Recent Developments in Exactly Solvable Quantum Mechanics Ryu - - PowerPoint PPT Presentation

▶
recent developments in exactly solvable quantum mechanics
SMART_READER_LITE
LIVE PREVIEW

Recent Developments in Exactly Solvable Quantum Mechanics Ryu - - PowerPoint PPT Presentation

Nov 14, 2022 135 likes •594 views

Introduction New Discovery General Recipe Multi-Indexed Orthogonal Polynomials Summary and Outlook Appendix Recent Developments in Exactly Solvable Quantum Mechanics Ryu SASAKI Department of Physics, National Taiwan University, Department

slide-1
SLIDE 1

Introduction New Discovery General Recipe Multi-Indexed Orthogonal Polynomials Summary and Outlook Appendix

Recent Developments in Exactly Solvable Quantum Mechanics

Ryu SASAKI

Department of Physics, National Taiwan University, Department of Physics, Shinshu University,

Colloquium Department of Physics, National Taiwan University Taipei (Taiwan), December 23, 2014

Sasaki New Orthogonal Polynomials

slide-2
SLIDE 2

Introduction New Discovery General Recipe Multi-Indexed Orthogonal Polynomials Summary and Outlook Appendix

Outline

1

Introduction

2

New Discovery

3

General Recipe Ordinary Quantum Mechanics

4

Multi-Indexed Orthogonal Polynomials Exceptional Jacobi Polynomials

5

Summary and Outlook

6

Appendix Fuchsian Differential Equations References

Sasaki New Orthogonal Polynomials

slide-3
SLIDE 3

Introduction New Discovery General Recipe Multi-Indexed Orthogonal Polynomials Summary and Outlook Appendix

Exactly Solvable Quantum Mechanics

1-d QM, given a Hamiltonian H = − d2 dx2 + U(x), x1 < x < x2, U(x) ∈ C∞,

  • Eigenvalue problem

Hφn(x) = Enφn(x), n = 0, 1, 2, . . . , x2

x1

φ2

n(x)dx < ∞,

  • all the discrete eigenvalues {En} and the corresponding

eigenfunctions {φn(x)} are exactly calculable ⇒ Exactly Solvable Quantum Mechanics

Sasaki New Orthogonal Polynomials

slide-4
SLIDE 4

Introduction New Discovery General Recipe Multi-Indexed Orthogonal Polynomials Summary and Outlook Appendix

Typical examples of exactly solvable QM I

  • harmonic oscillator, H = − d2

dx2 + x2 − 1, −∞ < x < +∞, En = 2n, φn(x) = φ0(x)Hn(x): Hermite polynomial, φ0(x) = e−x2/2, +∞

−∞

φ2

0(x)Hn(x)Hm(x)dx ∝ δn m

  • radial oscillator, H = − d2

dx2 + x2 + g(g − 1) x2 − (1 + 2g), 0 < x < +∞, g > 1, En = 4n, φn(x) = φ0(x)L(g−1/2)

n

(x2):Laguerre polynomial, φ0(x) = e−x2/2xg, +∞ φ2

0(x)L(g−1/2) n

(x2)L(g−1/2)

m

(x2)dx ∝ δn m

Sasaki New Orthogonal Polynomials

slide-5
SLIDE 5

Introduction New Discovery General Recipe Multi-Indexed Orthogonal Polynomials Summary and Outlook Appendix

Typical examples of exactly solvable QM II

  • P¨
  • schl-Teller potential,

H = − d2 dx2 + x2 + g(g − 1) sin2 x + h(h − 1) cos2 x − (g + h)2, 0 < x < π/2, g > 1, h > 1, En = 4n(n + g + h), φn(x) = φ0(x)P(g−1/2,h−1/2)

n

(cos 2x):Jacobi polynomial, φ0(x) = (sin x)g(cos x)h, π/2 φ2

0(x)P(g−1/2,h−1/2) n

(cos 2x)P(g−1/2,h−1/2)

m

(cos 2x)dx ∝ δn m

Sasaki New Orthogonal Polynomials

slide-6
SLIDE 6

Introduction New Discovery General Recipe Multi-Indexed Orthogonal Polynomials Summary and Outlook Appendix

Motivations for exactly solvable QM I

1 cornerstones of modern quantum physics 2 Heisenberg operator formalism 1

creation, annihilation operators

2

coherent states

3

dynamical symmetry algebras

3 Schr¨

  • dinger eq. i.e. eigenvalue problem of a self-adjoint

Hamiltonian real eigenvalues and mutually orthogonal eigenfunctions → unified framework of classical orthogonal polynomials

4 orthogonality weight function = φ2

0(x): square of the ground

state eigenfunction

Sasaki New Orthogonal Polynomials

slide-7
SLIDE 7

Introduction New Discovery General Recipe Multi-Indexed Orthogonal Polynomials Summary and Outlook Appendix

Motivations for exactly solvable QM II

new exactly solvable QM ⇒ new orthogonal polynomials with your name on? like Hermite, Laguerre or Jacobi? (My na¨ ıvest motivation for this research) Not so⇐ = Bochner’s Theorem

  • rthogonal polynomials satisfying second order differential

equations are Classical orthogonal polynomials; Hermite, Laguerre, Jacobi & Bessel

