + b − ν , for some0 ≤ µ ≤ a and 0 ≤ ν ≤ b . In particular n ( x, a/b ) + n ( x, a/b ) = ax + b .In Section 2 we prove Theorem 1.1.
For each q ∈ (0 , ∞ ) there is a Hilbert space of analyticfunctions H q on which the operators P ± q are bounded, self-adjoint and iso-spectral. Their common spectrum is given by { } ∪ (0 , , with (0 , purelyabsolutely continuous. Remark 1.2.
From thermodynamic formalism it follows that P + q for q ∈ ( −∞ , , when acting on a suitable Banach space has a leading eigenvalue λ ( q ) ≥ which is a differentiable and monotonically decreasing functionwith lim q → − λ ( q ) = 1 (and λ ( q ) = 1 for all q ≥ , see [PS]). From theabove theorem we see that corresponding eigenfunction does not belong tothe space H q (for q = 1 it is just the invariant density /x ). Moreover λ ( q ) = lim n →∞ n log( P + q n . Note that by (1.4) we can write ( P + q n X ab ∈F n \{ } b − q and the above sum is equal to the partition function Z n − (2 q ) at (inverse)temperature q of the number-theoretical spin chain introduced by AndreasKnauf in [Kn]. Remark 1.3.
One easily checks that the function f ( x ) = (1 − x ) /x is aneigenfunction of P − q for q = 1 / and eigenvalue . But, again, this functiondoes not belong to H . There are interesting functional symmetries related to the eigenvalue equa-tion for P ± q , which can be rephrased in terms of Hankel transforms. Theconstruction of Section 2 allows for a complete account of the correspondingself-reciprocal functions in L ( R + ), discussed in Section 3. Finally, in Sec-tion 4 we characterise all polynomial eigenvectors of P ± q when q = − k/ k ≥ P ± q for real positive q In this section we give the proof of Theorem 1.1, hence in the sequel werestrict ourselves to the case q ∈ (0 , ∞ ). The proof of the theorem followsfrom the results of the following subsections.4 .1 An invariant Hilbert space In this subsection we introduce a family of Hilbert spaces H q , where q ∈ (0 , ∞ ), and give the representation of the operators P ± q on H q . Definition 2.1.
For q ∈ (0 , ∞ ) we denote by H q the Hilbert space of allcomplex-valued functions f which can be represented as a generalised Boreltransform f ( x ) = B q [ ϕ ]( x ) := 1 x q Z ∞ e − tx e t ϕ ( t ) m q ( dt ) ϕ ∈ L ( m q ) (2.1) with inner product ( f , f ) = Z ∞ ϕ ( t ) ϕ ( t ) m q ( dt ) if f i = B q [ ϕ i ] (2.2) and measure ( p = 2 q − ) m q ( dt ) = t p e − t dt (2.3)Function spaces related to that introduced above have been used in [Is], [GI]and [Pre]. In [Is] an explicit connection between the approach presentedhere and Mayer’s work on the transfer operator for the Gauss map [Ma] isestablished by means of a suitable operator-valued power series. Remark 2.2.
For q ∈ C , Re q > , the space H q can be regarded as acomplex Hilbert space. Setting χ p ( x ) := x p ( p = 2 q −
1) (2.4) an alternative representation for f ∈ H q can be obtained by a simple changeof variable when x is real and positive: ( χ p · f )( x ) = Z ∞ e − s ( χ p · ϕ )( sx ) ds (2.5) Note that a function f ∈ H q is analytic in the disk D = (cid:8) x ∈ C : Re x > (cid:9) = { x ∈ C : | x − | < } (2.6) In particular, ( χ p · ϕ )( t ) = ∞ X n =0 a n n ! t n = ⇒ ( χ p · f )( x ) = ∞ X n =0 a n x n (2.7) in the sense of formal power series. So the power series of χ p · ϕ is obtainedBorel transforming that of χ p · f , in the usual sense. This justify the nameof the integral transform (2.1). emark 2.3. The invariant density /x for the Farey map, that is the fixedfunction of P +1 , is the generalised Borel transform (for q = 1 ) of the function ϕ ( t ) = 1 /t which, however, does not belong to L ( m ) . Let us now study the Hilbert space L ( m q ). First of all we notice that themeasure m q ( dt ) is finite, indeed Z ∞ m q ( dt ) = Γ(2 q ) (2.8)Second, for the linearly independent family of functions f n ( t ) := t n n ! ( n ≥ f n , f m ) = Γ( n + m + 2 q ) n ! m ! (2.9)This implies that the (generalised) Laguerre polynomials L pn ( t ) ( n ≥ p > −
1) by e n ( t ) := L pn ( t ) = n X m =0 (cid:18) n + pn − m (cid:19) ( − t ) m m ! (2.10)form a complete orthogonal basis in L ( m q ), with( e n , e m ) = Γ( n + 2 q ) n ! δ n,m (2.11)Moreover, using ([GR], p.850) and (2.11) we get for m ≤ n ( f n , e m ) = ( − m Γ( n + 2 q ) m !( n − m )! = ( − m (cid:18) nm (cid:19) k e n k = ( − m Γ( n + 2 q )Γ( m + 2 q ) ( n − m )! k e m k = ( − m (cid:18) n + pn − m (cid:19) k e m k (2.12)In particular ( f n , e n ) = ( − n k e n k . Also note that ( f n , e m ) = 0 for m > n .Comparing to (2.10) we obtain the following result Lemma 2.4.
