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

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

Introduction New Discovery General Recipe Multi-Indexed Orthogonal Polynomials Summary and Outlook Appendix Outline Introduction 1 New Discovery 2 General Recipe 3 Ordinary Quantum Mechanics Multi-Indexed Orthogonal Polynomials 4 Exceptional Jacobi Polynomials Summary and Outlook 5 Appendix 6 Fuchsian Differential Equations References Sasaki New Orthogonal Polynomials

Introduction New Discovery General Recipe Multi-Indexed Orthogonal Polynomials Summary and Outlook Appendix Exactly Solvable Quantum Mechanics 1-d QM, given a Hamiltonian H = − d 2 dx 2 + U ( x ), x 1 < x < x 2 , U ( x ) ∈ C ∞ , • Eigenvalue problem � x 2 φ 2 H φ n ( x ) = E n φ n ( x ) , n = 0 , 1 , 2 , . . . , n ( x ) dx < ∞ , x 1 • all the discrete eigenvalues {E n } and the corresponding eigenfunctions { φ n ( x ) } are exactly calculable ⇒ Exactly Solvable Quantum Mechanics Sasaki New Orthogonal Polynomials

Introduction New Discovery General Recipe Multi-Indexed Orthogonal Polynomials Summary and Outlook Appendix Typical examples of exactly solvable QM I • harmonic oscillator, H = − d 2 dx 2 + x 2 − 1, −∞ < x < + ∞ , E n = 2 n , φ n ( x ) = φ 0 ( x ) H n ( x ): Hermite polynomial, φ 0 ( x ) = e − x 2 / 2 , � + ∞ φ 2 0 ( x ) H n ( x ) H m ( x ) dx ∝ δ n m −∞ • radial oscillator, H = − d 2 dx 2 + x 2 + g ( g − 1) − (1 + 2 g ), x 2 0 < x < + ∞ , g > 1, E n = 4 n , φ n ( x ) = φ 0 ( x ) L ( g − 1 / 2) ( x 2 ):Laguerre polynomial, n φ 0 ( x ) = e − x 2 / 2 x g , � + ∞ 0 ( x ) L ( g − 1 / 2) ( x 2 ) L ( g − 1 / 2) φ 2 ( x 2 ) dx ∝ δ n m n m 0 Sasaki New Orthogonal Polynomials

Introduction New Discovery General Recipe Multi-Indexed Orthogonal Polynomials Summary and Outlook Appendix Typical examples of exactly solvable QM II • P¨ oschl-Teller potential, H = − d 2 dx 2 + x 2 + g ( g − 1) + h ( h − 1) − ( g + h ) 2 , sin 2 x cos 2 x 0 < x < π/ 2, g > 1, h > 1, E n = 4 n ( n + g + h ), φ n ( x ) = φ 0 ( x ) P ( g − 1 / 2 , h − 1 / 2) (cos 2 x ):Jacobi polynomial, n φ 0 ( x ) = (sin x ) g (cos x ) h , � π/ 2 0 ( x ) P ( g − 1 / 2 , h − 1 / 2) (cos 2 x ) P ( g − 1 / 2 , h − 1 / 2) φ 2 (cos 2 x ) dx ∝ δ n m n m 0 Sasaki New Orthogonal Polynomials

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 creation, annihilation operators 1 coherent states 2 dynamical symmetry algebras 3 3 Schr¨ odinger 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 = φ 2 0 ( x ): square of the ground state eigenfunction Sasaki New Orthogonal Polynomials

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 orthogonal polynomials satisfying second order differential equations are Classical orthogonal polynomials; Hermite, Laguerre, Jacobi & Bessel Sasaki New Orthogonal Polynomials

Introduction New Discovery General Recipe Multi-Indexed Orthogonal Polynomials Summary and Outlook Appendix Bochner’s Theorem ’29 If polynomials { p n ( x ) } satisfy three term recurrence relations and a second order differential equation σ ( x ) y ′′ + τ ( x ) y ′ + λ n y = 0 , they must be one of the Classical orthogonal polynomials, i.e., the Hermite, Laguerre, Jacobi and Bessel. For y = p 0 ( x ) =const, ⇒ λ 0 = 0. For y = p 1 ( x ) ⇒ degree( τ ( x )) ≤ 1. For y = p 2 ( 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

Introduction New Discovery General Recipe Multi-Indexed Orthogonal Polynomials Summary and Outlook Appendix Avoiding Bochner’ constraints polynomials satisfying difference Schr¨ odinger 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 option) polynomials satisfying difference Schr¨ odinger equation and starting at degree ℓ ≥ 1 and having holes in the degree Sasaki New Orthogonal Polynomials

