Lie Algebras and the Schrödinger equation: (quasi-exact-solvability, symmetric coordinates)

Transcription

1 Lie Algebras and the Schrödinger equation: (quasi-exact-solvability, symmetric coordinates) Alexander Turbiner Nuclear Science Institute, UNAM, Mexico September 18, 2010

2 Let us consider the Hamiltonian = the Schrödinger operator H = + V(x), x R d A problem of quantum mechanics is to solve the Schrödinger equation HΨ(x) = EΨ(x), Ψ(x) L 2 (R d ) finding the spectra (the energies and eigenfunctions). The Hamiltonian is an infinite-dimensional matrix To solve the Schrödinger equation diagonalize the infinite-dimensional matrix It is transcendental problem, the characteristic polynomial is of infinite order and it has infinitely-many roots.

3 One-dimensional Anharmonic Oscillator H = d2 dx 2 + m2 x 2 + gx 4 Ground State Energy E 0 (m 2,1) (m 2 m2 g 1/3,g 1) m 2 Infinitely-many square-root branch points (level crossings, Bender-Wu 69, Gabrielov-Eremenko 09) Infinitely-sheeted Riemann surface on E in m 2 Studying one eigenstate implies a study of the whole spectra (via analytic continuation)

4 Do exist Hamiltonians H with algebraic substructure det(h E) = det(h E) n n det(h E) n n (factorization) It implies the existence of the basis where matrix H has block-diagonal (block-triangular) form. Factorization does not depend on basis! (invariant property) How to find it (if exist)?

5 Example 1: Take Ψ 0 = e ax4 4 - ground state (no nodes) Find a potential for which it s the lowest eigenfunction V 0 = Ψ 0 Ψ 0 = a 2 x 6 3ax 2 and E 0 = 0 Define H 0 = 2 x +a2 x 6 3ax 2 where one eigenstate is known exactly det(h 0 E) = E det(h 0 E) { 1} { 1}

6 2 4 V(x)= 3x + x x Figure: The double-well potential V = a 2 x 6 3ax 2 (at a = 1) with Ψ 0 = e ax4 4 and E 0 = 0

7 Example 2: Take Ψ ± = ( 2ax 2 ±1)e ax4 4 (+) ground state (no nodes) and ( ) 2nd excited state (two nodes) There is a potential(!) for which Ψ ± are the eigenfunctions Ψ ± Ψ ± = a 2 x 6 7ax 2 and E ± = 2 2a Define H 1 = 2 x +a2 x 6 7ax 2 where two eigenstates are known exactly (either Ψ ± or E ± form two-sheeted Riemann surface in a), det(h 1 E) = (E 2 8a)det(H 1 E) { 2} { 2}

8 Example n: Take Ψ n = P n (x 2 )e ax4 4 P n (x 2 ) is a set of (n+1) polynomials having from zero up to 2n nodes There is a single potential(!) in which all Ψ n are the eigenfunctions Ψ n Ψ n = a 2 x 6 (4n+3)ax 2, Q n+1 (E,a) = 0, where Q n+1 (E,a) is (n+1)th degree polynomial in E with real roots at a > 0 Define H n = 2 x +a2 x 6 (4n +3)ax 2 where (n + 1) eigenstates are known exactly (algebraically) (either Ψ n or E n form (n+1)-sheeted Riemann surface in a), det(h n E) = Q n+1 (E,a)det(H n E) { n 1} { n 1}

9 V n,p = a 2 x 6 +2abx 4 +[b 2 (4n+2p+3)a]x 2, p = 0,1 the unique polynomial 1D quasi-exactly-solvable potential with three possible patterns: single-well, double-well, triple-well Eigenfunctions: Ψ n,p = x p P n (x 2 )e ax4 4 b x2 2 What is behind quasi-exact-solvability?

