Countable infinity of isospectral potential families

Abstract

Given any one-dimensional potential V(x) with n bound states, we construct an infinite number of isospectral potential families V^(a)(λ₀,λ₁,...,λ_n-1;x). The parameter a takes integer values and the n parameters λ_i take real continuous values. All potentials have identical bound-state eigenvalues, but the reflection and transmission coefficients depend on the index a.