For each n ∈ N the numbers a n,m := ( − m (cid:18) n + pn − m (cid:19) if m ≤ n m > n re the Fourier coefficients of f n w.r.t the basis ( e m ) , i.e. a n,m = ( f n , e m ) k e m k Moreover f n = n X m =0 a n,m e m e n = n X m =0 a n,m f m Remark 2.5.
In particular, the ( n + 1) × ( n + 1) lower triangular ma-trix A n := ( a i,j ) ≤ i,j ≤ n satisfies A n = I n +1 . Therefore, the operator Π n :L ( m q ) → L ( m q ) acting as Π n : ∞ X s =0 c s e s −→ ∞ X s =0 c s n X r =0 ( f r , e s ) k e s k f r = n X r =0 d r f r with d r := r X s =0 a r,s c s or d ( n ) = A n c ( n ) where we have set c ( n ) = ( c , c , . . . , c n ) T and similarly for d ( n ) , is the or-thogonal projection onto the linear subspace spanned by (1 , t, t , . . . , t n n ! ) . Let us now consider the action of the transform B q on the functions ( e n )and ( f m ). We have B q [ e n ]( x ) = n X m =0 Γ(2 q + m ) (cid:18) n + pn − m (cid:19) ( − x ) m m != ( n + 1) p (1 − x ) n (2.13)where ( a ) p := Γ( a + p ) / Γ( a ) = a ( a + 1) · · · ( a + p −
1) is the shifted factorial,and B q [ f n ]( x ) = ( n + 1) p x n . (2.14)The next result describes the action of P ± q on the Hilbert space H q . Proposition 2.6.
For q ∈ (0 , ∞ ) the space H q is invariant for P ± q and P ± q : H q → H q are positive operators, isomorphic to self-adjoint compactperturbations of the multiplication operator M : L ( m q ) → L ( m q ) given by ( M ϕ )( t ) = e − t ϕ ( t )7 ore specifically P ± q B q [ ϕ ] = B q [ P ± ϕ ] where P ± = M ± N and N : L ( m q ) → L ( m q ) is the symmetric integraloperator given by ( N ϕ )( t ) = Z ∞ J p (cid:0) √ st (cid:1) ( st ) p/ ϕ ( s ) m q ( ds ) where J p denotes the Bessel function of order p .Proof. The representation of P ± q on H q follows from a direct computation(see [Is], [GI]). The positivity amounts to(( M ± N ) ϕ, ϕ ) ≥ ∀ ϕ ∈ L ( m q ) , k ϕ k = 1 (2.15)and can be checked expanding ϕ on the basis of (normalised) Laguerre poly-nomials. Indeed, a calculation using ([GR], pp.849-850) yields( M e n , e n ) k e n k = 2 − n − q (cid:18) n + pn (cid:19) and ( N e n , e n ) k e n k = 2 − n − q (cid:18) n + pn (cid:19) F ( − n, n + 2 q ; 2 q ; 1 / − n − q P ( p, n (0)where P ( a,b ) n ( x ) denotes the Jacobi polynomial ([AAR], p.99). Since P ( p, n (0) = ( − − n n X k =0 ( − k (cid:18) n + pk (cid:19) (cid:18) nk (cid:19) and (cid:18) n + pn (cid:19) = n X k =0 (cid:18) n + pk (cid:19) (cid:18) nk (cid:19) we get (( M ± N ) e n , e n ) k e n k = 12 n +2 q n X k =0 (1 ± ( − n − k ) (cid:18) n + pk (cid:19)(cid:18) nk (cid:19) N ϕ can be written as R ∞ k ( s, t ) ϕ ( s ) m q ( ds ) withsymmetric kernel k ( s, t ) = J p (cid:0) √ st (cid:1) ( st ) p/ (2.16)From the estimates J p ( t ) ∼ − p t p / Γ( p + 1) as t → + and J p ( t ) = O ( t − / )as t → ∞ ([E], vol. II), we see that the kernel k ( s, t ) is bounded andcontinuous.We can now describe the action of P ± on ( e n ) and ( f n ). Applying theintegral representation (see [E], Vol. II, p.190) n ! e − t L pn ( t ) = Z ∞ J p (cid:0) √ st (cid:1) ( st ) p/ s n m q ( ds )we get M − N f n = e n M − N e n = f n (2.17) Let introduce an isometry which turns out to be useful for the characterisa-tion of eigenfunctions of the operators P ± q . Let J q be the involution definedby ( J q f )( x ) := 1 x q f (cid:18) x (cid:19) (2.18)and consider its action on the Hilbert space H q . We have the following Proposition 2.7.
For any ϕ ∈ L ( m q ) it holds J q B q [ ϕ ] = B q [ J ϕ ] (2.19) where J := N M − is a bounded operator in L ( m q ) with k J k ≤ π . Ifmoreover P ± q f = λ f for some λ = 0 then f satisfies the functional equation J q f = ± f (2.20) Proof.
