Quasi-Exact Solvability of Heun and related equations

Transcription

1 Quasi-Exact Solvability of Heun and related equations M. A. González León 1, J. Mateos Guilarte and M. de la Torre Mayado 1 Departamento de Matemática Aplicada and IUFFyM (Universidad de Salamanca, Spain) Departamento de Física Fundamental and IUFFyM (Universidad de Salamanca, Spain) CRM-ICMAT Workshop on Exceptional Orthogonal Polynomials and exact Solutions in Mathematical Physics Segovia (Spain), September 7 1, 014. M.A.G.L. - J. Mateos - M. de la Torre (USAL) Quasi-Exact Solvability of Heun and related equations Segovia CRM-ICMAT workshop 1 / 8

2 Outline 1 Introduction Confluent Forms of Heun Equation 3 Quasi-Exact Solvable Systems 4 Normalizable QES systems 5 Some applications M.A.G.L. - J. Mateos - M. de la Torre (USAL) Quasi-Exact Solvability of Heun and related equations Segovia CRM-ICMAT workshop / 8

3 Introduction The Heun Equation (HEq) ( u γ (z) + z + δ z 1 + ɛ ) u (z) + z a αβz q z(z 1)(z a) u(z) = 0 is analyzed in the context of Quasi-Exactly Solvable systems associated with the Lie algebra sl(, R). Goals: To prove that the Heun equation is QES under certain conditions of the parameters. The four confluent cases: Confluent (CHEq), Bi-confluent (BCHEq), Double-confluent (DCHEq) and Tri-confluent (TCHEq) Heun equations inherit this property. The four confluent cases are the canonical forms for QES spectral problems that admit normalizable solutions. A. Ronveaux, Heun s Differential Equations (Oxford University Press, Oxford, U.K., 1995). S.Y. Slavyanov and W. Lay, Special Functions: A Unified Theory Based on Singularities (Oxford University Press, Oxford, U.K., 000). M.A.G.L. - J. Mateos - M. de la Torre (USAL) Quasi-Exact Solvability of Heun and related equations Segovia CRM-ICMAT workshop 3 / 8

4 Heun Equation Heun equation is the Fuchsian equation with four singularities, i.e. the second-order linear differential equation defined in the Riemann sphere with four regular singular points. It is written in canonical natural form, with the finite regular singular points located at z = 0, z = 1 and z = a, as D H u(z) = q u(z) D H = z(z 1)(z a) d dz + (γ(z 1)(z a) + δz(z a) + ɛz(z 1)) d dz + αβz Fuchs condition: 0 1 a α z; q 1 γ 1 δ 1 ɛ β γ + δ + ɛ = α + β + 1 M.A.G.L. - J. Mateos - M. de la Torre (USAL) Quasi-Exact Solvability of Heun and related equations Segovia CRM-ICMAT workshop 4 / 8

5 Confluent Forms of Heun Equation CHEq: Confluence of z = a and z = into an irregular singular point at z = of rank-1 confluent type. D CH u(z) = q u(z) D CH = z(z 1) d dz + (γ(z 1) + δz + ɛz(z 1)) d dz + αz DCHEq: Confluence of z = a with z = and z = 1 with z = 0 into two irregular singular points at z = and z = 0 of rank-1 confluent type. D DCH u(z) = q u(z) D DCH = z d dz + ( z + γz + δ) d dz + αz M.A.G.L. - J. Mateos - M. de la Torre (USAL) Quasi-Exact Solvability of Heun and related equations Segovia CRM-ICMAT workshop 5 / 8

6 Confluent Forms of Heun Equation BCHEq: Confluence of z = 1, z = a and z = into an irregular singular point at z = of rank- confluent type. D BCH u(z) = q u(z) D BCH = z d dz + ( γ + δz + z ) d dz + αz TCHEq: Confluence of z = 0 with the z = point of the BCHeq all the original singular points reduce to an irregular singular point at z = of rank-3 confluent type. D TCH u(z) = q u(z) D TCH = d dz + ( γ + z ) d dz + αz M.A.G.L. - J. Mateos - M. de la Torre (USAL) Quasi-Exact Solvability of Heun and related equations Segovia CRM-ICMAT workshop 6 / 8

