Minimally relativistic Schrödinger equation and the linearly rising potential

In place of the usual approximation $\text{T = (m}{}^2\text{c}{}^4\text{+ p}{}^2\text{c}{}^2\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{? mc}{}^2\text{= p}{}^2\text{/2m ? p}{}^4\text{/8m}{}^3\text{c}{}^2\text{+}\text{O}\text{(1/c}{}^4\text{)}$. in the “minimally relativistic” Schrödinger equation, we suggest to employ the alternative formula $\text{T = p}{}^2\text{c}{}^2\text{[(m}{}^2\text{c}{}^4\text{+ p}{}^2\text{c}{}^2\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{+ mc}{}^2\text{]}{}^{\mathrm{?1}}\text{= p}{}^2\text{(2m + p}{}^2\text{/2mc}{}^2\text{)}{}^{\mathrm{?1}}\text{+}\text{O}\text{(1/c}{}^4\text{)}$. Its merits are illustrated on the particular linear potentials V(x) = μx + const, where a small O(c^?3) modification of the coupling (if needed) enables us to construct the ground and first M ( = O(c³)) excited states ψ(x) = e^-cx × polynomial (x) exactly.