Quasi-exactly-solvable Problems

A linear differential operator ''L'' is called quasi-exactly-solvable (QES) if it has a finite-dimensional invariant subspace of functions $\_n$ such that $L: \_n \rightarrow \_n,$ where ''n'' is a dimension of $\_n$. There are two important cases:
# $\_n$ is the space of multivariate polynomials of degree not higher than some integer number; and
# $\_n$ is a subspace of a Hilbert space. Sometimes, the functional space $\_n$ 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 $sl(2)$-Lie-algebraic quasi-exactly-solvable (Schrödinger) operators. The best known example is the sextic QES anharmonic oscillator with the Hamiltonian

\ = -\frac + a^2 x^6 + 2abx^4 + ^2 - (4 n + 3 + 2p) ax^2, \ a \geq 0\ ,\ n\in\mathbb\ ,\ p=\,

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

$\Psi (x)\ =\ x^p P_n(x^2) e^ \ ,$

where $P_n(x^2)$ 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

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External links

*{{citation, title=A Quasi-Exactly Solvable Travel Guide, first=Peter, last=Olver, url=http://math.umn.edu/~olver/q_/qes21.pdf

Differential operators
Mathematical analysis
Multivariable calculus