Quasi-exact solvability
A linear differential operator L is called quasi-exactly-solvable (QES) if it has a finite-dimensional invariant subspace of functions [math]\displaystyle{ \{\mathcal{V}\}_n }[/math] such that [math]\displaystyle{ L: \{\mathcal{V}\}_n \rightarrow \{\mathcal{V}\}_n, }[/math] where n is a dimension of [math]\displaystyle{ \{\mathcal{V}\}_n }[/math]. There are two important cases:
- [math]\displaystyle{ \{\mathcal{V}\}_n }[/math] is the space of multivariate polynomials of degree not higher than some integer number; and
- [math]\displaystyle{ \{\mathcal{V}\}_n }[/math] is a subspace of a Hilbert space. Sometimes, the functional space [math]\displaystyle{ \{\mathcal{V}\}_n }[/math] 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 [math]\displaystyle{ sl(2) }[/math]-Lie-algebraic quasi-exactly-solvable (Schrödinger) operators. The best known example is the sextic QES anharmonic oscillator with the Hamiltonian
[math]\displaystyle{ \{\mathcal{H}\} = -\frac{d^2}{dx^2} + a^2 x^6 + 2abx^4 + [b^2 - (4 n + 3 + 2p) a] x^2, \ a \geq 0\ ,\ n\in\mathbb{N}\ ,\ p=\{0,1\}, }[/math]
where (n+1) eigenstates of positive (negative) parity can be found algebraically. Their eigenfunctions are of the form
[math]\displaystyle{ \Psi (x)\ =\ x^p P_n(x^2) e^{-\frac{a x^4}{4} - \frac{b x^2}{2} } \ , }[/math]
where [math]\displaystyle{ P_n(x^2) }[/math] 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 (Elsevier BV) 126 (3): 181–183. doi:10.1016/0375-9601(87)90456-7. ISSN 0375-9601. Bibcode: 1987PhLA..126..181T.
- Turbiner, A. V. (1988). "Quasi-exactly-solvable problems and [math]\displaystyle{ sl(2,R) }[/math] algebra". Communications in Mathematical Physics (Springer Science and Business Media LLC) 118 (3): 467–474. doi:10.1007/bf01466727. ISSN 0010-3616.
- 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., 160, Providence, RI: Amer. Math. Soc., pp. 113–140
- Turbiner, A.V. (1996), "Quasi-exactly-solvable differential equations", in Ibragimov, N.H., CRC Handbook of Lie Group Analysis of Differential Equations, 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
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