Abstract
We construct a new example of 2 × 2-matrix quasi-exactly solvable (QES) Hamiltonian which is associated to a potential depending on the Jacobi elliptic functions. We establish three necessary and sufficient algebraic conditions for the previous operator to have an invariant vector space whose generic elements are polynomials. This operator is called quasi-exactly solvable.
Highlights
In quantum physics, one of the main mathematical problems consists in constructing the spectrum of a linear operator defined on a suitable domain of Hilbert space
We construct a new example of 2 × 2-matrix quasi-exactly solvable (QES) Hamiltonian which is associated to a potential depending on the Jacobi elliptic functions
We will apply in a systematic way the previous QES analytic method in order to construct a 2 × 2-matrix QES Hamiltonian associated to a potential depending on the Jacobi elliptic functions [11,12]
Summary
One of the main mathematical problems consists in constructing the spectrum of a linear operator defined on a suitable domain of Hilbert space. In most cases, this type of problem cannot be explicitly solved, in other words the eigenvalues of the Hamiltonian cannot be computed algebraically. The two major examples of this kind are the celebrated harmonic quantum oscillator and the hydrogen atom (i.e. 3dimensional Schrödinger equation coupled to an external Coulomb potential) These examples are called exactly solvable in the sense that the full spectrum of the Hamiltonian is found explicitly.
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