Exact polynomial and non-polynomial bound states in a one-dimensional quasi-exactly solvable sl(2) potential

This study presents a systematic investigation of one-dimensional quasi-exactly solvable (QES) potentials with [Formula: see text] algebra that preserves finite-dimensional polynomial subspace. We derive analytical solutions expressed through confluent Heun functions (CHFs) and establish a fundamental equivalence between the polynomial conditions for CHFs and the underlying Lie algebraic constraints. When model parameters deviate from conventional polynomial solutions, we demonstrate the existence of an infinite family of non-polynomial bound state solutions extending beyond the invariant polynomial subspace. These novel solutions exhibit exceptional properties under specific parametric conditions, significantly expanding the spectrum of known analytical solutions in QES systems.