Sasaki New Orthogonal Polynomials

slide-8
SLIDE 8

Introduction New Discovery General Recipe Multi-Indexed Orthogonal Polynomials Summary and Outlook Appendix

Bochner’s Theorem ’29

If polynomials {pn(x)} satisfy three term recurrence relations and a second order differential equation σ(x)y′′ + τ(x)y′ + λny = 0, they must be one of the Classical orthogonal polynomials, i.e., the Hermite, Laguerre, Jacobi and Bessel. For y = p0(x) =const, ⇒ λ0 = 0. For y = p1(x) ⇒ degree(τ(x)) ≤ 1. For y = p2(x) ⇒ degree(σ(x)) ≤ 2. deg(σ(x)) = 2, two equal roots (x = 0) ⇒ Bessel deg(σ(x)) = 2, two distinct roots (x = ±1) ⇒ Jacobi deg(σ(x)) = 1, root at x = 0 ⇒ Laguerre deg(σ(x)) = 0 ⇒ Hermite

Sasaki New Orthogonal Polynomials

slide-9
SLIDE 9

Introduction New Discovery General Recipe Multi-Indexed Orthogonal Polynomials Summary and Outlook Appendix

Avoiding Bochner’ constraints

polynomials satisfying difference Schr¨

  • dinger equation

differential eq. ⇒ difference eq. ⇒ Wilson, Askey-Wilson, Racah, q-Racah polynomials polynomials having holes (three term recurrence is broken) in the degree

polynomials starting at degree ℓ ≥ 1

(completeness not obvious ⇒ experts did not think this

  • ption)

polynomials satisfying difference Schr¨

  • dinger equation and

starting at degree ℓ ≥ 1 and having holes in the degree

Sasaki New Orthogonal Polynomials

slide-10
SLIDE 10

Introduction New Discovery General Recipe Multi-Indexed Orthogonal Polynomials Summary and Outlook Appendix

Discovery of ∞ Multi-Indexed Orthogonal Polynomials

Infinitely many orthogonal polynomials satisfying second order differential equations, discovered after Hermite, Laguerre and Jacobi polynomials (Gomez-Ullate,Kamran,Milson, Quesne, ’08, Odake-RS ’09 and others) Multi-Indexed orthogonal polynomials PD,n(x), D = {d1, . . . , dM}, dj ∈ N: degrees of polynomial type seed solutions (virtual state wave functions) employed by multiple Darboux transformations, (n counts nodes in (x1, x2)) x2

x1

PD,n(x)PD,m(x)WD(x)dx = hD,nδn m degree ℓ + n polynomial in x, but forming a complete set,

Sasaki New Orthogonal Polynomials

slide-11
SLIDE 11

Introduction New Discovery General Recipe Multi-Indexed Orthogonal Polynomials Summary and Outlook Appendix

Discovery of ∞ Multi-Indexed Orthogonal Polynomials II

No three term recurrence relations

main part of the eigenfunctions of exactly solvable Schr¨

  • dinger eq.

when eigenfunctions are employed, D = {d1, . . . , dM}, dj ∈ N: degrees of the holes global solutions of (confluent) Fuchsian differential equations with 3 + ℓ regular singularities, all the ℓ extra singularities are apparent and located outside of the orthogonality interval

Sasaki New Orthogonal Polynomials

slide-12
SLIDE 12

Introduction New Discovery General Recipe Multi-Indexed Orthogonal Polynomials Summary and Outlook Appendix Ordinary Quantum Mechanics

Basic Ingredients

Exactly Solvable Quantum Mechanical System Hφn(x) = Enφn(x), E0 = 0, n = 0, 1, 2, . . . , Factorised positive semi-definite Hamiltonian H = A†A ≥ 0 Multiple Darboux-Crum-Krein-Adler transformation Hψ(x) = Eψ(x), Hϕ(x) = ˜ Eϕ(x), ⇒ H(1)ψ(1)(x) = Eψ(1)(x), H(1) def = H − 2∂2

x log ϕ(x),

ψ(1)(x) def = ∂xψ(x) − ∂xϕ(x) ϕ(x) ψ(x) = W[ϕ, ψ](x) ϕ(x) , Virtual State solutions, H ˜ ϕv(x) = ˜ Ev ˜ ϕv(x), ˜ Ev < 0, ˜ ϕv(x) > 0, v ∈ V

Sasaki New Orthogonal Polynomials

slide-13
SLIDE 13

Introduction New Discovery General Recipe Multi-Indexed Orthogonal Polynomials Summary and Outlook Appendix Ordinary Quantum Mechanics

Factorised Hamiltonians

Starting point: H with complete set of eigenvalues and

eigenfunctions Hφn(x) = Enφn(x), (φn, φm) = hnδn m, hn > 0, n = 0, 1, 2, . . . , by adjusting the const. of H ⇒ E0 = 0 ⇒ Positive Semi-Definite Hamiltonian H (Hermitian Matrix) 0 = E0 < E1 < E2 < · · · , ⇒ H = A†A A = d/dx − ∂xφ0(x)/φ0(x), A† = −d/dx − ∂xφ0(x)/φ0(x), φ0(x): ground state wavefunction, no node (φ0(x) > 0), square integrable Aφ0(x) = 0 H = −d2/dx2 + V (x), V (x) = ∂2

xφ0(x)

