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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 such that where n is a dimension of . There are two important cases: 1.
* is the space of multivariate polynomials of degree not higher than some integer number; and 2.
* is a subspace of a Hilbert space. Sometimes, the functional space 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 -Lie-algebraic quasi-exactly-solvable (Schrödinger) operators. The best known example is the sextic QES anharmonic oscillator with the Hamiltonian where (n+1) eigenstates of positive (negative) parity can be found algebraically. Their eigenfunctions are of the form where 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. (en)
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- 4094 (xsd:nonNegativeInteger)
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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 such that where n is a dimension of . There are two important cases: The most studied cases are one-dimensional -Lie-algebraic quasi-exactly-solvable (Schrödinger) operators. The best known example is the sextic QES anharmonic oscillator with the Hamiltonian where (n+1) eigenstates of positive (negative) parity can be found algebraically. Their eigenfunctions are of the form (en)
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- Quasi-exact solvability (en)
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