The representation of J q in H q is easily checked by first noting thatfor any f ∈ H q the function J q f can be written as an ordinary Laplacetransform, i.e. f ( x ) = B q [ ϕ ]( x ) = ⇒ ( J q f )( x ) = Z ∞ e − tx ( χ p · ϕ )( t ) dt (2.21)9nd then using Tricomi’s theorem ([Sne], p.165). Let us prove the bound on k J k . Adapting formula (33) of [RS], vol. IV, to our situation we get for all ϕ ∈ L ( m q ) and λ ∈ [0 , k N ( M − λ ) − ϕ k ≤ Z k N ( M − λ ) − ϕ k dλ ≤ π Z ∞−∞ k N e iτM ϕ k dτ (2.22)On the other hand we claim that Z ∞−∞ k N e iτM ϕ k dτ ≤ π Z ∞ e − t (cid:18)Z ∞ | J p (2 √ st ) | | ϕ ( s ) | s p e − s ds (cid:19) dt (2.23)To prove (2.23) we write( N e iτM ϕ )( t ) = Z ∞ J p (2 √ st )( st ) p/ e iτe − s ϕ ( s ) s p e − s ds so that interchanging the order of integration k N e iτM ϕ k = Z ∞ | G ( t, τ ) | e − t dt where we have set G ( t, τ ) = Z ∞ J p (2 √ st ) e iτe − s s p/ ϕ ( s ) e − s ds = − Z J p (2 √− t ln u ) e iτu ( − ln u ) p/ ϕ ( − ln u ) du Equation (2.23) now follows by applying Fourier-Plancherel theorem: Z ∞−∞ | G ( t, τ ) | dτ = 2 π Z | J p (2 √− t ln u ) ϕ ( − ln u ) | ( − ln u ) p du = 2 π Z ∞ | J p (2 √ st ) | | ϕ ( s ) | s p e − s ds Hence, putting together (2.22) and (2.23), we have k N ( M − λ ) − ϕ k ≤ π Z ∞ e − t (cid:18)Z ∞ | J p (2 √ st ) | | ϕ ( s ) | s p e − s ds (cid:19) dt The right hand side is bounded above by4 π k ϕ k Z ∞ e − t sup st ≥ | J p (2 √ st ) | dt =: 4 π C k ϕ k x ≥ | J p (2 √ x ) | = 1 we get C = 1. Therefore k N ( M − λ ) − k ≤ π ∀ λ ∈ [0 , λ = 0 we get k J k ≤ π as claimed.To finish the proof, we note that if ϕ ∈ L ( m q ) the functions M ϕ and
N ϕ are bounded at infinity. Therefore, if f ∈ H q satisfies P ± q f = λ f with λ = 0,then f extends analytically from the disk D to the half-plane { Re x > } . Inaddition the expression ( P ± q f )( x ) reproduces itself times ± /x for x and dividing through x q . Hence (2.20) holds. Remark 2.8.
Note that (2.18) is only a necessary condition for f to be aneigenfunction (with λ = 0 ). For instance the function f ( x ) = x − q (whichdoes not belong to H q ) although plainly satisfying (2.18) for all q ∈ (0 , ∞ ) is an eigenfunction of P + q only for q = 1 (with λ = 1 ). Remark 2.9.
Applying Proposition 2.7, the eigenvalue equations P ± q f = λf , with λ = 0 , can be rewritten as the three-term functional equations, λ f ( x ) − f ( x + 1) = ± x q f (cid:18) x (cid:19) (2.24) which for λ = 1 are studied in [Le] and [LeZa]. P ± in L ( m q ) We are now reduced to study the spectrum of the operators P ± in L ( m q ).Let us start studying the operators Q ± = M − P ± = I ± M − N (2.25)We first show that they are bounded in L ( m q ). Lemma 2.10.
We have k Q ± k ≤ π .Proof. The adjoint of the operator J = N M − dealt with in the previoussubsection exists and equals J ∗ = M − N . A priori it is defined only on D ( M − ). Recall however that J ∗ is continuous if and only if J is such and k J ∗ k = k J k . The assertion now follows from Proposition 2.7.Recall now the orthogonal basis of L ( m q ) given by e n ( t ) (see (2.10)) andthe independent family of functions f n ( t ) = t n n ! . We introduce the familiesof functions ℓ ± n ( t ) := e n ( t ) ± f n ( t ) , h ± n ( t ) := e − t ( e n ( t ) ± f n ( t )) (2.26)and consider the linear manifolds spanned by them.11 roposition 2.11. The linear manifolds E ± ⊂ L ( m q ) defined by E ± := ( m X n =0 c n h ± n : c n ∈ C , ≤ n ≤ m, m ≥ ) (2.27) have the following properties:1. they are fixed by the operators ± J , i.e. ± J ϕ = ϕ , ∀ ϕ ∈ E ± ;2. their intersection is the trivial subspace, i.e. E + ∩ E − = { } ;3. they are dense, i.e. E ± ≡ Span { h ± n } n ≥ = L ( m q ) .Proof. We first use (2.13) and (2.14) to get B q [ h ± n ]( x ) = ( n + 1) p ± x n (1 + x ) n +2 q (2.28)hence J q B q [ h ± n ]( x ) = ±B q [ h ± n ]( x ). Now the first property follows upon ap-plication of Proposition 2.7.The second property follows at once from the fact that the operator J is aninvolution.Finally, from the proof of Proposition 2.6 and (2.17) one readily gets that( h ± n , e n ) > ∀ n ≥
0. This yields the density of E ± in L ( m q ).Let us now consider the functions ( ℓ ± n ). From the definition it follows thatthe function ℓ + n ( t ) is a polynomial of degree 2 k for n = 2 k and n = 2 k + 1,( k ≥ ℓ − n ( t ) has degree 2 k + 1 for n = 2 k + 1 and n = 2 k + 2,( k ≥ ℓ ± n , e n ) = (1 ± ( − n ) k e n k so that( ℓ +2 k +1 , e k +1 ) = ( ℓ − k +2 , e k +2 ) = 0( ℓ +2 k , e k ) = 2 k e k k ( ℓ − k +1 , e k +1 ) = 2 k e k +1 k (2.29) Proposition 2.12.