φ0(x)

Sasaki New Orthogonal Polynomials

slide-14
SLIDE 14

Introduction New Discovery General Recipe Multi-Indexed Orthogonal Polynomials Summary and Outlook Appendix Ordinary Quantum Mechanics

Use virtual state solutions

rewrite H by using ˆ Ad1, d1 ∈ N, ( ˆ Ad1 annihilates ˜ ϕd1(x), ˆ Ad1 ˜ ϕd1(x) = 0): ˆ Ad1

def

= d/dx − ∂x log ˜ ϕd1(x), ˆ A†

d1 = −d/dx − −∂x log ˜

ϕd1(x), ˆ A†

d1 ˆ

Ad1 = − d2 dx2 + ˜ ϕ′

d1(x)

˜ ϕd1(x) 2 + d dx ˜ ϕ′

d1(x)

˜ ϕd1(x)

  • = − d2

dx2 + ˜ ϕ′′

d1(x)

˜ ϕd1(x) = − d2 dx2 + V (x) − ˜ Ed1, H = ˆ A†

d1 ˆ

Ad1 + ˜ Ed1, ˆ Ad1 : non-singular, define a new Hamiltonian by changing the order of ˆ Ad1 and ˆ A†

d1: H(1) d1 def

= ˆ Ad1 ˆ A†

d1 + ˜

Ed1 = H − 2∂2

x log ˜

ϕd1(x)

Sasaki New Orthogonal Polynomials

slide-15
SLIDE 15

Introduction New Discovery General Recipe Multi-Indexed Orthogonal Polynomials Summary and Outlook Appendix Ordinary Quantum Mechanics

new exactly solvable Hamiltonian

intertwining relation H(1)

d1 ˆ

Ad1 = ( ˆ Ad1 ˆ A†

d1 + ˜

Ed1) ˆ Ad1 = ˆ Ad1( ˆ A†

d1 ˆ

Ad1 + ˜ Ed1) = ˆ Ad1H H(1)

d1 : new exactly solvable isospectral Hamiltonian

φd1,n(x) def = ˆ Ad1φn(x) = W[ ˜ ϕd1, φn](x) ˜ ϕd1 , n = 0, 1, . . . , ˜ ϕd1,v(x) def = ˆ Ad1 ˜ ϕv(x) = W[ ˜ ϕd1, ˜ ϕv](x) ˜ ϕd1 , v ∈ D\d1 H(1)

d1 φd1,n(x) = Enφd1,n(x),

H(1)

d1 ˜

ϕd1,v(x) = ˜ Ev ˜ ϕd1,v(x), (φd1,n, φd1,m) = (φn, ˆ A†

d1 ˆ

Ad1φm) = (En − ˜ Ev)hnδn m

Sasaki New Orthogonal Polynomials

slide-16
SLIDE 16

Introduction New Discovery General Recipe Multi-Indexed Orthogonal Polynomials Summary and Outlook Appendix Ordinary Quantum Mechanics

new exactly solvable Hamiltonian 2

repeat M times by using virtual state solutions specified by D = {d1, d2, . . . , dM} ⇒ new exactly solvable Hamiltonian with multi-index D H(M)

D def

= H − 2∂2

x log W[ ˜

ϕd1, . . . , ˜ ϕdM](x) φD,n(x) def = W[ ˜ ϕd1, . . . , ˜ ϕdM, φn](x) W[ ˜ ϕd1, . . . , ˜ ϕdM](x) H(M)

D φD,n(x) = EnφD,n(x),

n = 0, 1, . . . , (φD,n, φD,m) =

M

  • j=1

(En − ˜ Edj) · hnδn m positive definite inner products En > 0, ˜ Edj < 0.

Sasaki New Orthogonal Polynomials

slide-17
SLIDE 17

Introduction New Discovery General Recipe Multi-Indexed Orthogonal Polynomials Summary and Outlook Appendix Exceptional Jacobi Polynomials

Example: P¨

  • schl-Teller potential⇒ Jacobi Polynomial

H = − d2 dx2 + U(x), U(x) = g(g − 1) sin2 x + h(h − 1) cos2 x − (g + h)2, regular sing. x = 0, g, 1 − g, x = π/2, h, 1 − h, λ = {g, h}, ground state wavefunct. φ0(x) = (sin x)g(cos x)h, g, h > 0, En(λ) = 4n(n + g + h), η(x) def = cos 2x φn(x; λ) = φ0(x)P(g−1/2,h−1/2)

n

(η(x)), Pn: Jacobi polynomial

Sasaki New Orthogonal Polynomials

slide-18
SLIDE 18

Introduction New Discovery General Recipe Multi-Indexed Orthogonal Polynomials Summary and Outlook Appendix Exceptional Jacobi Polynomials

Multi-Indexed Orthogonal Polynomials 1

P¨

  • schl-Teller potential has virtual state solutions, type I and

II, generated by the discrete symmetry of the potential: g → 1 − g, or h → 1 − h negative energy and non-square integrable H˜ φv(x) = ˜ Ev ˜ φv(x), ˜ Ev < 0 (˜ φv, ˜ φv) = (1/˜ φv, 1/˜ φv) = ∞ they have no zeros in x ∈ (0, π/2) use these seed solutions D def = {dI