7 Quasi-Exact Solvable Systems Quasi-Exactly-Solvable (QES) systems are spectral problems characterized by the fact that part of the eigenvalues and eigenfunctions, but not the whole spectrum, can be found algebraically. An important class of QES systems are characterized by a Hamiltonian that is an element of the enveloping algebra of a finite-dimensional Lie algebra of differential operators, which in turn admits a finite dimensional invariant module of smooth functions. Turbiner AV "Quasi-exactly-solvable problems and sl(,r) algebra", Commun. Math. Phys Ushveridze AG "Quasi-exactly solvable models in quantum mechanics", Sov. J. Part. Nucl Shifman M and Turbiner AV "Quantal problems with partial algebraization of the spectrum", Commun. Math. Phys Shifman M "New Findings in Quantum Mechanics (Partial Algebraization of the Spectral Problem)", ITEP Lectures on Particle Physics and Field Theory, vol. II, chapter VII. World Scientific Lecture Notes in Physics 6. (Singapore: World Scientific). M.A.G.L. - J. Mateos - M. de la Torre (USAL) Quasi-Exact Solvability of Heun and related equations Segovia CRM-ICMAT workshop 7 / 8

8 Quasi-Exact Solvable Systems For a generic second-order (Lie-algebraic) QES spectral problem, Hf (z) = λf (z), the Hamiltonian operator H is written (possibly after an adequate gauge" transformation) as a quadratic combination: H = c + J + + c 0 J 0 + c J + c ++ (J + ) + c 00 (J 0 ) + c (J ) + ( + c +0 J + J 0 + J 0 J +) ( + c + J + J + J J +) ( + c 0 J 0 J + J J 0) of J, J + and J 0, the generators of the sl(, R) Lie algebra: J = d dz, J0 = z d dz n, J+ = z d dz nz. [ J 0, J +] [ = J +, J 0, J ] = J [, J +, J ] = J 0. n is a non-negative integer that determines the dimension of the invariant module. M.A.G.L. - J. Mateos - M. de la Torre (USAL) Quasi-Exact Solvability of Heun and related equations Segovia CRM-ICMAT workshop 8 / 8

9 HEq as a QES system D H is a quadratic combination of J +, J and J 0 if and only if one of the characteristic exponents associated to the singularity at infinity, i.e. α or β, is a non-positive integer. This QES condition corresponds with a very well known result in the theory of Heun equation: Heun polynomials exists if and only if α or β belongs to Z. Case α = n: D H = ( β + n 1 ) J + + ( a(γ + δ + n 1) β + δ) J 0 + a (γ + n 1) J + (a(γ + δ + n) + β δ + 1) (J 0 ) + 1 (J + J 0 + J 0 J +) n + a((γ + δ 1) + n) + β δ n ( + J + J + J J +) + a (J 0 J + J J 0) (n + ) M.A.G.L. - J. Mateos - M. de la Torre (USAL) Quasi-Exact Solvability of Heun and related equations Segovia CRM-ICMAT workshop 9 / 8

10 HEq as a QES system The polynomial solutions are obtained in the standard way: Considering the Frobenius solution at z = 0: Q(z) = k=0 P k(q) k! a k (γ) k z k the following three-term recurrence is verified: P k+1 = (q+k(k n+β δ +a(k+γ +δ 1)))P k a(k n 1)k(k+β 1)(k+γ 1)P k 1 Q(z) truncates to a polynomial of degree n if q is chosen as one of the n + 1 roots of P n+1(q). Q j(z) = n k=0 P k(q j) k! a k (γ) k z k, j = 1,..., n + 1 n+1 P n+1(q) = (q q j) j=1 M.A.G.L. - J. Mateos - M. de la Torre (USAL) Quasi-Exact Solvability of Heun and related equations Segovia CRM-ICMAT workshop 10 / 8