Let H ± := Span { ℓ ± n } n ≥ . Then1. L ( m q ) = H + ⊕ H − ;2. Q ± | H ± = 2 I and Q ± | H ∓ = 0 .Proof.
1. By the relations (2.29), H + and H − do not have non-zero commonvectors, thus H + ∩ H − = { } . Moreover, let ϕ ∈ L ( m q ) be such that ϕ ⊥ H + ⊕ H − . Since ( ℓ ± n , e n ) = (1 ± ( − n ) k e n k we get ϕ = 0.12. We recall (2.17), M − N f n = e n M − N e n = f n From it we get Q ± ℓ ± n = 2 ℓ ± n and Q ± ℓ ∓ n = 0For ϕ = P mn =0 c n ℓ ± n we have by linearity Q ± ϕ = 2 ϕ so that k Q ± ϕ k =2 k ϕ k , independently of m . This implies Q ± ϕ = 2 ϕ for all ϕ ∈ H ± .Hence Q ± H ± ⊆ H ± and Q ± | H ± = 2 I . In the same way one provesthat Q ± | H ∓ = 0. Remark 2.13.
From the above it follows that the operators Q ± are boundedin L ( m q ) with k Q ± k = 2 . The operators P ± are self-adjoint and positive on L ( m q ), hence the spec-trum is real and positive. Moreover k P ± k ≤ k Q k k M k = 2. Hence σ ( P ± ) ⊆ [0 , σ p ( P ± ). Corollary 2.14. In L ( m q ) it holds Ker P ± = H ∓ and σ p ( P ± ) = { } withinfinite multiplicity.Proof. We first observe that since Ker M = { } we have by Proposition 2.12Ker P ± = Ker ( M Q ± ) = Ker Q ± = H ∓ Now suppose that P ± ϕ = λϕ for some 0 < λ ≤ ϕ
0. Then ϕ ∈ H ± and hence P ± ϕ = M Q ± ϕ = 2 M ϕ . Therefore we would have (2 M − λ ) ϕ = 0which implies ϕ ≡ N . Proposition 2.15.
For Re q > the operator N : L ( m q ) → L ( m q ) isnuclear (and hence of the trace class). Its spectrum is given by σ ( N ) = { } ∪ n ( − k α q + k ) o k ≥ (2.30) where α = ( √ − / is the golden mean. Each eigenvalue λ k ∈ σ ( N ) issimple and the corresponding (normalised) eigenfunction ψ k is given by ψ k ( t ) = s q k !Γ( k + 2 q ) L pk ( √ t ) exp ( − αt ) (2.31)13 orollary 2.16. For Re q > it holds tr( N ) = 1 √ α p and k N k = α q < Proof of Proposition 2.15.
Expanding the kernel of N (see (2.16)) on thebasis ( e n ) n ≥ , one get (see [Sze], p.102) J p (cid:0) √ st (cid:1) ( st ) p/ = ∞ X n =0 e n ( s ) e − t t n Γ( n + 2 q )This yields N ϕ = X n ≥ ( ϕ, e n ) g n where g n ( t ) = N e n ( t ) = e − t t n /n !. Since k e n k = r Γ( n + 2 q ) n ! k g n k = p Γ(2 n + 2 q ) n ! 3 n + q we have X n k e n k k g n k < ∞ and therefore N is nuclear. To compute the spectrum of N we use thefollowing Hankel transform (see [E], vol. II) Z ∞ x p + e − bx L pk ( ax ) J p ( xy ) √ xy dx = ( b − a ) k y p + p +1 b p + k +1 e − y b L pk (cid:18) ay b ( a − b ) (cid:19) which can be recast in terms of the operator N as N h L pk (2 at ) e − (2 b − t i = ( b − a ) k q b q + k e − t b L pk (cid:18) at b ( a − b ) (cid:19) This becomes an eigenvalue equation in L ( m q ) provided 2 b = α − and2 a = √
5. The normalisation constant results from (2.11) noting that k L pk ( √ t ) exp ( − αt ) k = 15 q k L pk ( t ) k This gives the eigenfunctions ψ k , and the proof is complete.We now put together the previous results. We have seen that for all q ∈ (0 , ∞ ) the operators P ± = M ± N when acting on L ( m q ) are self-adjointand positive with k M k = 1 and k N k = α q .14he operator M is spectrally absolutely continuous ([Ka], p.520). Its spec-trum, being the essential range of the multiplying function, coincides with[0 , ( m q ) = H ac ( M ) ⊕ H s ( M ) of the Hilbert space into the subspace of absolute continuity H ac ( M ) =Π ac ( M )L ( m q ) and that of singularity H s ( M ) = Π s ( M )L ( m q ), we haveH s ( M ) = 0 (and thus Π ac ( M ) = I ).On the other hand N q is of the trace class. Therefore, applying the Kato-Rosenblum theorem (see [Ka], p.542, or [RS], vol. III, p.26), it holds Proposition 2.17.