1, . . . , dI M, dII 1 , . . . , dII N},

dI,II

j

≥ 1

Sasaki New Orthogonal Polynomials

slide-19
SLIDE 19

Introduction New Discovery General Recipe Multi-Indexed Orthogonal Polynomials Summary and Outlook Appendix Exceptional Jacobi Polynomials

Multi-Indexed Orthogonal Polynomials 2

explicit forms of type I virtual states (h → 1 − h) ˜ φI

v(x) def

= (sin x)g(cos x)1−hξI

v(η(x); g, h),

ξI

v(η; g, h) def

= Pv(η; g, 1 − h), v = 0, 1, . . . , [h − 1

2]′,

˜ EI

v def

= −4(g + v + 1

2)(h − v − 1 2),

˜ δ

I def

= (−1, 1) explicit forms of type II virtual states (g → 1 − g) ˜ φII

v(x) def

= (sin x)1−g(cos x)hξII

v (η(x); g, h),

ξII

v (η; g, h) def

= Pv(η; 1 − g, h), v = 0, 1, . . . , [g − 1

2]′,

˜ EII

v def

= −4(g − v − 1

2)(h + v + 1 2), ˜

δ

II def

= (1, −1) They are Not symmetries of Jacobi polynomials S.Odake & R. Sasaki, Phys. Lett. B702 (2011) 164-170, arXiv:1105.0508[math-ph]

Sasaki New Orthogonal Polynomials

slide-20
SLIDE 20

Introduction New Discovery General Recipe Multi-Indexed Orthogonal Polynomials Summary and Outlook Appendix Exceptional Jacobi Polynomials

Schematic Picture of Virtual States Deletion

Sasaki New Orthogonal Polynomials

slide-21
SLIDE 21

Introduction New Discovery General Recipe Multi-Indexed Orthogonal Polynomials Summary and Outlook Appendix Exceptional Jacobi Polynomials

Eigenfunctions etc after Virtual States Deletion

H[M]φ[M]

n

(x) = Enφ[M]

n

(x) (n ∈ Z≥0), H[M] ˜ φ[M]

v

(x) = ˜ Ev ˜ φ[M]

v

(x) (v ∈ V\D), φ[M]

n

(x) def = W[˜ φd1, ˜ φd2, . . . , ˜ φdM, φn](x) W[˜ φd1, ˜ φd2, . . . , ˜ φdM](x) , (φ[M]

m , φ[M] n

) =

M

  • j=1

(En − ˜ Edj) · hnδm n, ˜ φ[M]

v

(x) def = W[˜ φd1, ˜ φd2, . . . , ˜ φdM, ˜ φv](x) W[˜ φd1, ˜ φd2, . . . , ˜ φdM](x) , U[M](x) def = U(x) − 2∂2

x log

  • W[˜

φd1, ˜ φd2, . . . , ˜ φdM](x)

  • .

shape inv. exactly solvable ⇒ shape inv. exactly solvable

Sasaki New Orthogonal Polynomials

slide-22
SLIDE 22

Introduction New Discovery General Recipe Multi-Indexed Orthogonal Polynomials Summary and Outlook Appendix Exceptional Jacobi Polynomials

Multi-Indexed Orthogonal Polynomials 3

Multi-Indexed Orthogonal Polynomials PD,n(η): φ[M,N]

n

(x) ≡ φD,n(x; λ) = (−4)M+NψD(x; λ)PD,n(η(x); λ), ψD(x; λ) def = φ0(x; λ[M,N]) ΞD(η(x); λ) , PD,0(η; λ) ∝ ΞD(η; λ + δ) λ[M,N] = (g + M − N, h − M + N), ΞD(η) has no node in −1 < η < 1;

  • rthogonality

1

−1

dη W (η; λ[M,N]) ΞD(η; λ)2 PD,m(η; λ)PD,n(η; λ) = hD,nδnm

Sasaki New Orthogonal Polynomials

slide-23
SLIDE 23

Introduction New Discovery General Recipe Multi-Indexed Orthogonal Polynomials Summary and Outlook Appendix Exceptional Jacobi Polynomials

Multi-Indexed Orthogonal Polynomials 4

Explicit Forms PD,n(η; λ) def = W[µ1, . . . , µM, ν1, . . . , νN, Pn](η) × 1 − η 2 (M+g+ 1

2 )N1 + η

2 (N+h+ 1

2 )M

ΞD(η; λ) def = W[µ1, . . . , µM, ν1, . . . , νN](η) × 1 − η 2 (M+g− 1

2 )N1 + η

2 (N+h− 1

2 )M

µj = 1 + η 2 1

2 −hξI

dI

j (η; g, h),

νj = 1 − η 2 1

2 −gξII

dII

j (η; g, h)

PD,n(η) degree ℓ + n, ΞD(η) degree ℓ; ℓ =

M

  • j=1

dI

j + N

  • j=1

dII

j − 1

2M(M − 1) − 1 2N(N − 1) + MN ≥ 1

Sasaki New Orthogonal Polynomials

slide-24
SLIDE 24

Introduction New Discovery General Recipe Multi-Indexed Orthogonal Polynomials Summary and Outlook Appendix Exceptional Jacobi Polynomials