10 Lets us make a gauge rotation of the Hamiltonian H n with Ψ 0,p and introduce new variable t = bx 2 : h n (t, t ) 1 4b (xp e bx2 /2 ax 4 /4 ) 1 H n (x p e bx2 /2 ax 4 /4 ) t=bx 2 = t 2 t +(at 2 +t p 1 2 ) t ant, It s Algebraic form of H n. h n has finite-dimensional invariant subspace P n =< 1,t,t 2,...t n > hence, it has (n + 1) polynomial eigenfunctions: P (k) n (t) = n i=0 γ (k) i t i, k = 0,1,...,n, Does P n have a Lie-algebraic meaning? It does!

11 gl 2 -algebra (in R) J = t J 0 = t t T 0 n = t t n J + n = tt 0 n = t(t t n) P n is finite-dimensional irreducible representation space of dimension (n+1) Generators J n +, J and Tn 0 +J0 span the algebra sl(2) P n form the infinite flag P 0 P 1 P 2... P n...p

12 gl(2)-representation h n (t, t ) = aj + n J 0 J + J 0 (p )J (degenerate) Quantum top in constant magnetic field # of known eigenstates dimension of finite-dimensional irreducible representation Finite-dimensional invariant subspace P n of h n (t, t ) finite-dimensional irreducible representation space of gl(2) Quasi-exactly-solvable Hamiltonian H n has a hidden Lie algebra gl(2), it implies it can be rewritten in terms of gl(2) generators. It is well-seen in the space t = x 2, where Z 2 -symmetry of the problem realized!

13 What would happen if a = 0 (highest-weight generator is absent)? h n (t, t ) has infinitely-many finite-dimensional invariant subspaces P n, n = 0,1,2,3,... It preserves the flag it implies block-diagonal structure of H n complete factorization of the characteristic equation to product of monomials and, exact-solvability h exact = t 2 t +(t p 1 2 ) t = 1 4b (xp e bx2 /2 ) 1 H exact (x p e bx2 /2 ) t=bx 2 H exact = 2 x + b2 x 2 We recover the harmonic oscillator!

14 h exact (t, t ) = J 0 J + J 0 (p )J It is the gl(2)-lie-algebraic form in generators of b gl(2). This Lie-algebraic form is different from the second-quantization form H exact = {a + a} They act in different spaces, H exact is generator of gl(2), h exact is non-linear combination of generators as well as h n...

15 How to find such a basis in general, if exists? - representation theory comes to mind! Take h n = α,β=±,0 a αβ J α J β + α=±,0 b α J α It has (n + 1)-finite-dimensional invariant subspace by construction h exact = α,β=,0 a αβ J α J β + α=,0 b α J α It has infinitely-many, finite-dimensional invariant subspaces by construction

16 Substitute J α in explicit form h n = Q 4 (t) 2 t +Q 3(t) t +Q 2 (t) where Q 4,3,2 are polynomials of degree 4,3,2. It is Heun operator. To Schrödinger form: change of variable t = t(x) and make gauge rotation g 1 h n g. Finally, g 1 h n g t=t(x) = 2 x + V(x) where x = dt Q4 (t) elliptic change of variable

17 CLASSFICATION All four known 1D exactly solvable problems (Harmonic Oscillator, Coulomb Problem, Morse Potential and Pöschle-Teller Potential) have gl(2) hidden algebra There exist 11 gl(2)-lie-algebraic Potentials One polynomial QES Potential in (, ) One finite-piece Laurant series (in r) QES Potential in [0, ) Two QES Coulomb Potentials Three QES Morse Potentials Two QES Pöschl-Teller Potentials One QES Mathieu Potential (Magnus-Winkler Potential) (generalized)lame Potential It exhausts all 1D Schrödinger eqs having reduced to polynomial eigenfunctions.

18 Is there anything else, beyond Lie algebras? Yes! (1995, 2009) Eigenfunctions are from (one monomial is missing) P n =< 1,t,t 2,...,t n 2,t n > Quantum algebra sl(2) q (in Azbel-Hofstadter Hamiltonian (Wiegmann-Zabrodin, 95)) Is there anything else with gl(2), perhaps, beyond differential operators? Let us make a quantum canonical transformation of quantum phase space.