The operator M is unitarily equivalent to the spectrallyabsolutely continuous part of P ± . Hence on L ( m q ) we have σ ac ( P ± ) =(0 , . Remark 2.18.
Such equivalence is gained by means of the one-parameterfamily of unitary operators W ( τ ) = e iτP e − iτM − ∞ < τ < ∞ The (strong) limits W ± of W ( τ ) as τ → ±∞ are called wave operators and S = W ∗ + W − the scattering operator , which is unitary on L ( m q ) to itselfand commutes with M . The Kato-Rosenblum theorem says that in this casethe wave operators W ± exist and are complete, meaning that they are partialisometries with initial domain L ( m q ) and range H ac ( P ) = Π ac ( P )L ( m q ) .Therefore we have W ∗± W ± = I , W ± W ∗± = Π ac ( P ) and P W ± = W ± M (see[RS], vol. III, pp. 17-19). Putting together Proposition 2.6, Corollary 2.14 and Proposition 2.17, weget Theorem 1.1. L ( R + ) Given a continuous function φ on R + and q ∈ C , with Re q > p > − J φ = N M − φ considered in Section 2.2 can be viewed asa version of its Hankel transform , i.e.
J φ ( t ) := Z ∞ J p (2 √ st ) (cid:16) st (cid:17) p/ φ ( s ) ds (3.1)We can also define the conjugate transform ˜ J as˜ J := χ q J χ − p (3.2)15r else ˜ J φ ( t ) = Z ∞ J p (2 √ st ) (cid:18) ts (cid:19) p/ φ ( s ) ds (3.3)From the asymptotic estimates on J p ( t ) we see that the conditions on φ sufficient to give the absolute convergence of the integral (3.1) are φ ( t ) = O ( t − a ) as t → + and φ ( t ) = O ( t − b ) as t → ∞ with a < q and b > Re q + . For the integral (3.3) we have the same conditions with b > − q and a < J q f = ± f for f = B q [ ϕ ] can be rephrased as a self-reciprocity property for the functions ϕ and ψ := χ p · ϕ , that is J q f = ± f = ⇒ J ϕ = ± ϕ and ˜ J ψ = ± ψ (3.4) Lemma 3.1. If ϕ ∈ L ( R + ) then ϕ ∈ L ( m q ) ∩ L ( R + ) provided Re p ≥ .Conversely, if ϕ ∈ L ( m q ) and J ϕ = ± ϕ then ϕ ∈ L ( R + ) . Proof.
The first implication is immediate. The second follows from theasymptotic estimates on J p ( t ).Therefore, we shall study self-reciprocal functions in L ( R + ). Moreover, bya change of variables the conditions (3.4) can be recast in the form that thefunction φ ( t ) = 2 − q + t p + ϕ (cid:18) t (cid:19) = 2 q − t − p + ψ (cid:18) t (cid:19) (3.5)satisfies K φ = ± φ where K is the symmetric version of the Hankel trans-form given by Kφ ( t ) := Z ∞ J p ( st ) √ st φ ( s ) ds. (3.6)For Re p > − K φ = φ is φ ( t ) = √ t − whichcorresponds to ϕ ( t ) = t − q and ψ ( t ) = t q − . This solution has been alreadyconsidered above and does not belong to L ( R + ). We refer to [Tit], Chap.9,for an analysis of the equation K φ = φ in L (0 , ∞ ).For a >
0, let S a : L ( R + ) → L ( R + ) be given by ( S a ϕ )( t ) := a q ϕ ( at ).Then J S a = S /a J . In particular, since J e − t = e − t we have that a q e − at and a − q e − t/a is a Hankel transform pair for all a >