How it started: X1 Jacobi polynomials

  • X1 Jacobi Hamiltonian (Gomez-Ullate et al, Quesne et al, ’08)

H = − d2 dx2 + g(g + 1) sin2 x + h(h + 1) cos2 x − (2 + g + h)2 + 8(g + h + 1) 1+g +h+(g −h) cos 2x − 8(2g + 1)(2h + 1) (1+g +h+(g −h) cos 2x)2 φ0(x) = (sin x)g+1(cos x)h+1 3 + g + h + (g − h) cos 2x 1 + g + h + (g − h) cos 2x P(g+2−3/2,−h−2−1/2)

1

(cos 2x) P(g+1−3/2,−h−1−1/2)

1

(cos 2x)

  • generalisation

wℓ(x; λ) = (g + ℓ) log sin x + (h + ℓ) log cos x + log ξℓ(η; λ + δ) ξℓ(η; λ) ξℓ(η; λ) def = Pℓ(g+ℓ−3/2,−h−ℓ−1/2)(η), η = cos 2x

Sasaki New Orthogonal Polynomials

slide-25
SLIDE 25

Introduction New Discovery General Recipe Multi-Indexed Orthogonal Polynomials Summary and Outlook Appendix Exceptional Jacobi Polynomials

Xℓ Jacobi Polynomials

shape invariance can be verified directly (∂xwℓ(x; λ))2−∂2

xwℓ(x; λ) = (∂xwℓ(x; λ + δ))2

+ ∂2

xwℓ(x; λ + δ) + 4(g + h + 2ℓ + 1)

eigenvalues Eℓ,n(g, h) = En(g + ℓ, h + ℓ) = 4n(n + g + h + 2ℓ) eigenfunctions φℓ,n(x; λ) = ψℓ(x; λ)Pℓ,n(η; λ) ψℓ(x; λ) def = ew0(x;λ+ℓδ) ξℓ(η; λ) , cn

def

= n + h + 1/2 Pℓ,n(η; λ) def = c−1

n

  • (h + 1

2)ξℓ(η; λ + δ)P(g+ℓ−3/2,h+ℓ+1/2)

n

(η) + (1 + η)ξℓ(η; λ)∂ηP(g+ℓ−3/2,h+ℓ+1/2)

n

(η)

  • degree ℓ + n polynomial

Sasaki New Orthogonal Polynomials

slide-26
SLIDE 26

Introduction New Discovery General Recipe Multi-Indexed Orthogonal Polynomials Summary and Outlook Appendix Exceptional Jacobi Polynomials

Xℓ Jacobi Polynomials 2: Fuchsian differential equation

lowest degree Pℓ,0(η; λ) ∝ ξℓ(η; λ + δ)

  • rthogonality

π/2 ψℓ(x; λ)2Pℓ,n(cos 2x; λ)Pℓ,m(cos 2x; λ)dx = hℓ,n(λ)δn m Fuchsian differential eq. (1 − η2)∂2

ηPℓ,n(η; λ)

+

  • h − g − (g + h + 2ℓ + 1)η − 2(1 − η2)∂ηξℓ(η; λ)

ξℓ(η; λ)

  • ∂ηPℓ,n(η; λ)

+

  • −2(h + 1

2)(1 − η)∂ηξℓ(η; λ + δ)

ξℓ(η; λ) + ℓ(ℓ + g − h − 1) + n(n + g + h + 2ℓ)

  • Pℓ,n(η; λ) = 0

Sasaki New Orthogonal Polynomials

slide-27
SLIDE 27

Introduction New Discovery General Recipe Multi-Indexed Orthogonal Polynomials Summary and Outlook Appendix Exceptional Jacobi Polynomials

Xℓ Jacobi Polynomials 3: Fuchsian differential equation 2

regular singularities at ℓ roots of ξℓ(η; λ), η = ηj: ξℓ(ηj; λ) = 0, j = 1, 2, . . . , ℓ in the neighbourhood of η = ηj: (1 − η2

j )y′′ − 2

(1 − η2

j )

η − ηj y′ − 2(h + 1/2)(1 − ηj) β η − ηj y + regular terms = 0 characteristic eq.: same exponents everywhere ρ(ρ − 1) − 2ρ = 0 ⇒ ρ = 0, 3 ρ = 0 corresponds to the polynomial solution

Sasaki New Orthogonal Polynomials

slide-28
SLIDE 28

Introduction New Discovery General Recipe Multi-Indexed Orthogonal Polynomials Summary and Outlook Appendix

Summary and Outlook

Question: Why the “New Polynomials” were Not discovered by the experts of the orthogonal polynomials? Answers: ‘physical thinking’ is more suitable for the problem

1

Schr¨

  • dinger equation is more general than the equations

governing orthogonal polynomials

2

g ↔ 1 − g, h ↔ 1 − h are Not the symmetries of the Jacobi (Laguerre) polynomial equations

3

they are equations for the eigenpolynomials, i e. non-square integrable solutions are discarded

4

Darboux transformations are defined most generally for the Schr¨

  • dinger equations

Sasaki New Orthogonal Polynomials

slide-29
SLIDE 29

Introduction New Discovery General Recipe Multi-Indexed Orthogonal Polynomials Summary and Outlook Appendix

Summary and Outlook 2

Infinitely many new orthogonal polynomials satisfying second

  • rder differential or difference equations are discovered.

