A recursive construction of joint eigenfunctions for the commuting - PowerPoint PPT Presentation

A recursive construction of joint eigenfunctions for the commuting hyperbolic Calogero-Moser Hamiltonians (Joint work with M. Hallnäs) Simon Ruijsenaars School of of Mathematics University of Leeds, UK Budapest, 4 April 2014

Introduction Reminders ◮ The hyperbolic ( A N − 1 ) Calogero-Moser systems are integrable systems describing N particles on the line with hyperbolic pair interaction. ◮ The nonrelativistic quantum version is defined by the Hamiltonian N H = − � 2 � � ∂ 2 x j + g ( g − � ) V ( x j − x l ) , 2 j = 1 1 ≤ j < l ≤ N with � > 0 (Planck’s constant), g ∈ R (coupling constant), and pair potential V ( x ) = µ 2 / 4 sinh 2 ( µ x / 2 ) , µ > 0 . ◮ The N = 2 Schrödinger equation can be solved via the conical function, a specialization of the Gauss hypergeometric function.

◮ Associated integrable system ( N commuting PDOs): N � � � ∂ x j , H 2 = − � 2 H 1 = − i � ∂ x j 1 ∂ x j 2 − g ( g − � ) V ( x j − x l ) , j = 1 1 ≤ j 1 < j 2 ≤ N 1 ≤ j < l ≤ N � H k = ( − i � ) k ∂ x j 1 · · · ∂ x jk + l . o ., k = 3 , . . . , N , 1 ≤ j 1 < ··· < j k ≤ N where l.o. = lower order in partials. Thus, the defining Hamiltonian is given by H = H 2 1 / 2 − H 2 . ◮ Integrable versions exist for Lie algebras B N , . . . , E 8 , F 4 , G 2 (Olshanetsky/Perelomov, Oshima) and BC N (Inozemtsev, Oshima). ◮ N > 2 eigenfunctions: Harish-Chandra, Heckman/Opdam, Felder/Varchenko, Chalykh,...

Introduction Goal ◮ By proceeding recursively in N , construct joint eigenfunctions Ψ N ( x , p ) of the Hamiltonians H k : H k ( x )Ψ N ( x , p ) = S k ( p )Ψ N ( x , p ) , k = 1 , . . . , N , where � S k ( p ) = p j 1 · · · p j N . 1 ≤ j 1 < ··· < j k ≤ N ◮ For convenience, we rewrite Ψ N ( x , p ) as Ψ N ( g ; ( x 1 , . . . , x N ) , ( p 1 , . . . , p N )) = W N ( g / � ; µ x / 2 ) 1 / 2 × F N ( g / � ; ( µ x 1 / 2 , . . . , µ x N / 2 ) , ( 2 p 1 / � µ, . . . , 2 p N / � µ )) with � [ 4 sinh 2 ( t j − t k )] λ . W N ( λ ; t ) ≡ 1 ≤ j < k ≤ N

Introduction Main results ◮ Assuming Re λ ≥ 1, u ∈ R N and | Im t j | < π/ 2, we obtain N − 1 1 ≤ j < k ≤ n [ 4 sinh 2 ( t nj − t nk )] λ � � � F N ( λ ; t , u ) = � λ n ! � n + 1 � n � 2 cosh ( t n + 1 , j − t nk ) R N ( N − 1 ) / 2 n = 1 j = 1 k = 1     N n n − 1 N − 1 n � � � � � × exp  i u n t nj − t n − 1 , j dt nj ,    n = 1 j = 1 j = 1 n = 1 j = 1 where t Nj ≡ t j , j = 1 , . . . , N . ◮ This integral can also be written N � N − 1 � � � � � � exp iu N t j exp i ( u n − u n + 1 )( t n 1 + · · · + t nn ) t nn < ··· < t n 1 j = 1 n = 1 1 ≤ j < k ≤ n [ 2 sinh ( t nj − t nk )] 2 λ n � � × dt nj . � λ � n + 1 � n � 2 cosh ( t n + 1 , j − t nk ) j = 1 j = 1 k = 1

Introduction Tools A crucial ingredient is an explicit description of the eigenvalue equations for F N . ◮ The starting point consists of the Lax matrix i λ L ( t , u ) jk ≡ δ jk u j + ( 1 − δ jk ) sinh ( t j − t k ) and the diagonal matrix E ( t ) ≡ diag ( w 1 ( t ) , . . . , w N ( t )) with � w j ( t ) ≡ − i λ coth ( t j − t k ) . k � = j