19 Three (minimal) realizations of h 3 : [a,b] = 1, a 0 = 0 a = d dt, b = t (I) a δ = D +δ, b δ = t δ t(1 δd δ ) = te δ d dt (II) 0 = 1 1+t d a q = D q, b q = t q (q 1)t dt q 1+t d dt 1 (III) f(t ±δ) f(t) where D ±δ f(t) = ±δ discrete or Jackson derivative. and D q f(t) = f(qt) f(t) t(q 1), (I) coordinate-momentum representation (II) is due to Yu.F. Smirnov and A.T. 95 (uniform lattice) (III) is due to C. Chryssomalakos and A.T. 01 (exponential lattice)

20 Basic object : b k+1 0 = t k+1 t (k+1) (I) (II) (k+1)! {k+1}! tk+1 (III) where t (k+1) t(t δ) (t kδ), {k} = qk 1 q 1 is a q-number, {k}! = {1}{2}...{k} is a q-factorial. t δ D +δ = td δ, t q D q = t d dt

21 sl(2)-algebra realized by finite-difference operators (on uniform lattice of spacing δ) J + = t (2) δd 2 δ +t[t δ(n +1)]D δ nt J 0 = td δ J = D +δ with P n = 1,t,t 2,...t n as common invariant subspace

22 sl(2)-algebra realized by finite-difference operators (on exponential lattice of dilation q) J + = t q t d dt nt q J 0 = t d dt J = D q with P n = 1,t,t 2,...t n as common invariant subspace

23 Take Laguerre operator h t = t 2 t +(t p 1 2 ) t Eigenstates: φ n = L (p 1 2 ) n (t) and ǫ n = n, n N, where L (p 1 2 ) n (t) is associated Laguerre polynomial. h t,δ (t,d ±δ ) = [t +(p )]D +δ +2tD δ The operator h t,δ (t,d ±δ ) is a non-local, three-point, finite-difference operator on uniform lattice in t-space φ(t δ) φ(t) φ(t +δ)

24 The corresponding spectral problem at δ = 1 [ t +(p )] φ(t +1)+ [ 3t +(p )] φ(t) 2tφ(t 1) = ǫφ(t) eigenfunctions are δ-modified associated Laguerre polynomials ˆL (p 1 2 ) n (t;δ) = n l=0 The operators h ht,δ and h t are isospectral. a (ν 1 2 ) l t (l)

25 h t,d (t, t,d q ) = t t D q +t t (p )D q The operator h t,d (t, t,d q ) is a non-local, two-point, differential-difference operator on exponential lattice in t-space. Its eigenfunctions can be called q-modified associated Laguerre polynomials ˆL (p 1 2 ) n (t,q) = n l=0 a (p 1 2 ) l l! {l}! tl, The operators h t,d, h ht,δ and h t are isospectral.

26 Immediate application of Lie-algebra formalism - algebraic perturbation theory. Take One-dimensional Anharmonic Oscillator 2 H = 1 2 x 2 +ω2 x 2 + g }{{ x 2 + 2λω 3 x 4 } A 1 rational (2-body Calogero) model ψ 0 = x ν e ω 2 x2, g = ν(ν 1), t = ωx 2 h = 1 2ω ψ 1 0 (H ω 2 (1+2ν))ψ 0 = t 2 t +(t ν 1/2) t +λt 2 t 2 P 2 (h 0 +λh 1 )φ = ǫφ

27 Perturbation theory: φ = λ n φ n, ǫ = λ n ǫ n Ground State: φ 0 = 1, ǫ 0 = 0 First correction: t t 2 φ 1 +(t ν 1/2) t φ 1 = ǫ 1 t 2 φ 1 = 1 2 t2 +(ν )t, E 1 = (ν )(ν ) Second correction: t t 2 φ 2 +(t ν 1/2) t φ 2 = ǫ 2 +(ǫ 1 t 2 )φ 1 [ ( t 4 11 φ 2 = ν )t ν +4ν2 + t 2 + (3+2ν)(7+4ν) ] t E 2 = (1+2ν)(3+2ν)(7+4ν) 2 In general, φ n = P 2n (t) and coeffs in front of leading terms can be found explicitly! (generalized Catalan numbers)

