On the Orthonormality and Spectrum of 1D Scattering States

The normalization of scattering states, far from being a rote step on the way to calculating expectation values (as is done in the bound state sector), contains important information regarding the density of the scattering spectrum (along with useful bound state information). For many applications, this information is more useful than the actual wavefunctions themselves. Even the simplest systems in 1D have nontrivial normalizations. In this paper we show that this normalization/density correspondence is a consequence of the completeness relation, and present formulas for calculating this spectrum for 1D, finite-range symmetric potentials. We then apply the formulas to the delta potential and the square well, and plot the corresponding spectrums. We further demonstrate the utility of such an analysis by showing how simple it is, once the normalization is known, to calculate the thermodynamics of a Yang-Gaudin gas in the dilute limit.Comment: 11 pages, 3 figure