Hopefully they will find many interesting applications. At least, they give infinitely many examples of exactly solvable Birth and Death Processes. Various concepts and methods of QM have much wider currency and utility in the theory of ordinary differential and difference equations than is usually regarded. Various properties of the Askey-scheme of hypergeometric

  • rthogonal polynomials can be understood in a unified

fashion, both of a continuous and a discrete variable. Multi-variable Multi-Indexed Orthogonal polynomials are the next challenge

Sasaki New Orthogonal Polynomials

slide-30
SLIDE 30

Introduction New Discovery General Recipe Multi-Indexed Orthogonal Polynomials Summary and Outlook Appendix Fuchsian Differential Equations References

Fuchsian Differential Equations 1: overview

y′′ + f (x)y′ + g(x)y = 0, f (x) = α x − x0 +

∞

  • n=0

fn(x − x0)n, g(x) = β (x − x0)2 + γ x − x0 +

∞

  • n=0

gn(x − x0)n x0 : regular singularity singular solutions yj = (x − x0)ρj(1 +

∞

  • n=1

an(x − x0)n) ρ1, ρ2: characteristic exponents ρ(ρ − 1) + αρ + β = 0 regular singularities only ⇒ Fuchsian equation local theory only

Sasaki New Orthogonal Polynomials

slide-31
SLIDE 31

Introduction New Discovery General Recipe Multi-Indexed Orthogonal Polynomials Summary and Outlook Appendix Fuchsian Differential Equations References

Fuchsian Differential Equations 2: examples

3 regular singularities at 0, 1, ∞: Gauss hypergeometric eq. x(1 − x)y′′ + (γ − (α + β + 1)x)y′ − αβy = 0 solutions around x = 0: ρ1 = 0, ρ2 = 1 − γ y1 = 2F1(α, β; γ|x) =

∞

  • n=0

(α)n(β)n (γ)n xn n! , y2 = x1−γ2F1(α − γ + 1, β − γ + 1; 2 − γ|x) hypergeometric function ⇒ globally continued 4 regular singularities 0, 1, ∞, t: Heun equation more than 4 regular singularities: global solution virtually unknown

Sasaki New Orthogonal Polynomials

slide-32
SLIDE 32

Introduction New Discovery General Recipe Multi-Indexed Orthogonal Polynomials Summary and Outlook Appendix Fuchsian Differential Equations References

Fuchsian Differential Equations 3

if ρ2 − ρ1 = n ∈ N: possible log terms (Frobenius) if ρ2 − ρ1 = n ∈ N and no log terms ⇒ apparent singularity apparent singularity of Schr¨

  • dinger eq. (at x = 0)

H = − d2 dx2 + α x2 + β x + regular terms ρ = (1 ± √ 1 + 4α)/2 α ρ2 − ρ1 1 regular 3/4 2 Painlev´ e case 2 3 Darboux trans. 15/4 4 ?? 6 5 Ho-Sasaki-Takemura . . . . . . if all extra singularities are apparent⇒ global solutions possible

Sasaki New Orthogonal Polynomials

slide-33
SLIDE 33

Introduction New Discovery General Recipe Multi-Indexed Orthogonal Polynomials Summary and Outlook Appendix Fuchsian Differential Equations References

Parallel History

shape invariant potentials in discrete QM with pure imaginary shifts; Wilson, Askey-Wilson polynomials etc (’04) Heisenberg operator solutions & dynamical symmetry algebras in discrete QM with pure imaginary shifts; Wilson, Askey-Wilson polynomials etc (’06) shape invariant potentials in discrete QM with real imaginary shifts; (q-)Racah, (dual) (q-)Hahn, etc (’08) Crum’s theorem for discrete QM (’09) Xℓ Wilson and Askey-Wilson polynomials (’09) Modified Crum’s theorem (Krein-Adler transformations) for discrete QM (’10) Xℓ Wilson and Askey-Wilson polynomials derived by Darboux transformations (’10)

Sasaki New Orthogonal Polynomials

slide-34
SLIDE 34

Introduction New Discovery General Recipe Multi-Indexed Orthogonal Polynomials Summary and Outlook Appendix Fuchsian Differential Equations References

Parallel History II

Xℓ (q)-Racah polynomials (’11) Multi-indexed (q)-Racah polynomials (’12) Multi-indexed Wilson and Askey-Wilson polynomials (’12) duality between pseudo virtual and eigenstates & Casoratian identities for Wilson and Askey-Wilson polynomials non-confining potentials (discrete analogues of Morse, Eckart potentials) (’14)

Sasaki New Orthogonal Polynomials

slide-35
SLIDE 35

Introduction New Discovery General Recipe Multi-Indexed Orthogonal Polynomials Summary and Outlook Appendix Fuchsian Differential Equations References