11 QES Solvability for the confluent equations The different confluences in the Heun equation respect the QES property: CHEq: QES condition: α = n ɛ. D CH = ɛ J + + (γ + δ + n ɛ 1) J 0 1 (γ + n 1) J + + (γ + δ + n ɛ) (J 0 ) n + (γ + δ ɛ 1) + n ( J + J + J J +) 1 (n + ) (J 0 J + J J 0) DCHEq: QES condition: α = n. D DCH = J + + (γ + n 1) J 0 + δ J + γ + n (n + ) ( J + J + J J +) (γ + n) n + (J0 ) M.A.G.L. - J. Mateos - M. de la Torre (USAL) Quasi-Exact Solvability of Heun and related equations Segovia CRM-ICMAT workshop 11 / 8

12 QES Solvability for the confluent equations BCHEq: QES condition: α = n. ( D BCH = J + + δ J 0 γ + n 1 ) J + δ n + (J0 ) δ ( J + J + J J +) + 1 (J 0 J + J J 0) n + TCHEq: QES condition: α = n. D TCH = J + + γ J + (J 0 ) M.A.G.L. - J. Mateos - M. de la Torre (USAL) Quasi-Exact Solvability of Heun and related equations Segovia CRM-ICMAT workshop 1 / 8

13 Normalizable QES systems Gonzalez-Kamran-Olver (1993) have classified QES spectral problems that admit normalizable wave functions. Let H be a general quasi-exactly solvable operator: H = P (z) d dz + P1(z) d dz + P0(z), The spectral Schrödinger problem associated to H admits normalizable wave functions if the function P (z) is of one of the following canonical forms: 1. P (z) = (z + 1), z (, ).. a) P (z) = (z 1), z [1, ), b) P (z) = (z 1), z (, 1]. 3. a) P (z) = z, z (0, ), b) P(z) = z, z (, 0). 4. P (z) = z, z [0, ). 5. P (z) = 1, z (, ). Gonzalez-Lopez A, Kamran N & Olver PJ "Normalizability of One-dimensional Quasi-exactly Solvable Schrödinger Operators", Commun. Math. Phys M.A.G.L. - J. Mateos - M. de la Torre (USAL) Quasi-Exact Solvability of Heun and related equations Segovia CRM-ICMAT workshop 13 / 8

14 Normalizable QES systems Any quasi-exactly solvable operator of the general form H = P (z) d dz + P1(z) d dz + P0(z), with P (z) > 0 on an interval I R, can be written in Schrödinger form: where x is determined by: µ(z) H and µ(z) is called the "gauge" factor. 1 µ(z) = d dx + V(x). ( ) dz = P (z) z = ζ(x) dx If ϕ(z) is a wave function in the original coordinates, then ψ(x) = µ(ζ(x)) ϕ(ζ(x)) will be the corresponding wave function in the "Schrödinger" coordinates. M.A.G.L. - J. Mateos - M. de la Torre (USAL) Quasi-Exact Solvability of Heun and related equations Segovia CRM-ICMAT workshop 14 / 8

15 Normalizable QES systems and Confluent Heun Eqs. CHEq. The change of variable: z z 1 moves the regular singular points in z = ±1. CHEq operator reads: D CH = ( ) z d ( ɛ ) d 1 dz + (z 1) + γ(z 1) + δ(z + 1) dz + α (z + 1) and it belongs to the canonical Case of normalizable canonical QES systems. Up to additive constants it is written in Schrödinger form as: µ(x)d CH 1 µ(x) = d dx a cosh x b cosh x c coth x csch x d csch x (1) where x = arccosh z, and µ(z) = (z 1) δ 1 4 (z + 1) γ 1 4 e ɛz 4. But general QES systems of type that admit normalizable wave-functions have necessarily a potential term of the form (1). Thus the Confluent Heun operator, in the QES case, represents the Canonical form of any QES spectral problem of type. M.A.G.L. - J. Mateos - M. de la Torre (USAL) Quasi-Exact Solvability of Heun and related equations Segovia CRM-ICMAT workshop 15 / 8

