Connection Formula for Heine s Hypergeometric Function with q = 1 Ryu SASAKI Department of Physics, Shinshu University based on arxiv:1411.307[math-ph], J. Phys. A in 48 (015) 11504, with S. Odake nd Numazu Meeting, Numazu National College of Technology Numazu, 10 March 015 Sasaki Connection Formula for φ 1, q = 1
Outline Introduction 1 Introduction Exactly solvable Schrödinger equations Exactly solvable difference Schrödinger equations 3 4 Sasaki Connection Formula for φ 1, q = 1
Exactly solvable Schrödinger equations Exactly solvable difference Schrödinger equations Target: Exactly solvable difference Schrödinger eq. with finitely many discrete eigenstates Background Exactly Solvable Quantum Mechanics orthogonal eigenfunctions provides unified theory of classical orthogonal polynomials; ordinary Schrödinger eq. Hermite, Laguerre, Jacobi, difference Schrödinger eq. Wilson, Askey-Wilson, Racah, q-racah etc, so-called Askey-scheme of hypergeometric orthogonal polynomials cf. S. Odake & RS: Discrete Quantum Mechanics, J. Phys. A44 (011) 353001 Exactly solvable difference Schrödinger eq. with non-confining potentials non-complete set of eigenfunctions continuous spectrum scattering problem connection formulas for (q-) hypergeometric functions, q = 1 Sasaki Connection Formula for φ 1, q = 1
Exactly solvable Schrödinger equations Exactly solvable difference Schrödinger equations Exactly solvable ordinary Schrödinger equations 1 degree of freedom, Schrödinger operator (Hamiltonian) H = d dx + U(x): all eigenvalues E n and eigenfunctions φ n (x) are exactly calculable Hφ n (x) = E n φ n (x), n = 0, 1,..., confining potentials, having discrete eigenlevels: Harmonic oscillator, Hermite polynomial U(x) = x 1, < x <, E n = n, φ n (x) = φ 0 (x)h n (x), n=0,1..., φ 0 (x) = e x /, Radial oscillator, Laguerre polynomial U(x) = x + g(g 1)/x (1 + g), 0 < x <, E n = 4n, φ n (x) = φ 0 (x)l n (α) (x ), n=0,1..., α = g 1/ φ 0 (x) = e x / x g, Sasaki Connection Formula for φ 1, q = 1
Exactly solvable Schrödinger equations Exactly solvable difference Schrödinger equations Exactly solvable ordinary Schrödinger eqs confining potentials, having discrete eigenlevels: continued Pöschl-Teller, Jacobi polynomial g(g 1) h(h 1) U(x) = sin + x cos (g + h), 0 < x < π/ x E n = 4n(n + g + h), φ n (x) = φ 0 (x)p n (α,β) (cos x), n=0,1..., α = g 1/, β = h 1/, φ 0 (x) = (sin x) g (cos x) h, ground state eigenfunction squared φ 0(x) provides the orthogonality weight function no scattering problem! Sasaki Connection Formula for φ 1, q = 1
Exactly solvable Schrödinger equations Exactly solvable difference Schrödinger equations Exactly solvable difference Schrödinger eqs confining potentials, having discrete eigenlevels: continued 4 j=1 Wilson poly. V (x) = (a j + ix) ix(ix + 1), 0 < x <, Re(a j) > 0, singular at x = 0, x =, E n = n(n + b 1 1), b 1 = 4 a j, 4 j=1 Askey-Wilson poly. V (x) = (1 a je ix ) (1 e ix )(1 qe ix ), 0 < x < π, a j < 1, singular at x = 0,π, E n = (q n 1)(1 b 4 q n 1 ), b 4 = a 1 a a 3 a 4 j=1 Sasaki Connection Formula for φ 1, q = 1
Exactly solvable Schrödinger equations Exactly solvable difference Schrödinger equations Exactly solvable ordinary Schrödinger eqs, non-confining g(g 1) U(x) = sin, confining Gegenbauer poly x x π/ ix U(x) 1/cosh x, lim x ± U(x) = 0, non-confining, soliton potential, or so called soliton potential finitely many discrete eigenstates Sasaki Connection Formula for φ 1, q = 1
