The Cauchy problem for a linear 2mth-order Schrödinger equation $$u_t=-i(-Δ)^mu, in R^N×R_+, u|_{t=0}=u_0; m≥ 1$$ is an integer, is studied, for initial data u0 in the weighted space $L^2_{ρ^*}(R^N)$, with $ρ^*(x)=e^{|x|^α}$ and $α=\frac{2m}{2m-1}∈(1,2]$. The following five problems are studied: (I) A sharp asymptotic behaviour of solutions as $t →+∞$ is governed by a discrete spectrum and a countable set Φ of the eigenfunctions of the linear rescaled operator $$B=-i(-Δ)^m+\frac{1}{2m}y\cdot∇+\frac{N}{2m}I, with the spectrum σ(B)={λ_β=-\frac{|β|}{2m}, |β|≥ 0}.$$ (II) Finite-time blow-up local structures of nodal sets of solutions as t?^- and a formation of "multiple zeros" are described by the eigenfunctions, being generalized Hermite polynomials, of the "adjoint" operator $$B^*=-i(-Δ)^m-\frac{1}{2m}y\cdot∇, with the spectrum σ(B^*)=σ(B).$$ Applications of these spectral results also include: (III) a unique continuation theorem, and (IV) boundary characteristic point regularity issues. Some applications are discussed for more general linear PDEs and for the nonlinear Schrödinger equations in the focusing ("+") and defocusing ("-") cases $$u_t=-i(-Δ)^mu ± i|u|^{p-1}u, in R^N×R_+, where P>1,$$ as well as for: (V) the quasilinear Schrödinger equation of a "porous medium type" $$u_t=-i(-Δ)^mu (|u|^nu), in R^N×R_+, where n>0.$$ For the latter one, the main idea towards countable families of nonlinear eigenfunctions is to perform a homotopic path $n→ 0^+$ and to use spectral theory of the pair ${B,B^*}$.