0. Now, the operator ˜ J isan adjoint to J in the sense that < ψ, J ϕ > = < ˜ J ψ, ϕ > with < φ , φ > := R ∞ φ ( t ) φ ( t ) dt . Whence, the identity Z ∞ a − q e − t/a ψ ( t ) dt = Z ∞ a q e − at ψ ( t ) dt , a > ψ and ψ is a pair w.r.t the Hankel transform ˜ J . Ifmoreover ˜ ψ is another Hankel transform of ψ then R ∞ e − at ( ψ − ˜ ψ ) dt = 0for all a > ψ = ˜ ψ almost everywhere. Therefore the identity(3.7) is a necessary and sufficient condition for ψ and ψ to be a pair w.r.tthe Hankel transform ˜ J . Let moreover ψ ∗ ( s ) := Z ∞ ψ ( t ) t s − dt (3.8)be the Mellin transform of ψ . If there are two constants a < b such that ψ ( t ) = O ( t − a ) as t → + and ψ ( t ) = O ( t − b ) as t → ∞ then the integral(3.3) converges for s in the strip a < Re s < b and ψ ∗ ( s ) is a holomorphicfunction in this strip. Remark 3.2. If P q + f = λ f then one easily checks that λ = 1 + f (1) f (0) , λ λ −
1) = f (2) f (0) Thus, if λ = 1 we have f (0) = 0 and f ( x ) ∼ f (0) x − q , x → ∞ Therefore, if Re q > then the Mellin transform f ∗ is analytic in the strip < Re s < q and in this region it holds ( J q f )( x ) = f ( x ) = ⇒ f ∗ ( s ) = f ∗ (2 q − s ) . Now, taking the Mellin transform of both sides in (3.7) we obtainΓ(1 − s ) ψ ∗ ( s ) = Γ( s + p ) ψ ∗ (1 − p − s )Note that if ψ = χ p · ϕ , then ψ ∗ ( s ) = ϕ ∗ ( s + p ). Moreover, Mellin trans-forming (3.5) gives φ ∗ ( s ) = 2 s − ϕ ∗ (cid:0) s + p + (cid:1) = 2 s − ψ ∗ (cid:0) s − p + (cid:1) Therefore, if we define the weighted transforms ¯ ϕ ∗ , ˜ ψ ∗ and ˆ φ ∗ as¯ ϕ ∗ ( s ) := ϕ ∗ ( s )Γ( s ) , ˜ ψ ∗ ( s ) := ψ ∗ ( s )Γ( s + p ) , ˆ φ ∗ ( s ) := 2 p +1 φ ∗ ( s )Γ (cid:0) s + p + (cid:1) and taking into account that 1 + p − (cid:0) s + p + (cid:1) = − s + p + , we havethe following result: 17 roposition 3.3. The functions ϕ, ψ, φ ∈ L ( R + ) , related to each other by(3.5), are jointly self-reciprocal, i.e. J ϕ = ± ϕ , ˜ J ψ = ± ψ and Kφ = ± φ ,if and only if ¯ ϕ ∗ ( s ) = ± ¯ ϕ ∗ (1+ p − s ) , ˜ ψ ∗ ( s ) = ± ˜ ψ ∗ (1 − p − s ) , ˆ φ ∗ ( s ) = ± ˆ φ ∗ (1 − s )The sequences h ± n introduced in (2.26) were our first example of self-reciprocalfunctions in L ( R + ), in that J h ± n = ± h ± n for all n ≥
0. Even more in-teresting self-reciprocal functions are provided by the conjugate sequences ϕ n , ψ n ∈ L ( R + ), n ≥
0, defined for Re p > − ϕ n ( t ) := q p +1 n !Γ( n + p +1) e − t L pn (2 t ) , ψ n ( t ) := ( χ p · ϕ n )( t ) , (3.9)and satisfying the condition < ϕ n , ψ m > = δ n,m . They are related to thesequences h ± n by (see [E], vol. II, p.192) ϕ n = ( − n q p +1 n !Γ( n + p +1) n X m =0 (cid:18) n + pn − m (cid:19) ( − m (cid:18) h + m + h − m (cid:19) Thus
J ϕ n = ( − n q p +1 n !Γ( n + p +1) n X m =0 (cid:18) n + pn − m (cid:19) ( − m (cid:18) h + m − h − m (cid:19) which, compared to (2.10), yields J ϕ n = ( − n ϕ n , ˜ J ψ n = ( − n ψ n . (3.10)Note that B q [ ϕ n ]( x ) = ( n + 1) p (1 − x ) n (1 + x ) n +2 q (3.11)so that J q B q [ ϕ n ] = ( − n B q [ ϕ n ], as expected (compare to (2.28)).Moreover ¯ ϕ ∗ n ( s ) = ( p + 1) n n ! F ( − n , s ; p + 1 ; 2) (3.12)which satisfies the functional equation of Proposition 3.3 because of Pfaff’sidentity ([AAR], Theorem 2.2.5) which implies F ( − n , b ; c ; 2) = ( − n F ( − n , c − b ; c ; 2) . Finally, the orthonormal family { φ n } of L ( R + ) given by φ n ( t ) := q n !Γ( n + p +1) e − t / t p + L pn ( t ) (3.13)18atisfies Kφ n = ( − n φ n , n ≥ . (3.14)Thus, the families ϕ n , ψ n , φ n furnish a complete characterization of self-reciprocal functions in L ( R + ) for the Hankel transforms J, ˜ J , K . Remark 3.4.