28 Do exist quantum systems with hidden algebra gl(d +1)?

29 gl d+1 -algebra (in R d ) (almost degenerate or totally symmetric, Young tableaux is a row) (n,0,0,...0) }{{} d 1 J i J ij 0 J 0 = J + i =, i = 1,2...d, t i = t i, i,j = 1,2...d, t j d t i n, t i i=1 d = t i J 0 = t i t j n, i = 1,2...d. t j (d +1) 2 generators j=1

30 if n = 0,1,2..., fin-dim irreps P (d) n = t 1 p 1 t 2 p 2...t d p d 0 Σp i n Remark. The flag P (d) is made out of finite-dimensional irreducible representation spaces P n (d) of the algebra gl d+1 taken in realization ( ). Any operator made out of generators ( ) has finite-dimensional invariant subspace which is finite-dimensional irreducible representation space and visa versa.

31 To the best of my knowledge almost all known explicitly (algebraically) eigenfunctions have a form Ψ(x) = (polynomial in φ(x)) factor with a non-singular function in the domain x as factor. What is a meaning (if any) of variables φ(x)?

32 Hamiltonian Reduction Method (Olshanetsky-Perelomov 77, Kazhdan-Kostant-Sternberg 78) Define Laplace-Beltrami operators on symmetric spaces of simple Lie groups (free/harmonic oscillator motion) Radial parts of L-B operators Olshanetsky-Perelomov Hamiltonians relevant from physical point of view. They can be associated with root systems. Rational case: H = 1 2 N [ 2 x 2 k=1 k +ω 2 x 2 k ] + 1 ν 2 α (ν α 1) α 2 (α x) 2 α R + where R + is a set of positive roots and ν α are coupling constants depending on the root length.

33 For all roots of the same length ν α = ν. They take discrete values but can be generalized to any value. Configuration space - Weyl chamber. Ground state wave function Ψ 0 (y) = (α y) ν α e ωy2/2 α R + The Hamiltonian is completely-integrable (super-integrable) and exactly-solvable for any value of ν > 1 2 and ω > 0. It is invariant wrt Weyl (Coxeter) group transformation (symmetry group of root space)

34 Trigonometric case: H = 1 2 N [ 2 y 2 k=1 k ] + β2 8 α 2 ν α (ν α 1) α R + sin 2 β 2 (α y) where R + is a set of positive roots and ν α are coupling constants depending on the root length. They take discrete values but can be generalized to any value. Configuration space - Weyl alcove. Ground state wave function Ψ 0 (y) = sin β ν α 2 (α y) α R + The Hamiltonian is completely-integrable and exactly-solvable for any value of ν > 1 2. It is invariant wrt Weyl group transformation + periodic.

35 Procedure: Gauging away ground state eigenfunction (similarity transformation) (Ψ 0 ) 1 (H E 0 )Ψ 0 = h Olshanetsky-Perelomov Hamiltonians (OPH) possess different symmetries (permutations, translation-invariance, reflections, periodicity etc). These symmetries correspond to the Weyl (Coxeter) group plus translations. By coding these symmetries to new coordinates (taking the Weyl (Coxeter) invariants as new coordinates) we find premature (undressed by symmetries) operators to these Hamiltonians. Example: Weyl(A n ) = S n +T

36 WHAT ARE THESE COORDINATES? Rational case Weyl (Coxeter)-invariant variables: t (Ω) a (x) = α Ω(α,x) a, where a s are the degrees of the Weyl (Coxeter) group W and Ω is an orbit. The invariants t are defined ambiguously, they depend on chosen orbit, but always lead to rational OPH h in a form of algebraic operator with polynomial coeffs. Trigonometric case trigonometric Weyl-invariant variables: τ (Ω) a (y) = α Ω a e iβ(α,y), where Ω a is an orbit generated by fundamental weight α a, a = 1,2,...,r (r rank of the root system) The invariants τ taken as coordinates always lead to trigonometric OPH h in a form of algebraic operator with polynomial coeffs.