Exactly solvable Schrödinger equations Exactly solvable difference Schrödinger equations Exactly solvable ordinary Schrödinger eqs, non-confining non-confining potentials, having finite discrete eigenlevels: all eigenvalues E n and eigenfunctions φ n (x) and scattering amplitudes t(k), r(k) are exactly calculable scattering problem by connection formula for F 1 e ikx + r(k)e ikx t(k)e ikx U(x) ψ k (x) e ikx, x analytically continue to x ψ k (x) A(k)e ikx + B(k)e ikx, t(k) = 1/A(k), r(k) = B(k)/A(k), unitarity t(k) + r(k) = 1 Sasaki Connection Formula for φ 1, q = 1
Problem: Find the discrete analogues Exactly solvable Schrödinger equations Exactly solvable difference Schrödinger equations discrete analogues x π/ ix in q-ultraspherical poly with q = 1, polynomials are OK discrete reflection potentials analogues of 1/ cosh x potential for scattering problem connection formula for φ 1 is needed φ 1 with q = 1 unknown scattering amplitudes obtained from conjectured connection formula for φ 1 several supporting evidence presented Sasaki Connection Formula for φ 1, q = 1
Solutions for h(h + 1)/ cosh x potential U(x) = h(h + 1)/ cosh x, < x <, E n = (h n), (n = 0, 1,... [h] ), invariant h (h + 1) φ n (x) = ( cosh x) h+n P n (h n,h n) (tanh x), Gegenbauer or ultraspherical polynomial Γ( h ik)γ(1 + h ik) t(k) =, Γ( ik)γ(1 ik) Γ(ik)Γ( h ik)γ(1 + h ik) r(k) =, invariant h (h + 1) Γ( ik)γ( h)γ(1 + h) rewrite by Gauss hypergeometric function ( n, n + h + 1 φ n (x) = ( cosh x) h+n F 1 1 tanh x ) h n + 1 introduce k by n h + ik: ( h ik, 1 + h ik ( cosh x) ik F 1 1 tanh x ) e ikx, x + 1 ik Sasaki Connection Formula for φ 1, q = 1
Gaussian Connection formula Scattering amplitudes analytically continue to x by connection formula ( α, β ) F 1 z γ Γ(γ)Γ(α + β γ) ( γ α, γ β ) = (1 z) γ α β F 1 1 z Γ(α)Γ(β) γ α β + 1 Γ(γ)Γ(γ α β) ( + Γ(γ α)γ(γ β) α, β ) F 1 1 z, α + β γ + 1 positive integer h N, 1/Γ( h) = 0 r(k) = 0 reflectionless general reflectionless potential of Schrödinger eq. = profile of KdV soliton, last year s talk Sasaki Connection Formula for φ 1, q = 1
general difference Schrödinger eqs Schrödinger operator (Hamiltonian): H = V (x)v (x iγ) e iγ + V (x)v (x + iγ) e +iγ V (x) V (x), e ±iγ ψ(x) = ψ(x ± iγ), γ R, confining potentials, having discrete eigenlevels: 4 j=1 V (x) = (1 a je ix ) (1 e ix )(1 q e ix ), a j < 1, 0 < x < π, Askey-Wilson polynomial E n = (q n 1)(1 b 4 q n 1 ), 4 0 < q = e γ < 1, b 4 = a j, φ n (x) = φ 0 (x)p n (cos x), j=1 P n (cos x) = p n (cos x ; a 1, a, a 3, a 4 q) ( def q = a1 n n (a, b 4 q n 1, a 1 e ix, a 1 e ix ) q 1a, a 1 a 3, a 1 a 4 ; q) n 4 φ 3 ; q, a 1 a, a 1 a 3, a 1 a 4 Sasaki Connection Formula for φ 1, q = 1
general difference Schrödinger eqs, squared ground state wave function orthogonality weight n def function: (a; q) n = (1 aq j 1 ), j=1 φ 0 (x) = (e ix ; q) (e ix ; q) 4 ( (aj e ix ; q) (a j e ix ) 1 ; q) non-confining potentials, having finite discrete eigenlevels: all eigenvalues E n and eigenfunctions φ n (x) are exactly calculable j=1 Sasaki Connection Formula for φ 1, q = 1
Explicit form of discrete analogue of 1/ cosh x potential V (x; h) = e iγh (1 + eiγh e x )(1 + e iγ(h 1) e x ) (1 + e x )(1 + e iγ e x 1, ) x ±, H: invariant h (h + 1), < x < +, q = e iγ, lim γ 0 γ H = d h(h + 1) dx cosh x, Sasaki Connection Formula for φ 1, q = 1
