The Puzzle of Anomalous Isotope Effect in Zr, Nb$_{3}$Sn, and YBa$_{2}$Cu$_{3}$O$_{7}$

Anomalously small isotope effect in some high and low T$_{c}$ superconductors such as Zr, Nb$_{3}$Sn, YBa$_{2}$Cu$_{3}$O$_{7}$ created a great challenge for understanding. To shed light on a clue to solve this puzzle, a new methodology was implemented by integrating first-principles calculations of electronic structures of the materials into the theory of many-body physics for superconductivity. The aim is to seek a unified methodology to calculate the electronic and superconducting properties of these materials. It is shown that the electronic structures of Zr, Nb$_{3}$Sn, YBa$_{2}$Cu$_{3}$O$_{7}$ are very complex. The electron densities of states around the Fermi level in Zr, Nb$_{3}$Sn, YBa$_{2}$Cu$_{3}$O$_{7}$ possess sharp variations that could have a significant contribution to the anomalous isotope effect in these superconductors. However, there still exist some differences between the calculated and experimental results that require further research work.