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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
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.
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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
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.