Integrable minisuperspace models with Liouville field: energy density self-adjointness and semiclassical wave packets

Abstract

The homogeneous cosmological models with a Liouville scalar field are investigated in classical and quantum contexts of Wheeler–DeWitt geometrodynamics. In the quantum case of quintessence field with potential unbounded from below and phantom field, the energy density operators are not essentially self-adjoint, and self-adjoint extensions contain ambiguities. Therefore the same classical actions correspond to a family of distinct quantum models. For the phantom field the energy spectrum happens to be discrete. The probability conservation and appropriate classical limit can be achieved with a certain restriction of the functional class. The appropriately localized wave packets are studied numerically using the Schrödinger’s norm and a conserved Mostafazadeh’s norm introduced from techniques of pseudo-Hermitian quantum mechanics. These norms give a similar packet evolution that is confronted with analytical classical solutions.