◮ We let : ˆ Σ k ( L + E )( t ) : denote the normal-ordered PDOs obtained from the symmetric functions � Σ k ( L ( t , u ) + E ( t )) ≡ det ( L ( t , u ) + E ( t )) I I ⊂{ 1 ,..., N } | I | = k by performing the substitutions u j → − i ∂ t j , j = 1 , . . . , N . ◮ The Hamiltonians H k ( λ ; t ) ≡ ( 2 / � µ ) k H k ( λ � ; 2 t /µ ) are given by (S. R.) H k ( λ ; t ) = W ( t ) 1 / 2 : ˆ Σ k ( L + E )( t ) : W ( t ) − 1 / 2 . ◮ It follows that F N ( t , u ) should satisfy the eigenvalue equations : ˆ Σ k ( L + E )( t ) : F N ( t , u ) = S k ( u ) F N ( t , u ) , k = 1 , . . . , N .

Another key ingredient is given by so-called kernel functions. ◮ Given a pair of operators H 1 ( v ) and H 2 ( w ) , a kernel function is a function Ψ( v , w ) satisfying H 1 ( v )Ψ( v , w ) = H 2 ( w )Ψ( v , w ) . Here, v and w may vary over spaces of different dimension. ◮ There exist elementary kernel functions that connect the PDOs : ˆ Σ k ( L + E )( t ) : to a sum of PDOs in variables s 1 , . . . , s N − ℓ . (Langmann for k=2, Hallnäs/S. R. for k>2.) ◮ For ℓ = 1 this connection can be used to set up a recursive scheme yielding the above explicit integral representations of the joint eigenfunctions F N . ◮ For λ = 1 / 2 recursive H -eigenfunctions were previously found by Gerasimov/Kharchev/Lebedev, and for λ = − 1 , − 2 , . . . by Felder/Veselov. (Relation unclear to date.)

N = 2 case From N = 1 to N = 2 ◮ For N = 1 we set F 1 ( t , u ) ≡ exp ( itu ) , which obviously satisfies − i ∂ t F 1 ( t , u ) = uF 1 ( t , u ) . ◮ Now consider � F 2 ( λ ; t , u ) ≡ e iu 2 ( t 1 + t 2 ) ds K ♯ 2 ( λ ; t , s ) F 1 ( s , u 1 − u 2 ) R with kernel function 2 K ♯ � [ 2 cosh ( t j − s )] − λ . 2 ( λ ; t , s ) ≡ j = 1

◮ If Re λ > 0 and u ∈ R 2 , then the integrand decays exponentially as | s | → ∞ . It has singularities only at s = t j ± i π 2 ( 2 n + 1 ) , j = 1 , 2 , n ∈ N . ◮ Hence F 2 ( λ ; t , u ) is well defined as long as u ∈ R 2 , Re λ > 0 , and t ∈ C 2 satisfies | Im t j | < π/ 2 , j = 1 , 2 .

N = 2 case Holomorphy ◮ F 2 ( λ ; t , u ) has analytic continuation in ( λ, t ) to { λ ∈ C | Re λ > 0 } × { t ∈ C 2 || Im ( t 1 − t 2 ) | < π } . ◮ Follows via contour shifts: r t 2 + i π/ 2 r t 1 + i π/ 2 R + i η ✲ ✲ r t 2 − i π/ 2 r t 1 − i π/ 2 where we can choose η = Im ( t 1 + t 2 ) / 2. ◮ Can allow u ∈ C 2 such that | Im ( u 1 − u 2 ) | < 2 Re λ .

N = 2 case Eigenfunction property We claim that F 2 ( λ ; t , u ) is a joint eigenfunction of the PDOs Σ ( 2 ) : ˆ 1 ( L + E )( t ) := − i ( ∂ t 1 + ∂ t 2 ) , : ˆ Σ ( 2 ) 2 ( L + E )( t ) := − ∂ t 1 ∂ t 2 + λ coth ( t 1 − t 2 )( ∂ t 1 − ∂ t 2 ) + λ 2 . ◮ Key point: eigenfunction identity Σ ( 2 ) : ˆ 2 ( L + E )( t ) : K ♯ 2 ( t , s ) = 0 , and kernel identity : ˆ Σ ( 2 ) 1 ( L + E )( t ) : K ♯ 2 ( t , s ) =: ˆ Σ ( 1 ) 1 ( L + E )( − s ) : K ♯ 2 ( t , s ) = i ∂ s K ♯ 2 ( t , s ) . ◮ By analyticity, need only consider t ∈ R and λ > 0 (say).

