Construction of a symmetric potential barrier from tunneling transmission coefficients

Abstract

For a symmetric potential barrier with two turning points there exists an inversion procedure which is closely related to the inversion of scattering phase shifts. The inversion formulas correct to first order in ${\ensuremath{\hbar}}^{2}$ are derived in the WKB approximation. To illustrate the applicability and limitation of the proposed procedure, we construct the quartic potential barrier $U(X)=\ensuremath{-}{X}^{4}$, from its transmission coefficients.