Explicit form of discrete analogue of 1/ cosh x potential Discrete eigenstates E n = 4 sin γ (h n), n = 0, 1,..., [h]. φ n (x) = φ 0 (x)p n ( η ), η = sinh x P n (η) p n (i sinh x; e i γ h, e i γ (h 1), e i γ h, e i γ (h 1) e iγ ) : AW C n (i sinh x; β q), q = e iγ, q = 1, β def = e iγh = q h = e nx φ 1 ( q n, β β 1 q 1 n q ; β 1 qe x iπ), q-ultraspherical polynomial well-defined for q = 1, Heine s q-hypergeometric functions Sasaki Connection Formula for φ 1, q = 1
Discrete eigenstates, Quantum Dilog as weight function (a; q) does Not converge for q = 1, ground state eigenfunction by Quantum Dilog Φ γ (x): φ 0 (x) = e hx (1 + e x ) Φ ( γ x + iγ(h + 1 ( )) x iγ(h + 1 )), Φ γ quantum dilog, integral rep q-gamma function for q = 1 ( e izt dt ) Φ γ (z) = exp ( Im z < γ + π), R+i0 4 sinh γt sinh πt t functional relations Φ γ (z + iγ) Φ γ (z iγ) = 1 1 + e z, Φ γ (z + iπ) Φ γ (z iπ) = 1 1 + e π γ z, Sasaki Connection Formula for φ 1, q = 1
Reflectionless potentials in difference Schrödinger eqs Scattering problem, trivial potential V (x) 1, γ 0 limit: p = i x, H H 0 = e γp + e γp, H 0 e ±kx = Ẽ k e ±kx, Ẽ k def = 4 sin kγ < 0, H 0 e ±ikx = E s k e±ikx, E s k def = 4 sinh kγ > 0. lim γ 0 γ H 0 = p, lim γ Ẽ k = k, γ 0 lim γ Ek s = k, γ 0 Sasaki Connection Formula for φ 1, q = 1
General discrete reflectionless potentials Darboux transformation with seed solutions ψ j (x) = e k j x + c j e k j x = ψ j (x), 0 < k 1 < k < < k N π γ, H 0 ψ j (x) = Ẽ kj ψ j (x), ( 1) j 1 c j > 0, first step Darboux transformation H 0 =  1Â1 + Ẽ k1, N-step Darboux transformation  1 def = i ( e γ p ˆV 1 (x) e γ p ˆV 1 (x) ), ˆV 1 (x) def = ψ 1(x iγ). ψ 1 (x) H [N] =A [N] A [N] + Ẽ kn, A [N] def = i ( e γ p V [N] (x) e γ p V [N] (x) ), V [N] (x) def = W γ[ψ 1,..., ψ N 1 ](x iγ) W γ [ψ 1,..., ψ N ](x + i γ ) W γ [ψ 1,..., ψ N 1 ](x) W γ [ψ 1,..., ψ N ](x i γ ) Sasaki Connection Formula for φ 1, q = 1
General discrete reflectionless potentials right moving wave Ψ [N] k (x) (k > 0) H [N] Ψ [N] k (x) = Es k Ψ[N] (x), k Ψ [N] k (x) = W γ [ψ 1,..., ψ N, e ikx ](x) ( Wγ [ψ 1,..., ψ N ](x i γ )W γ[ψ 1,..., ψ N ](x + i γ )) 1 discrete eigenfunctions Φ [N] j (x) (j = 1,,..., N) H [N] Φ [N] j Φ [N] j (x) = (x) = Ẽ kj Φ [N] (x), j W γ [ψ 1,..., ψ j,..., ψ N ](x) (, Wγ [ψ 1,..., ψ N ](x i γ )W γ[ψ 1,..., ψ N ](x + i γ )) 1 Sasaki Connection Formula for φ 1, q = 1
General discrete reflectionless potentials 3 Casoratian W γ [f 1,..., f n ](x) def = i 1 n(n 1) det x (n) j ( f k ( x (n) j def = x + i( n+1 j)γ, ) ) 1 j,k n, lim γ 0 γ 1 n(n 1) W γ [f 1, f,..., f n ](x) = W [f 1, f,..., f n ](x), Casoratian of exponentials W γ [e k 1x, e k x,..., e knx ](x) = 1 i<j n sin γ (k j k i ) e P n j=1 k j x transmission and reflection amplitudes: N t [N] sin γ (k) = (ik k j) N sin γ (ik + k j) = sinh γ (k + ik j) sinh γ (k ik j), r [N] (k) = 0 j=1 j=1 Sasaki Connection Formula for φ 1, q = 1
Discrete analogue of 1/ cosh x potentials choose integer wavenumbers k j = j, reflectionless potential c j = ( 1) j 1 V [N] (x) = e iγn (1 + eiγn e x )(1 + e iγ(n 1) e x ) (1 + e x )(1 + e iγ e x ) ground state eigenfunction Φ [N] N (N 1 (x) = ) ( sin γ 1 N j j=1 j=1 V (x; N) 4 cosh(x i γ j) cosh(x + i γ j) ) 1 quantum dialog degenerates to elementary functions for h = N. Sasaki Connection Formula for φ 1, q = 1