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 P D , n ( x ), D = { d 1 , . . . , d M } , d j ∈ N : degrees of polynomial type seed solutions (virtual state wave functions) employed by multiple Darboux transformations, ( n counts nodes in ( x 1 , x 2 )) � x 2 P D , n ( x ) P D , m ( x ) W D ( x ) dx = h D , n δ n m x 1 degree ℓ + n polynomial in x , but forming a complete set, Sasaki New Orthogonal Polynomials

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¨ odinger eq. when eigenfunctions are employed, D = { d 1 , . . . , d M } , d j ∈ 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

Introduction New Discovery General Recipe Ordinary Quantum Mechanics Multi-Indexed Orthogonal Polynomials Summary and Outlook Appendix Basic Ingredients Exactly Solvable Quantum Mechanical System H φ n ( x ) = E n φ n ( x ) , E 0 = 0 , n = 0 , 1 , 2 , . . . , Factorised positive semi-definite Hamiltonian H = A † A ≥ 0 Multiple Darboux-Crum-Krein-Adler transformation H ϕ ( x ) = ˜ H ψ ( x ) = E ψ ( x ) , E ϕ ( x ) , H (1) def ⇒ H (1) ψ (1) ( x ) = E ψ (1) ( x ) , = H − 2 ∂ 2 x log ϕ ( x ) , = ∂ x ψ ( x ) − ∂ x ϕ ( x ) ϕ ( x ) ψ ( x ) = W[ ϕ, ψ ]( x ) ψ (1) ( x ) def , ϕ ( x ) ϕ v ( x ) = ˜ ϕ v ( x ), ˜ Virtual State solutions, H ˜ E v ˜ E v < 0, ϕ v ( x ) > 0, v ∈ V ˜ Sasaki New Orthogonal Polynomials

Introduction New Discovery General Recipe Ordinary Quantum Mechanics Multi-Indexed Orthogonal Polynomials Summary and Outlook Appendix Factorised Hamiltonians Starting point: H with complete set of eigenvalues and eigenfunctions H φ n ( x ) = E n φ n ( x ) , ( φ n , φ m ) = h n δ n m , h n > 0 , n = 0 , 1 , 2 , . . . , by adjusting the const. of H ⇒ E 0 = 0 ⇒ Positive Semi-Definite Hamiltonian H (Hermitian Matrix) H = A † A 0 = E 0 < E 1 < E 2 < · · · , ⇒ A † = − d / dx − ∂ x φ 0 ( x ) /φ 0 ( x ) , A = d / dx − ∂ x φ 0 ( x ) /φ 0 ( x ) , φ 0 ( x ): ground state wavefunction, no node ( φ 0 ( x ) > 0), square integrable A φ 0 ( x ) = 0 V ( x ) = ∂ 2 x φ 0 ( x ) H = − d 2 / dx 2 + V ( x ) , φ 0 ( x ) Sasaki New Orthogonal Polynomials

Introduction New Discovery General Recipe Ordinary Quantum Mechanics Multi-Indexed Orthogonal Polynomials Summary and Outlook Appendix Use virtual state solutions rewrite H by using ˆ A d 1 , d 1 ∈ N , ( ˆ A d 1 annihilates ˜ ϕ d 1 ( x ), ˆ A d 1 ˜ ϕ d 1 ( x ) = 0): def ˆ A † ˆ A d 1 = d / dx − ∂ x log ˜ ϕ d 1 ( x ) , d 1 = − d / dx − − ∂ x log ˜ ϕ d 1 ( x ) , � ˜ � ˜ � 2 ϕ ′ ϕ ′ A d 1 = − d 2 d 1 ( x ) d 1 ( x ) � + d A † ˆ d 1 ˆ dx 2 + ϕ d 1 ( x ) ˜ ϕ d 1 ( x ) ˜ dx ϕ ′′ = − d 2 ϕ d 1 ( x ) = − d 2 ˜ d 1 ( x ) dx 2 + V ( x ) − ˜ dx 2 + E d 1 , ˜ H = ˆ A † d 1 ˆ A d 1 + ˜ ˆ E d 1 , A d 1 : non-singular , define a new Hamiltonian by changing the order of ˆ A d 1 and d 1 : H (1) def A † ˆ = ˆ A d 1 ˆ A † d 1 + ˜ E d 1 = H − 2 ∂ 2 x log ˜ ϕ d 1 ( x ) d 1 Sasaki New Orthogonal Polynomials