 Parity-constrained variational bound

An even one-dimensional confining potential preserves even and odd parity. Restricting a normalized trial state to the odd sector makes its Rayleigh quotient an upper bound on the lowest odd energy, which is the first excited energy for the usual nonsingular one-dimensional bound-state problem. This avoids contamination by the even ground state. For an adjustable-width odd Gaussian, optimize its own width rather than borrowing the ground-state optimum.