Scattering Wavefunction scattering wave function is obtained by setting n h + ik in eigenfunction: Ψ k (x) def = e ikx e hx ( ( Φ 1 + e x γ x + iγ(h + 1 ( )) ) 1 x iγ(h + 1 )) Φ γ φ 1 ( q h ik, q h q 1 ik q ; q 1+h e x iπ), infinite series: convergence unclear for q = 1, Sasaki Connection Formula for φ 1, q = 1
Connection Formula φ 1, 0 < q < 1 connection formula for Heine s q-hypergeometric function known only for 0 < q < 1 (Watson 1910) φ 1 ( a, b c ) q ; z = (b, c/a; q) (c, b/a; q) (az, q/(az); q) (z, q/z; q) + (a, c/b; q) (c, a/b; q) (bz, q/(bz); q) (z, q/z; q) (a; q) does Not converge for q = 1, φ 1 ( a, aq/c aq/b φ 1 ( b, bq/c bq/a q ; cq ) abz q ; cq abz ), Sasaki Connection Formula for φ 1, q = 1
How to solve the scattering problem?? starting from H 0, construct general reflectionless potentials by Darboux transformations choose special integer wavenumbers to construct reflectionless analogue of difference 1/ cosh x potential. Calculate the transmission amplitude t(k). for the generic h, calculate the transmission and reflection amplitude based on conjectured connection formula of φ 1 check various properties: everything seems OK Sasaki Connection Formula for φ 1, q = 1
Scattering amplitudes scattering wave function n h + ik Ψ k (x) def = e ikx e hx ( ( Φ 1 + e x γ x + iγ(h + 1 ( )) x iγ(h + 1 )) Φ γ φ 1 ( q h ik, q h q 1 ik q ; q 1+h e x iπ), ) 1 exclude q root of unity, assume q = 1 is approached from below q 1 (a; q) replaced by quantum dilogarithm function (e z ; q) const./φ (+) γ (z), Φ (+) γ q = e iγ (z) def = Φ γ ( z + i γ + iπ), Sasaki Connection Formula for φ 1, q = 1
conjectured connection formula for φ 1 φ 1 ( e λ, e µ e ν q ; e z ) = Φ (+) γ Φ (+) γ + (ν)φ (+) γ (µ λ) (µ)φ (+) γ (ν λ) Φ (+) γ Φ (+) γ Φ (+) γ Φ (+) γ (z)φ (+) γ ( iγ z) (λ + z)φ (+) γ ( iγ λ z) φ 1 ( e λ, q e λ ν q e λ µ q ; q e ν λ µ z ) (ν)φ (+) γ (λ µ) (λ)φ (+) γ (ν µ) Φ (+) γ Φ (+) γ (z)φ (+) γ ( iγ z) (µ + z)φ (+) γ ( iγ µ z) φ 1 ( e µ, q e µ ν q e µ λ q ; q e ν λ µ z ) Sasaki Connection Formula for φ 1, q = 1
Scattering amplitudes Φ (+) Ψ k (x) e ikx e i γ h(h+1) γ ( kγ iγ)φ (+) γ ( kγ) Φ (+) γ ( kγ + iγh)φ (+) γ ( kγ iγ(h + 1)) Φ (+) + e ikx e i γ h(h+1)+ kγ (1 ik) γ ( kγ iγ)φ (+) γ (kγ) Φ (+) γ (iγh)φ (+) γ ( iγ(h + 1)). Scattering amplitudes t(k) = e i γ h(h+1) Φ γ ( kγ + iγ(h + 1 ) + iπ)φ γ ( kγ iγ(h + 1 ) + iπ) Φ γ ( kγ + i γ + iπ)φ γ ( kγ i γ + iπ), r(k) = e kγ (1 ik) Φ γ (kγ + i γ +iπ)φ γ ( kγ + iγ(h + 1 )+iπ)φ γ ( kγ iγ(h + 1 )+iπ) Φ γ ( kγ + i γ +iπ)φ γ (iγ(h + 1 ) + iπ)φ γ ( iγ(h + 1 ) + iπ) Sasaki Connection Formula for φ 1, q = 1
Scattering amplitude 4 t(k), r(k): invariant under h (h + 1) unitarity satisfied t(k) + r(k) = 1 h: integer: reproduces the reflectionless case t(k) = N j=1 sinh γ (k + i j) sinh γ r(k) = 0. (k i j), some more supporting evidence for the conjectured connection formula I do hope experts to provide an analytic proof of the connection formula. Sasaki Connection Formula for φ 1, q = 1
Further Challenge?? Introduction There are other difference analogues of exactly solvable potentials, Morse potential, hyperbolic Pöschl-Teller potential, etc. Do they lead to new problems?? new challenges??? Thank you!! Sasaki Connection Formula for φ 1, q = 1