The functions φ n are also solutions of the differential equa-tion: φ ′′ n − (cid:18) p − / t + t − n − p − (cid:19) φ n = 0 (3.15) as one can check using, e.g., [E], vol. II, p.188. More specifically, the secondorder differential operator H given by H := 12 (cid:18) − d dt + p − / t + t (cid:19) (3.16) has for real p ≥ a unique self-adjoint extension on C ∞ ( R + ) which has aninteger-spaced spectrum so that H φ n = (2 n + p + 1) φ n , n ≥ . (3.17) For − < p < there is more than one self-adjoint extension, one of which,however, still satisfies (3.17). Comparing (3.14) and (3.17) one may regardthe unitary mapping K of L ( R + ) onto itself as a hyperdifferential operatorof the form ( q = p + 1 ) K = e iπq exp ( − iπ H ) (3.18) and acting on a suitable class of analytic functions (see [Bar] and [Wo] fora discussion on this and related correspondences). P ± q for q = − k/ . Although the eigenfunction f ( q ) ( x ) corresponding to the leading eigenvalue λ ( q ) does not belong to the space H q (see Remark 1.2), we shall see thatexplicit expressions for λ ( q ) and f ( q ) ( x ) can be obtained when q = − k/ k a non-negative integer. Note that these values correspond exactlyto the simple poles of Γ(2 q ) and thus, by (2.8), to the q -values where the In quantum mechanics this corresponds to the Schr¨odinger operator for a two-dimensional isotropic harmonic potential (see [RS], vol. II, p.161). m q has an infinite mass. On the other hand, for q = − k/ P ± q take the form P ±− k f ( x ) = ( x + 1) k (cid:20) f (cid:18) xx + 1 (cid:19) ± f (cid:18) x + 1 (cid:19)(cid:21) so that they leave invariant the vector space ⊕ kn =0 C x n of polynomials ofdegree ≤ k . In particular we expect f ( − k ) ( x ) is a polynomial of degree k with real coefficients.To warm up, a direct calculation yields f (0) ( x ) = 1 , λ (0) = 2 ,f ( − ) ( x ) = x + 1 , λ ( − ) = 3 ,f ( − ( x ) = x + √ − x + 1 , λ ( −
1) = √ ,f ( − ) ( x ) = x + 2 x + 2 x + 1 , λ ( − ) = 7 ,f ( − ( x ) = x + √ − x + 3 x + √ − x + 1 , λ ( −
2) = √ To say more we first need the following result.
Lemma 4.1.
The ( k + 1) × ( k + 1) real positive matrix M k defined as M k ( i, j ) := (cid:0) k − ij − i (cid:1) if i < j if i = j (cid:0) ij (cid:1) if i > j (0 ≤ i, j ≤ k ) has the following properties:1. the symmetry M k ( i, j ) = M k ( k − i, k − j ) holds for all ≤ i, j ≤ k ;2. the sum S i of the entries in row i equals S i = 2 i + 2 k − i ;3. the sum R j of the entries in column j equals R j = (cid:0) k +2 j +1 (cid:1) ;4. if M k Φ = λ Φ with C k +1 ∋ Φ := ( b , b , · · · , b k ) T and λ = 0 then Φ is either a palindrome or a skew-palindrome, i.e. b i = ± b k − i for ≤ i ≤ k ;5. σ ( M k ) ⊂ R for all k ∈ N ∪ { } ;6. ∈ σ ( M k ) for all k ∈ N . roof. λ b i = k X j =0 M ( i, j ) b j (0 ≤ i ≤ k )which yields, using the symmetry 1), λ b k − i = k X j =0 M ( k − i, j ) b j = k X j =0 M ( k − i, k − j ) b k − j = k X j =0 M ( i, j ) b k − j so that M k Φ = λ Φ if and only if M k Φ ′ = λ Φ ′ with Φ ′ := ( b k , b k − , · · · , b ) T .If λ is (geometrically) simple then Φ = ± Φ ′ because k Φ k = k Φ ′ k where k k is the euclidean norm in C k +1 . In other words, Φ = ± Φ ′ is a necessary andsufficient condition for Φ and Φ ′ to be linearly dependent. Assume now that M k Φ = λ Φ with λ of geometric (and thus algebraic) multiplicity bigger than1. If Φ and Φ ′ are linearly dependent then we are done. Suppose they arenot. Then the vectors Ψ ± := Φ ± Φ ′ should also be two linearly independenteigenvectors. But this is impossible because Ψ ′± = ± Ψ ± . This concludesthe proof of 4).As for 5) we observe that M is symmetric and for each k ≥ k + 1) × ( k + 1) matrix N k such that the product M k N k is symmetric aswell. For instance with k = 4 one gets M = N = Then apply Theorem 1 in [DH]. Finally, the vector Φ = (1 , , · · · , , − T always satisfies M k Φ = Φ, which yields 6).
Remark 4.2.
If one defines a pseudo-scalar product of
Φ = ( b , b , · · · , b k ) T and Ψ = ( c , c , · · · , c k ) T as < Φ , Ψ > := P ki =0 b i c k − i then the symmetrystated in 1) amounts to < M k Φ , Ψ > = < Φ , M Tk Ψ > . Moreover, 4) impliesthat if M k Φ = λ Φ with λ = 0 then < Φ , Φ > = ±k Φ k . Theorem 4.3.
Let q = − k/ , k ≥ . The polynomial f ( x ) = k X i =0 (cid:18) ki (cid:19) b i x i (4.1)21 atisfies P ± q f = λf with λ = 0 if and only if the vector Φ = ( b , b , · · · , b k ) T satisfies M k Φ = λ Φ and is either a palindrome (if P + q f = λf ) or a skew-palindrome (if P − q f = λf ). Corollary 4.4.
