Poschl-Teller potentials based solution to Hilbert's tenth problem (2006)
Hypercomputers compute functions or numbers, or in general solve problems that cannot be computed or solved by a Turing machine. An adaptation of the hypercomputing quantum algorithm proposed by Tien D. Kieu is presented, to the dynamic algebra su (1, 1) performed on the Pöschl-Teller potentials. The classically incomputable problem that is solved with this hypercomputing algorithm is Hilbert's tenth problem. It is pointed out that a fundamental mathematical condition for these algorithms is the existence of an irreducible infinite dimensional unit representation of low-dimension algebras that admit the construction of coherent states of the Barut-Girardello type. Additionally, the hypercomputational algorithm on the infinite potential box previously constructed by the authors is presented as a limiting case of the proposed algorithm on the Pöschl-Teller potentials.