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Experiments have revealed multiple quantum oscillation frequencies inunderdoped high temperature superconductor$\mathrm{YBa_{2}Cu_{3}O_{6+\delta}}$, corresponding to approximately 10%doping, which contain $\mathrm{CuO}$ bilayers in the unit cell. These unitcells are further coupled along the c-axis by a tunneling matrix element. Amodel of the energy dispersion that has its roots in the previously determinedelectronic structure, combined with two-fold commensurate density waves,reveals multiple electron and hole pockets. To the extent that quasiparticlesof the reconstructed Fermi surface have finite residues, however small, theformation of Landau levels are the cause of these oscillations and the bilayersplitting and warping of the electronic dispersion along the directionperpendicular to the CuO-planes are firm consequences. We explore thispossibility in detail and find overall consistency with experiments. Animportant conclusion is that bilayer splitting is severely renormalized fromthe value obtained from band structure calculations. It would be extremelyinteresting to perform these experiments for higher values of doping. Weroughly expect the splitting of the frequencies to increase with doping, butthe full picture may be more complex because the density wave order parameteris also expected to decrease with doping, vanishing around the middle of thesuperconducting dome.

Multiple quantum oscillation frequencies in $\mathrm{YBa_{2}Cu_{3}O_{6+\delta}}$ and bilayer splitting