Work by Satoru Odake and R. Sasaki

“Equilibria of ’discrete’ integrable systems and deformations

  • f classical orthogonal polynomials,” J. Phys. A37 (2004)

11841-11876. “Shape Invariant Potentials in ‘Discrete Quantum Mechanics,’” J. Nonlinear Mathematical Physics 12 Supplement 1 (2005) 507-521. “Calogero-Sutherland-Moser Systems, Ruijsenaars-Schneider-van Diejen Systems and Orthogonal Polynomials,” Prog. Theor. Phys. 114 (2005) 1245–1260. “Unified Theory of Annihilation-Creation Operators for Solvable (‘Discrete’) Quantum Mechanics,” J. Math. Phys. 47 (2006) 102102. “Exact solution in the Heisenberg picture and annihilation-creation operators,” Phys. Lett. B641 (2006) 112–117.

Sasaki New Orthogonal Polynomials

slide-36
SLIDE 36

Introduction New Discovery General Recipe Multi-Indexed Orthogonal Polynomials Summary and Outlook Appendix Fuchsian Differential Equations References

Work by Satoru Odake and R. Sasaki, 2

“Exact Heisenberg operator solutions for multi-particle quantum mechanics,” J. Math. Phys. 48 (2007) 082106. “Multi-Particle Quasi Exactly Solvable Difference equations,”

  • J. Math. Phys. 48 (2007) 122105.

‘q-oscillator from the q-Hermite polynomial,” Phys. Lett. B663 (2008) 141-145. “Orthogonal Polynomials from Hermitian Matrices,” J. Math.

  • Phys. 49 (2008) 053503.

“Exactly solvable ‘discrete’ quantum mechanics; shape invariance, Heisenberg solutions, annihilation-creation

  • perators and coherent states,” Prog. Theor. Phys. 119

(2008) 663-700.

Sasaki New Orthogonal Polynomials

slide-37
SLIDE 37

Introduction New Discovery General Recipe Multi-Indexed Orthogonal Polynomials Summary and Outlook Appendix Fuchsian Differential Equations References

Work by Satoru Odake and R. Sasaki, 3

“Unified theory of exactly and quasi-exactly solvable ‘Discrete’ quantum mechanics: I. Formalism,” J. Math. Phys. 51 (2010) 083502 (24 pp). “Crum’s Theorem for Discrete Quantum Mechanics,” Prog.

  • Theor. Phys. 122 (2009) 1067-1079.

“Infinitely many shape invariant potentials and new

  • rthogonal polynomials,” Phys. Lett. B679 (2009) 414-417.

“Infinitely many shape invariant discrete quantum mechanical systems and new exceptional orthogonal polynomials related to the Wilson and the Askey-Wilson polynomials,” Phys. Lett. B682 (2009) 130-136. “Another set of infinitely many exceptional (Xℓ) Laguerre polynomials,” Phys. Lett. B684 (2010) 173-176.

Sasaki New Orthogonal Polynomials

slide-38
SLIDE 38

Introduction New Discovery General Recipe Multi-Indexed Orthogonal Polynomials Summary and Outlook Appendix Fuchsian Differential Equations References

Work by Satoru Odake and R. Sasaki, 4

“Infinitely many shape invariant potentials and cubic identities

  • f the Laguerre and Jacobi polynomials.” J. Math. Phys. 51

(2010) 053513. “Exceptional Askey-Wilson type polynomials through Darboux-Crum transformations,” J. Phys. A43 (2010) 335201 (18pp), arXiv:1004.0544[math-ph]. “A new family of shape invariantly deformed Darboux-P¨

  • schl-Teller potentials with continuous ℓ,” J.

Phys. A 44 (2011) 195203, arXiv:1007.3800 [math-ph]. “Dual Christoffel transformations,” Prog. Theor. Phys. 126 (2011) 1-34 arXiv:1101.5468[math-ph]. “The Exceptional (Xℓ) (q)-Racah Polynomials,” Prog. Theor.

  • Phys. 125 (2011) 851-870. arXiv:1102.0812[math-ph].

Sasaki New Orthogonal Polynomials

slide-39
SLIDE 39

Introduction New Discovery General Recipe Multi-Indexed Orthogonal Polynomials Summary and Outlook Appendix Fuchsian Differential Equations References

Work by Satoru Odake and R. Sasaki, 5

“Exactly solvable quantum mechanics and infinite families of multi-indexed orthogonal polynomials,” Phys. Lett. B702 (2011) 164-170. arXiv:1105.0508[math-ph]. “Discrete quantum mechanics,” (Topical Review) J. Phys. A44 (2011) 353001 (47 pp). arXiv:1104.0473[math-ph]. “Multi-indexed (q-)Racah polynomials,” J. Phys. A45 (2012) 385201 (21 pp). arXiv:1203.5868[math-ph]. “Multi-indexed Wilson and Askey-Wilson polynomials,” J.

  • Phys. A46 (2013) 045204 (22 pp).

arXiv:1207.5584[math-ph]. “Extensions of solvable potentials with finitely many discrete eigenstates,” J. Phys. A 46 (2013) 235205 (15pp) arXiv:1301.3980[math-ph].