37 Calogero Model (A N 1 Rational model) (F. Calogero, 69) N identical particles on a line with singular pairwise interaction.... x < x x x < < < N H Cal = 1 2 N i=1 ) ( 2 x 2 +ω2 2 x i +g i N i>j 1 (x i x j ) 2

38 Ψ 0 (x) = i<j x i x j ν e ω 2 x 2 i, g = ν(ν 1) h Cal = 2Ψ 1 0 (H Cal E 0 )Ψ 0 Y = x i, y i = x i 1 N Y, i = 1,...,N (x 1,x 2,...x N ) ( Y,t n (x) = σ n (y(x)) n = 2,3...N ) σ k (x) = x i1 x i2...x ik i 1 <i 2 <...<i k A ij = 2 h Cal = A ij (t) +B i (t) t i t j t i (N i +1)(1 j) t i 1 t j 1 + (2l j +i)t i+l 1 t j l 1 N l max(1,j i) B i = 1 N (1+νN)(N i +2)(N i +1)t i 2 +2ω(i 1)t i

39 Eigenvalues: ǫ n = 2ω N (i 1) n i the spectra of anisotropic harmonic oscillator, linear in quantum numbers. Hamiltonian: i=2 h = Pol(J i, J ij 0 ) gl(n 1) is the hidden algebra of N-body Calogero model. Eigenfunctions: they are elements of the flag P (N 1).

40 Sutherland Model (A N 1 Trigonometric model) (B Sutherland, 69) N identical particles on a circle with singular pairwise interaction x 1 x 2 x 3... x N H Suth = 1 2 N 2 x 2 k=1 k + g 4 k<l 1 sin 2 ( 1 2 (x k x l ))

41 Ψ 0 (x) = i<j ( ) 1 sin ν 2 (x i x j ), g = ν(ν 1) h Suth = 2Ψ 1 0 (H Suth E 0 )Ψ 0 Y = x i, y i = x i 1 N Y, i = 1,...,N (x 1,x 2,...x N ) ( e iy,η n (x) = σ n (e iy(x) ) n = 1,2...(N 1) ) A ij = 2 h Suth = A ij (η) +B i (η) η i η j η i (N i)j η i η j + (j i 2l)η i+l η j l N l max(1,j i) B i = ( 1 N +ν)i (N i)η i

42 Eigenvalues: quadratic in quantum numbers. Hamiltonian: ǫ n = Pol 2 (n i ) h = Pol(J i, J ij 0 ) gl(n 1) is the hidden algebra of N-body Sutherland model. Eigenfunctions: they are elements of the flag P (N 1).

43 BC N Rational model H BCN = 1 2 N ( 2 i=1 + g 2 2 N x 2 i=1 i x i 2 ω2 x 2 i 1 ) +g [ ] 1 (x i x j ) (x i +x j ) 2 i<j Ψ 0 = N x i x j ν x i +x j ν x i ν N 2 e ω 2 i=1 x2 i, i<j i=1 g = ν(ν 1),g 2 = ν 2 (ν 2 1), h BCN = (Ψ 0 ) 1 (H BCN E 0 )Ψ 0 (x 1,x 2,...x N ) ( σ k (x 2 ) ) k=1,2,...,n σ k (x) = x i1 x i2 x ik i 1 <i 2 < <i k

44 2 h BCN = A ij (σ) +B i (σ) σ i σ j σ i A ij = 2 l 0(2l +1+j i)σ i l 1 σ j+l B i = [1+ν 2 +2ν(N i)](n i +1)σ i 1 +2ωi σ i

45 Eigenvalues: ǫ n = 2ω N i n i the spectra of anisotropic harmonic oscillator, linear in quantum numbers. Hamiltonian: i=1 h = Pol(J i, J ij 0 ) gl(n) is the hidden algebra of BC N -rational model. Eigenfunctions: they are elements of the flag P (N).