16 Normalizable QES systems and Confluent Heun Eqs. DCHEq. DCHEq belongs to the canonical Case 3 of normalizable canonical QES systems. Up to additive constants his Schrödinger form is: µ(x)d DCH 1 µ(x) = d dx a e x b e x c e x d e x with x = ln z, and µ(z) = z γ 1 e 1 (z δ z ). This expression corresponds with the general QES systems of type 3 that admit normalizable solutions. BCHEq and TCHEq Cases 4 and 5. What about Case 1? It can be reduced to CHEq, after a Mœbius transformation that moves the regular singular points to z = ±i. M.A.G.L. - J. Mateos - M. de la Torre (USAL) Quasi-Exact Solvability of Heun and related equations Segovia CRM-ICMAT workshop 16 / 8

17 Summary Heun equation is a Quasi-Exactly Solvable system associated to the Lie Algebra sl(, R) if one of the characteristic exponents of z = is a non-positive integer. The processes that convert Heun Equation into the four confluent Heun equations preserves the QES character, leading to other four different QES systems associated to sl(, R). The four Confluent QES systems encompasses all the canonical cases of QES systems admitting normalizable wave-functions in the associated Schrödinger spectral problems. M.A.G.L. - J. Mateos - M. de la Torre (USAL) Quasi-Exact Solvability of Heun and related equations Segovia CRM-ICMAT workshop 17 / 8

18 Applications: 1) Demkov wave functions "Demkov" wave functions are Elementary solutions to the quantum problem of diatomic molecular ions, in the Born-Oppenheimer approximation (i.e. the Two fixed Coulombian centers quantum problem) Eigenfunctions = irrational factors exponentials polynomials. Energy levels of "hydrogenoid" type. The Schrödinger equation: [ 1 ( x 1 ) + + Z1 x x3 r 1 ] Z Ψ = E Ψ r is separable in spheroidal coordinates, i.e. the rotation about the focal axis of the D elliptic coordinates r1 + r r r1 ξ = (1, + ), η = ( 1, 1) R R x 1 = R ξ η, x = R ξ 1 1 η cos ϕ, x 3 = R ξ 1 1 η sin ϕ Demkov YN "Elementary Solutions of the Quantum Problem of the Motion of a Particle in the Field of Two Coulomb Centers", JETP Lett M.A.G.L. - J. Mateos - M. de la Torre (USAL) Quasi-Exact Solvability of Heun and related equations Segovia CRM-ICMAT workshop 18 / 8

19 The search for separated solutions: Ψ(ξ, η, ϕ) = F(ξ) G(η) e imϕ lead to the "Radial" and "Angular" equations: [ d (ξ 1) df(ξ) ] [ ] R E + dξ dξ (ξ 1) + R(Z 1 + Z )ξ + λ m F(ξ) = 0 () ξ 1 [ d (1 η ) dg(η) ] [ ] R E + dη dη (1 η ) + R(Z Z 1)η λ m G(η) = 0 (3) 1 η with separation constants: m Z and λ R. Both equations are the Generalized Spheroidal Equation (GSEq), but with different intervals of definition: ξ (1, + ), η ( 1, 1) and not identical parameters. Two interesting sub-cases: Z 1 = Z Angular GSEq reduces to Spheroidal equation. Z = 0 Both equations are the same GSEq. The Demkov approach consists in searching for eigen-functions corresponding to the energy levels of a hydrogenoid atom, i.e. E = (Z 1+Z ), for the radial equation, and simultaneously in the angular one: E = (Z 1 Z ), with n 1, n N. n n 1 M.A.G.L. - J. Mateos - M. de la Torre (USAL) Quasi-Exact Solvability of Heun and related equations Segovia CRM-ICMAT workshop 19 / 8

