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Quasi-exact solvability

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A linear differential operator L is called quasi-exactly-solvable (QES) if it has a finite-dimensional invariant subspace of functions { V } n {\displaystyle \{{\mathcal {V}}\}_{n}} such that L : { V } n → { V } n , {\displaystyle L:\{{\mathcal {V}}\}_{n}\rightarrow \{{\mathcal {V}}\}_{n},} where n is a dimension of { V } n {\displaystyle \{{\mathcal {V}}\}_{n}} . There are two important cases:

  1. { V } n {\displaystyle \{{\mathcal {V}}\}_{n}} is the space of multivariate polynomials of degree not higher than some integer number; and
  2. { V } n {\displaystyle \{{\mathcal {V}}\}_{n}} is a subspace of a Hilbert space. Sometimes, the functional space { V } n {\displaystyle \{{\mathcal {V}}\}_{n}} is isomorphic to the finite-dimensional representation space of a Lie algebra g of first-order differential operators. In this case, the operator L is called a g-Lie-algebraic Quasi-Exactly-Solvable operator. Usually, one can indicate basis where L has block-triangular form. If the operator L is of the second order and has the form of the Schrödinger operator, it is called a Quasi-Exactly-Solvable Schrödinger operator.

The most studied cases are one-dimensional s l ( 2 ) {\displaystyle sl(2)} -Lie-algebraic quasi-exactly-solvable (Schrödinger) operators. The best known example is the sextic QES anharmonic oscillator with the Hamiltonian

{ H } = − d 2 d x 2 + a 2 x 6 + 2 a b x 4 + [ b 2 − ( 4 n + 3 + 2 p ) a ] x 2 ,   a ≥ 0   ,   n ∈ N   ,   p = { 0 , 1 } , {\displaystyle \{{\mathcal {H}}\}=-{\frac {d^{2}}{dx^{2}}}+a^{2}x^{6}+2abx^{4}+x^{2},\ a\geq 0\ ,\ n\in \mathbb {N} \ ,\ p=\{0,1\},}

where (n+1) eigenstates of positive (negative) parity can be found algebraically. Their eigenfunctions are of the form

Ψ ( x )   =   x p P n ( x 2 ) e − a x 4 4 − b x 2 2   , {\displaystyle \Psi (x)\ =\ x^{p}P_{n}(x^{2})e^{-{\frac {ax^{4}}{4}}-{\frac {bx^{2}}{2}}}\ ,}

where P n ( x 2 ) {\displaystyle P_{n}(x^{2})} is a polynomial of degree n and (energies) eigenvalues are roots of an algebraic equation of degree (n+1). In general, twelve families of one-dimensional QES problems are known, two of them characterized by elliptic potentials.

References

  • Turbiner, A.V.; Ushveridze, A.G. (1987). "Spectral singularities and quasi-exactly solvable quantal problem". Physics Letters A. 126 (3). Elsevier BV: 181–183. Bibcode:1987PhLA..126..181T. doi:10.1016/0375-9601(87)90456-7. ISSN 0375-9601.
  • Turbiner, A. V. (1988). "Quasi-exactly-solvable problems and s l ( 2 , R ) {\displaystyle sl(2,R)} algebra". Communications in Mathematical Physics. 118 (3). Springer Science and Business Media LLC: 467–474. doi:10.1007/bf01466727. ISSN 0010-3616. S2CID 121442012.
  • González-López, Artemio; Kamran, Niky; Olver, Peter J. (1994), "Quasi-exact solvability", Lie algebras, cohomology, and new applications to quantum mechanics (Springfield, MO, 1992), Contemp. Math., vol. 160, Providence, RI: Amer. Math. Soc., pp. 113–140
  • Turbiner, A.V. (1996), "Quasi-exactly-solvable differential equations", in Ibragimov, N.H. (ed.), CRC Handbook of Lie Group Analysis of Differential Equations, vol. 3, Boca Raton, Fl.: CRC Press, pp. 329–364, ISBN 978-0849394195
  • Ushveridze, Alexander G. (1994), Quasi-exactly solvable models in quantum mechanics, Bristol: Institute of Physics Publishing, ISBN 0-7503-0266-6, MR 1329549

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