Quantum cosmology with dynamical vacuum in a minimal-length scenario

In this work, we consider effects of the dynamical vacuum in quantum cosmology in presence of a minimum length introduced by the GUP (generalized uncertainty principle) related to the modified commutation relation [X^,P^]:=iħ1-βP^2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[{\hat{X}},{\hat{P}}] := \frac{i\hbar }{ 1 - \beta {\hat{P}}^2 }$$\end{document}. We determine the wave function of the Universe ψqp(ξ,t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \psi _{qp}(\xi ,t)$$\end{document}, which is solution of the modified Wheeler–DeWitt equation in the representation of the quasi-position space, in the limit where the scale factor of the Universe is small. Although ψqp(ξ,t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi _{qp}(\xi ,t)$$\end{document} is a physically acceptable state it is not a realizable state of the Universe because ψqp(ξ,t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \psi _{qp}(\xi ,t)$$\end{document} has infinite norm, as in the ordinary case with no minimal length.