20 These two hypotheses are compatible only if the two energies coincide, i.e. there exist solutions only for values of n 1 and n that solve the diophantine equation: (Z 1 + Z ) n 1 = (Z1 Z) n Moreover, the value of the separation constant λ in the two equations obviously must be the same. The corresponding wave functions are "elementary": Ψ(ξ, η, ϕ) = (ξ 1) m (1 η ) m u r (ξ) u a (η) e R(Z 1 +Z ) n ξ 1 e R(Z 1 Z ) n η e imϕ where u r (ξ) and u a (η) are polynomials up to n 1 m 1 and n m 1 order. There exists Demkov solutions only for several concrete charge distributions. Having fixed the charges, Demkov wave functions occur only for quantum numbers n 1 and n that solve the diophantine equation. Forcing the same value of the separation constant λ in the radial and angular equations is only possible for concrete values of the internuclear distance R. M.A.G.L. - J. Mateos - M. de la Torre (USAL) Quasi-Exact Solvability of Heun and related equations Segovia CRM-ICMAT workshop 0 / 8

21 A similar situation can be achieved in the corresponding Planar problem: [ 1 ( ) ] + Z1 Z Ψ = E Ψ x1 x r 1 r The equation is separable in D elliptic coordinates: (ξ, η). (ξ 1) d F(ξ) + ξ df(ξ) [ ] ER + dξ dξ ξ + R(Z 1 + Z )ξ + λ F(ξ) = 0 (4) (1 η ) d G(η) dη η dg(η) dη [ ] ER η + R(Z Z 1)η + λ G(η) = 0 (5) (4) and (5) can be easily identified as the algebraic version of Razavy (REq) and Whittaker-Hill (WHEq) equations. The existence of Demkov solutions in both the three and two dimensional cases is completely explained having into account that: GSEq, REq and WHEq are particular cases of CHEq, The hydrogenoid-type hypothesis represents exactly the QES condition for the corresponding CH equations. Miller W and Turbiner AV, 014. J. Phys. A: Math. Theor MAGL, Mateos J, Moreno A, de la Torre M, 014. arxiv: [math-ph] M.A.G.L. - J. Mateos - M. de la Torre (USAL) Quasi-Exact Solvability of Heun and related equations Segovia CRM-ICMAT workshop 1 / 8

22 Applications: ) Stability of Kinks Another interesting example is the Heun equation arising in the stability analysis of a kink solution in a (1 + 1)-dimensional field theory: Let us consider the field theory in dimensions determined by the lagrangian density: L = 1 νφ ν φ U(φ) ; U(φ) = The theory admits a well known kink solution: φ K(x) = µ 8(1 + ε ) sinh ( ) µx ε + sinh ( µx ) (φ + ε ) ( 1 φ ) Graphical representation of U(φ) (left) and φ K(x) (right), for µ = 1, ε = 1. Christ NH and Lee TD "Quantum expansion of soliton solutions", Phys. Rev. D M.A.G.L. - J. Mateos - M. de la Torre (USAL) Quasi-Exact Solvability of Heun and related equations Segovia CRM-ICMAT workshop / 8

23 And the stability analysis around φ K(x) leads to the Schrödinger equation: ] [ d dx + V(x) ψ(x) = λ ψ(x), V(x) = d U(φ) dφ with: ( ) V(x) = µ sinh4 µx 5 ε sinh ( ) µx ε + (1+ε )(1 ε ) 4ε ( sinh ( )) µx ε φ=φk (x) Translational invariance allows to calculate the ground state (λ 0 = 0) in a simple way: ψ 0(x) dφ K (x), i.e. dx µ ( ) ( ε cosh µx ) ( ) ψ 0(x) (( ) 1 + ( )) 1 ε + sinh µx 3/ 1 φ K(x) φ K (x) + ε Christ & Lee identify, for the particular case ε = 1, the second excited state: λ = 3 ( 4 µ ; ψ (x) 1 φ K (x) φ K(x) 1 ) 4 Thus, for this value of ε, two eigen-states are known, but not the whole spectrum, we have a QES problem is it a sl()-qes system? M.A.G.L. - J. Mateos - M. de la Torre (USAL) Quasi-Exact Solvability of Heun and related equations Segovia CRM-ICMAT workshop 3 / 8