The eigenvector corresponding to the simple positive max-imal eigenvalue λ ( − k/ of M k is always palindromic and we have thebounds: s − h ≤ λ ≤ S − g where S := max i S i = 2 k + 1 , s := min i S i = k +1 + 2 k − , k even k +12 + 2 k − , k oddand h = − s + 2 + p s + 4( S − s )2 , g = S − p S − S − s )2( s − Proof.
Put together the above and [MM], p.155, eq.(9).
Proof of Theorem 4.3.
Setting f ( x ) = a k x k + a k − x k − + · · · + a x + a (4.2)the conditions J q f = ± f implies that the sequence of coefficients a i is eithera palindrome or a skew-palindrome, i.e. a i = ± a k − i , (0 ≤ i ≤ k ). Insertingthe function f ( x ) written above into (2.24) with q = − k/ k ≥ λ k X i =0 a i x i = k X i =0 a i i X j =0 (cid:18) ij (cid:19) ( x j ± x k − j )= k X j =0 k X l = j (cid:18) lj (cid:19) a l ( x j ± x k − j ) (4.3)= k X i =0 " k X l = i (cid:18) li (cid:19) a l ± k X l = k − i (cid:18) lk − i (cid:19) a l x i which in both cases yields λ a i = i X l =0 (cid:18) k − lk − i (cid:19) a l + k X l = i (cid:18) li (cid:19) a l (0 ≤ i ≤ k ) (4.4)22efining new coefficients b i so that a i = (cid:18) ki (cid:19) b i (0 ≤ i ≤ k ) (4.5)and using the identities (cid:18) k − lk − i (cid:19)(cid:18) kl (cid:19) = (cid:18) ki (cid:19)(cid:18) il (cid:19) and (cid:18) li (cid:19)(cid:18) kl (cid:19) = (cid:18) ki (cid:19)(cid:18) k − il − i (cid:19) we see that the above recursion becomes λ b i = i X l =0 (cid:18) il (cid:19) b l + k X l = i (cid:18) k − il − i (cid:19) b l (0 ≤ i ≤ k ) (4.6)and the proof is complete. Example 4.5.
For k = 4 we find sp( M ) = n √ , , −√ , − , − o and the corresponding eigenvectors are Φ = √ − / √ − Φ = − Φ = √ / √ Φ = − Φ = − Therefore the spectrum of M yields three eigenfunctions for P + − : h ( x ) = x + √ − x + 3 x + √ − x + 1 h ( x ) = x + √
113 + 14 x + 3 x + √
113 + 14 x + 1 and h ( x ) = − x + 12 x − nd two eigenfunction for P −− : h ( x ) = x − and h ( x ) = 4 x (1 − x ) . Remark 4.6.
Eigenvectors of M k to the eigenvalue are related to theperiod functions for the modular group (see [CM]). In particular, for k ∈ N the eigenvectors (1 , , · · · , , − T correspond to the fixed functions x k − of P −− k/ which yield the even part of the period functions corresponding toholomorphic Eisenstein forms of weight k + 2 . The odd parts are computedbelow in Proposition 4.7. Other linearly independent (skew-palindromic andpalindromic) eigenvectors with eigenvalue are expected for k ≥ , as theyare related to (even and odd part of ) holomorphic cusp forms [A]. For the sake of completeness we end with the following result, a version ofwhich is contained in [CM].
Proposition 4.7.
Let B m denote the m -th Bernoulli number. For k ∈ N ∪ { } the function f k ( x ) ∈ ⊕ k +1 n = − C x n given by f k ( x ) := ζ ( − k )2 (1 + x k ) + ( − k k ! X − ≤ n ≤ k +1 B n +1 B k +1 − n ( n + 1)!( k + 1 − n )! x n satisfies P + − k/ f k = f k for k even and f k ≡ for k odd. Two examples are f ( x ) = 112 (cid:20) x + 1 x − (cid:21) f ( x ) = 1360 (cid:20) x − (cid:18) x + 1 x (cid:19)(cid:21) Note that for k ≥
1, the odd parts of the period functions mentioned inRemark 4.6 can be expressed as ( − k k ! ( f k ( x ) − ζ ( − k )2 (1 + x k )) [Za1]. Proof.
Consider the function ψ q ( x ) defined for Re q > ψ q ( x ) = ζ (2 q )2 (1 + x − q ) + X n,m ≥ ( nx + m ) − q It is shown in [Za2] that the function ψ q ( x ) has an analytic extension into thecomplex q -plane with a simple pole at q = 1, and the analytic continuation24atisfies (2.24) with the sign + and λ = 1 for all q ∈ C \ { } . Note that ifRe q > ψ q ( ∞ ) = ζ (2 q ).The proof then amounts to show that for q = − k/ ψ q is precisely f k . This is achieved using standard Mellintransform techniques: start from the identity X n,m ≥ ( nx + m ) − q = 1Γ(2 q ) Z ∞ X n,m ≥ e − t ( nx + m ) t q − dt = 1Γ(2 q ) Z ∞ t q − ( e t − e tx − dt Recalling that 1 e t − ∞ X r = − B r +1 ( r + 1)! t r = 1 t −
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