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/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Applying certain known theorems about one-dimensional periodic potentials, weshow that the energy spectrum of the associated Lam\'{e} potentials$$a(a+1)m~{\rm sn}^2(x,m)+b(b+1)m~{\rm cn}^2(x,m)/{\rm dn}^2(x,m)$$ consists ofa finite number of bound bands followed by a continuum band when both $a$ and$b$ take integer values. Further, if $a$ and $b$ are unequal integers, we showthat there must exist some zero band-gap states, i.e. doubly degenerate stateswith the same number of nodes. More generally, in case $a$ and $b$ are notintegers, but either $a + b$ or $a - b$ is an integer ($a \ne b$), we againshow that several of the band-gaps vanish due to degeneracy of states with thesame number of nodes. Finally, when either $a$ or $b$ is an integer and theother takes a half-integral value, we obtain exact analytic solutions forseveral mid-band states.Comment: 18 pages, 2 figure

Some exact results for mid-band and zero band-gap states of associated Lamé potentials