24 Using as independent variable z = φ K(x) the Schrödinger equation is written in algebraic form: ( ) ) z(z 1) (z + ε ) d ψ dz + (z 1) z (1 3ε z ε dψ dz 1 4 (15z 6( ε )z + 1 ε )ψ(z) = 1 + ε µ λψ(z) (6) 0 1 ε 0 1 λ µ 0 3 z; 1 1 λ 1 5 µ 1+ε µ λ Equation (6) is Fuchsian but it is not a canonical form of Heun equation. We know that Heun equation (in canonical natural form and for specific values of the characteristic exponents) is a sl()-qes system, but it is not true for equation (6). This fact has induced to several authors to present this example as a case of Quasi-Exactly solvability not associated to sl() and even to conjecture that in general Heun equation is not a sl() QES system. The correct way to clarify this problem is obviously to convert equation (6) into a canonical natural Heun equation. The standard change of variable, determined by the characteristic exponents at z = 1 is: ψ(z) = (1 z) 1 λ µ f (z) M.A.G.L. - J. Mateos - M. de la Torre (USAL) Quasi-Exact Solvability of Heun and related equations Segovia CRM-ICMAT workshop 4 / 8

25 ( ) z(z 1)(z + ε )f (z) + γ(z 1)(z + ε ) + δz(z + ε ) + ɛz(z 1) f (z) + (αβz q)f (z) = 0 (7) γ = 1 ; δ = λ µ ; ɛ = 1 ; α = 3 + q = (1 + ε ) λ µ + ε (1 1 λ µ ) ε 1 λ µ ; β = λ µ z; q 1 1 λ µ + λ µ 1 λ µ Equation (7) is QES only if α or β are a non-positive integer, but this impossible for β and only available for α in the specific case: λ = 3 4 µ α = 1, β = 3 Equation (7) is QES only for λ = 3 4 µ, i.e. exactly the "magic" eigenvalue detected previously. We have a problem because to fix this value of λ automatically imposes that the spectral parameter q is necessarily: q = 1 + ε. M.A.G.L. - J. Mateos - M. de la Torre (USAL) Quasi-Exact Solvability of Heun and related equations Segovia CRM-ICMAT workshop 5 / 8

26 However we can calculate the set of two first-order polynomials that corresponds to the Heun equation with α = n = 1, considering a free valued spectral parameter q. In this specific case the allowed values of q are the two roots of P (q): P (q) = q q q1 = 1, q = 3 4 and these values are compatible with the required q = 1 + ε if and only if ε = 1 q = 1 = q1, an thus it is explained the "magic" value of ε previously obtained by trial and error methods. Explicit calculation lead to the polynomial solution: f (z) = z 1 4 and inverting the changes of variables, the wave-function: ( ψ (x) 1 φ K (x) φ K(x) 1 ) 4, λ = 3 4 µ M.A.G.L. - J. Mateos - M. de la Torre (USAL) Quasi-Exact Solvability of Heun and related equations Segovia CRM-ICMAT workshop 6 / 8

27 Graphical representation of V(x) (left) for µ = 1, ε = 1, and ψ0(x) (green) and ψ(x) (blue). In conclusion: The original Schrödinger equation is transformable into a Heun equation that is a QES system only for a concrete value of the Schrödinger eigenvalue. QES condition also fixes the spectral Heun parameter. Compatibility of these requirements determines that only one of the polynomial solutions of the Heun equation is associated to a genuine eigenfunction of the physical problem. M.A.G.L. - J. Mateos - M. de la Torre (USAL) Quasi-Exact Solvability of Heun and related equations Segovia CRM-ICMAT workshop 7 / 8

28 Thank you very much M.A.G.L. - J. Mateos - M. de la Torre (USAL) Quasi-Exact Solvability of Heun and related equations Segovia CRM-ICMAT workshop 8 / 8