Sasaki New Orthogonal Polynomials

slide-40
SLIDE 40

Introduction New Discovery General Recipe Multi-Indexed Orthogonal Polynomials Summary and Outlook Appendix Fuchsian Differential Equations References

Work by Satoru Odake and R. Sasaki, 6

“Discrete quantum mechanics,” (Topical Review) J. Phys. A44 (2011) 353001, arXiv:1104.5669 “Exactly solvable quantum mechanics and infinite families of multi-indexed orthogonal polynomials,” Phys. Lett. B702 (2011) 164-170, arXiv:1105.0508. Leonor Garc´ ıa-Guti´ errez, S. Odake and R. Sasaki, “Modification of Crum’s Theorem for ‘Discrete’ Quantum Mechanics,” Prog. Theor. Phys. 124 (2010) 1-24, arXiv:1004.0289[math-ph]. “Multi-indexed (q)-Racah polynomials,” J. Phys. A45 (2012) 385201 (21 pp), arXiv:1203.5868[math-ph]. “Multi-indexed Wilson & Askey-Wilson polynomials,” J. Phys. A46 (2013) 045204 (22 pp) arXiv:1207.5584[math-ph].

Sasaki New Orthogonal Polynomials

slide-41
SLIDE 41

Introduction New Discovery General Recipe Multi-Indexed Orthogonal Polynomials Summary and Outlook Appendix Fuchsian Differential Equations References

Work by Satoru Odake and R. Sasaki, 7

“Extensions of solvable potentials with finitely many discrete eigenstates,” J. Phys. A 46 (2013) 235205 (15pp) arXiv:1301.3980[math-ph]. “Krein-Adler transformations for shape-invariant potentials and pseudo virtual states,” arXiv:1212.6595[math-ph]. “Non-polynomial extensions of solvable potentials ´ a la Abraham-Moses,” J. Math. Phys. 54 (2013) 102106 (19pp) arXiv:1307.0931[math-ph]. “Casoratian Identities for the Wilson and Askey-Wilson Polynomials,” J. Approximation Theory (2014) arXiv:1308.4240[math-ph]. “Solvable Discrete Quantum Mechanics: q-Orthogonal Polynomials with |q| = 1 and Quantum Dilogarithm,” arXiv:1406.2768[math-ph].

Sasaki New Orthogonal Polynomials

slide-42
SLIDE 42

Introduction New Discovery General Recipe Multi-Indexed Orthogonal Polynomials Summary and Outlook Appendix Fuchsian Differential Equations References

Related Works

  • R. Sasaki, “Exactly Solvable Birth and Death Processes,” J.
  • Math. Phys. 50 (2009) 103509.

C-L. Ho, S. Odake and R. Sasaki, “Properties of the exceptional (Xℓ) Laguerre and Jacobi polynomials,” SIGMA 7 (2011) 107 (24pp) arXiv:0912.5477[math-ph].

  • R. Sasaki, S. Tsujimoto and A. Zhedanov, “Exceptional

Laguerre and Jacobi polynomials and the corresponding potentials through Darboux-Crum transformations,” J. Phys. A 43 (2010) 315204 (20pp), arXiv:1004.4711[math-ph].

  • R. Sasaki, “Exactly and Quasi-Exactly Solvable ‘Discrete’

Quantum Mechanics,” Phil. Trans. R. Soc. A369 (2011) 1301-1318, arXiv:1004.4712[math-ph].

Sasaki New Orthogonal Polynomials

slide-43
SLIDE 43

Introduction New Discovery General Recipe Multi-Indexed Orthogonal Polynomials Summary and Outlook Appendix Fuchsian Differential Equations References

Related Works, 2

C-L. Ho and R. Sasaki, “Zeros of the exceptional Laguerre and Jacobi polynomials,” ISRN Mathematical Physics 2012 920475 (27pp), doi:10.5402/2012/920475. arXiv:1102.5669[math-ph].

  • R. Sasaki and K. Takemura, “Global solutions of certain

second order differential equations with a high degree of apparent singularity,” SIGMA 8 (2012) 085 (18pp), arXiv:1207.5302[math.CA]. C.-L. Ho, R. Sasaki and K. Takemura, “Confluence of apparent singularities in multi-indexed orthogonal polynomials: the Jacobi case,” J. Phys. A 46 (2013) 115205 (21pp) arXiv:1210.0207[math.CA].

Sasaki New Orthogonal Polynomials

slide-44
SLIDE 44

Introduction New Discovery General Recipe Multi-Indexed Orthogonal Polynomials Summary and Outlook Appendix Fuchsian Differential Equations References

Related Works, 3

J-C. Lee, C-L. Ho and R. Sasaki, “Scattering Amplitudes for Multi-indexed Extensions of Solvable Potentials,” Annals of Physics 343 (2014) 115–131, arXiv:1309.5471.

  • R. Sasaki, “Exactly solvable potentials with finitely many

discrete eigenvalues of arbitrary choice,” J. Math. Phys. 55 (2014) 062101 (11pp). arXiv:1402.5474[math-ph].

  • R. Sasaki, “Exactly solvable potentials with finitely many

discrete eigenvalues of arbitrary choice,” J. Math. Phys. 55 (2014) 062101 (11pp). arXiv:1402.5474[math-ph]

Sasaki New Orthogonal Polynomials