46 BC N Trigonometric model (Inozemtsev model) H BCN = 1 2 N i=1 + g x i 2 + g 4 N i=1 [ ] N 1 sin 2( 1 2 (x i x j ) ) + 1 sin 2( 1 2 (x i +x j ) ) i<j 1 sin 2 + g 3 x i 4 N i=1 1 sin 2x i 2 N Ψ 0 = sin( 1 2 (x i x j )) ν sin( 1 2 (x i +x j )) ν i<j N i=1 sin(x i ) ν 2 sin( x i 2 ) ν 3, g = ν(ν 1),g 2 = ν 2 (ν 2 1), g 3 = ν 3 (ν 3 +2ν 2 1), h BCN = 2(Ψ 0 ) 1 (H BCN E 0 )Ψ 0 (x 1,x 2,...x N ) (ˆσ k (x) = σ k (cos(x)) k = 1,2...N ).

47 A ij =N ˆσ i 1 ˆσ j 1 l 0 2 h BCN = A ij (ˆσ) +B i (ˆσ) ˆσ i ˆσ j ˆσ i [ (i l)ˆσ i l ˆσ j+l +(l +j 1)ˆσ i l 1 ˆσ j+l 1 (i 2 l)ˆσ i 2 l ˆσ j+l (l +j +1)ˆσ i l 1 ˆσ j+l+1 ] B i = ν 3 2 (i N 1)ˆσ i 1 [ ν 2 + ν ν(2N i 1) ] i ˆσ i ν(n i +1)(N i +2)ˆσ i 2

48 Eigenvalues: quadratic in quantum numbers. Hamiltonian: ǫ n = Pol 2 (n i ) h = Pol(J i, J ij 0 ) gl(n) is the hidden algebra of BC N trigonometric model. Eigenfunctions: they are elements of the flag P (N)

49 Both A N and BC N rational and trigonometric models possess algebraic forms associated with preservation of the same flag of polynomials P (N). The flag is invariant wrt linear transformations in space of orbits t t +A. Their Hamiltonians (as well as higher integrals) can be written in the algebraic form h = P 2 (J(b gl ( ) N+1 )) where P 2 is a polynomial of second degree in the generators J of the maximal affine subalgebra of the algebra gl N+1 in realization ( ). Hence gl N+1 is their hidden algebra. From this viewpoint all four models are different faces of a single model. Supersymmetric A N and BC N rational and trigonometric models possess algebraic forms, preserve the same flag of (super)polynomials and their hidden algebra is the superalgebra gl(n + 1 N).

50 Do exist other quantum systems with gl(n) hidden algebra, with different Young tableau, different realization? (i) BC N -Elliptic model (ii) Matrix systems (iii) Discrete systems on uniform, exponential, mixed uniform-exponential lattices

51 For the OPH Hamiltonians for all exceptional root spaces G 2,F 4,E 6,7,8 (for both rational and trigonometric) and non-crystallographic H 3,4,I 2 (k) the eigenfunctions are polynomials in their invariants (in symmetric variables). Their hidden algebras are new infinite-dimensional but finite-generated algebras of differential operators. All of them have finite-dimensional invariant subspaces in polynomials. Generating elements of any such hidden algebra can be grouped in even number of (conjugated) Abelian algebras L i, L i and one Lie algebra B.

52 B L P p (B) L Figure: Triangular diagram relating the subalgebras L, L and B. p is integer. It is a generalization of Gauss decomposition for semi-simple algebras (p = 1).

53 G 2 -case: three-body problem with 2- and 3-body interaction p = 2 B [gl(2)] : J 1 = x, J 2 = x x n 3, J3 = 2y y n 3, J 4 = xj 0 x(x x +2y y n), L : R 0 = y, R 1 = x y, R 2 = x 2 y L : T 2 = y 2 xx, T 1 = yj 0 x, T 0 = yj 0 (J 0 +1) P (2) n = x p 1 y p 2 0 p 1 +2p 2 n

54 To the best of my knowledge almost all known explicitly (algebraically) eigenfunctions have a form Ψ(x) = (polynomial in φ(x)) factor with a non-singular function in the domain x as factor. What is a meaning (if any) of variables φ(x)? Invariants of the discrete group of symmetry of the system.