About: 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 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

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rdfs:label	Quasi-exact solvability (en)
rdfs:comment	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)
dcterms:subject	Differential operators Mathematical analysis Multivariable calculus
Wikipage page ID	48720870 (xsd:integer)
Wikipage revision ID	1117698814 (xsd:integer)
Link from a Wikipage to another Wikipage	Algebraic equation Invariant subspace Operator (mathematics) One-dimensional Eigenfunction Eigenvalue Function (mathematics) Lie algebra Space (mathematics) Algebra Finite-dimensional First-order differential operator Hilbert space Differential operators Mathematical analysis Eigenstate Polynomial Integer Multivariable calculus Second-order differential equation Multivariate polynomial Schrödinger operator Linear differential operator Functional space Representation space Dimension (mathematics) Isomorphism (algebra)
Link from a Wikipage to an external page	http://math.umn.edu/~olver/q_/qes21.pdf
sameAs	Quasi-exact solvability Quasi-exact solvability Quasi-exact solvability
dbp:wikiPageUsesTemplate	dbt:Citation dbt:Cite_journal dbt:Multiple_issues dbt:No_footnotes dbt:Orphan dbt:Reflist dbt:Technical
has abstract	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)
prov:wasDerivedFrom	wikipedia-en:Quasi-exact_solvability?oldid=1117698814&ns=0
page length (characters) of wiki page	4094 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Quasi-exact_solvability
is Link from a Wikipage to another Wikipage of	Quasi-exact-solvability Quasi-exactly-solvable problems Quasi-exactly solvable
is Wikipage redirect of	Quasi-exact-solvability Quasi-exactly-solvable problems Quasi-exactly solvable
is foaf:primaryTopic of	wikipedia-en:Quasi-exact_solvability

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