English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Open

Presents the pdf analysis as premium feature

PDF Analysis

Insights

Presents the summarisation as premium feature

Summarisation

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

This work reports and classifies the most general construction of rationalquantum potentials in terms of the generalized Hermite polynomials. This isachieved by exploiting the intrinsic relation between third-ordershape-invariant Hamiltonians and the fourth Painlev\'e equation, such that thegeneralized Hermite polynomials emerge from the $-1/x$ and $-2x$ hierarchies ofrational solutions. Such a relation unequivocally establishes the discretespectrum structure, which, in general, is composed as the union of a finite-and infinite-dimensional sequence of equidistant eigenvalues separated by agap. The two indices of the generalized Hermite polynomials determine thedimension of the finite sequence and the gap. Likewise, the complete set ofeigensolutions can be decomposed into two disjoint subsets. In this form, theeigensolutions within each set are written as the product of a weight functiondefined on the real line times a polynomial. These polynomials fulfill asecond-order differential equation and are alternatively determined from athree-term recurrence relation (second-order difference equation), the initialconditions of which are also fixed in terms of generalized Hermite polynomials.

On the general family of third-order shape-invariant Hamiltonians related to generalized Hermite polynomials