To establish the eigenfunction property for : ˆ Σ ( 2 ) 1 ( L + E )( t ) : (e. g.), we use the following 7 steps. 1. Recall that � F 2 ( λ ; t , u ) ≡ e iu 2 ( t 1 + t 2 ) ds K ♯ 2 ( λ ; t , s ) F 1 ( s , u 1 − u 2 ) . R Σ ( 2 ) 2. Act with : ˆ 1 ( L + E )( t ) : , and shift through plane wave: � e iu 2 ( t 1 + t 2 ) : ˆ Σ ( 2 ) ds K ♯ 1 ( L + E + u 2 1 2 )( t ) : 2 ( λ ; t , s ) F 1 ( s , u 1 − u 2 ) . R 3. Note the expansion : ˆ Σ ( 2 ) 1 ( L + E + u 2 1 2 )( t ) :=: ˆ Σ ( 2 ) 1 ( L + E )( t ) : + 2 u 2 . 4. Act with PDO under the integral sign and invoke kernel identity: � Σ ( 1 ) e iu 2 ( t 1 + t 2 ) dsF 1 ( s , u 1 − u 2 ) : ˆ 1 ( L + E )( − s ) : K ♯ 2 ( λ ; t , s ) R � ds K ♯ + 2 u 2 e iu 2 ( t 1 + t 2 ) 2 ( λ ; t , s ) F 1 ( s , u 1 − u 2 ) . R

5. Recall that : ˆ Σ ( 1 ) 1 ( L + E )( − s ) := i ∂ s , and integrate by parts: � Σ ( 1 ) e iu 2 ( t 1 + t 2 ) ds K ♯ 2 ( λ ; t , s ) : ˆ 1 ( L + E )( s ) : F 1 ( s , u 1 − u 2 ) R � ds K ♯ + 2 u 2 e iu 2 ( t 1 + t 2 ) 2 ( λ ; t , s ) F 1 ( s , u 1 − u 2 ) . R 6. Use eigenfunction property for F 1 : � ds K ♯ ( u 1 − u 2 + 2 u 2 ) e iu 2 ( t 1 + t 2 ) 2 ( λ ; t , s ) F 1 ( s , u 1 − u 2 ) . R 7. Conclude that Σ ( 2 ) 1 ( L + E )( t ) : F 2 ( t , u ) = S ( 2 ) : ˆ 1 ( u ) F 2 ( t , u ) , where S ( 2 ) 1 ( a 1 , a 2 ) ≡ a 1 + a 2 . �

N = 2 case A bound ◮ Let u ∈ R 2 . For Re λ > 0 and | Im ( t 1 − t 2 ) | < π , we have the F 2 -bound | F 2 ( λ ; t , u ) | < C ( λ, | Im ( t 1 − t 2 ) | ) Re ( t 1 − t 2 ) × exp ( − Im ( t 1 + t 2 )( u 1 + u 2 ) / 2 ) sinh ( Re λ Re ( t 1 − t 2 )) . ◮ This bound readily follows from the integral evaluation � ds z δ =+ , − 2 cosh ( s + δ z / 2 ) = 2 sinh z . � R

Recursion scheme Kernel function ◮ The function N N − 1 K ♯ � � [ 2 cosh ( t j − s k )] − λ , N ( λ ; t , s ) ≡ N > 1 , j = 1 k = 1 satisfies the eigenfunction identity : ˆ Σ ( N ) N ( L + E )( t ) : K ♯ N ( t , s ) = 0 , and the kernel identities : ˆ Σ ( N ) ( L + E )( t ) : − : ˆ Σ ( N − 1 ) K ♯ � � ( L + E )( − s ) : N ( t , s ) = 0 , k < N . k k ◮ This connection between the N and N − 1 variable cases can be used to recursively construct the joint eigenfunctions F N of the N Σ ( N ) PDOs : ˆ ( L + E )( t ) : , k = 1 , . . . , N . k

Recursion scheme Formal structure ◮ Assume the function F N − 1 ( t , u ) , t , u ∈ C N − 1 , has been constructed. ◮ Then F N ( t , u ) , t , u ∈ C N , is formally given by P N F N ( t , u ) ≡ e iu N j = 1 t j � R N − 1 dsW N − 1 ( s ) K ♯ N ( t , s ) ( N − 1 )! × F N − 1 ( s , ( u 1 − u N , . . . , u N − 1 − u N )) . ◮ Have shown F N ( λ ; t , u ) is well defined for Re λ ≥ 1 and t ∈ C N such that | Im t j | < π/ 2 (and u ∈ R N ), and continues analytically to { λ ∈ C | Re λ > 1 } × { t ∈ C N | 1 ≤ j < k ≤ N | Im ( t j − t k ) | < π } . max