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Asymptotic large- and short-time behavior of solutions of the linear dispersion equation u_t = u_xxx in R×R_+, and its (2k+1)th-order extensions are studied. Such a refined scattering is based on a "Hermitian" spectral theory for a pair B,B^∗ of non self-adjoint rescaled operators B=D^3_y+13yD_y+13I, and the adjoint one B^∗=D^3_y-13yD_y, with the discrete spectrum σ(B)=σ(B^∗)=λ_l=-l/3, l=0,1,2,... and eigenfunctions for B, ψ_l(y)=[(-1)^l/l!]D^l_yAi(y), l ≥ 0, where Ai(y) isAiry's classic function. Eigenfunctions of B^∗ are then generalized Hermite polynomials. Applications to very singular similarity solutions (VSSs) of the semilinear dispersion equation with absorption, u_S(x,t)=t^-1p-1 f(xt^13 ): u_t=u_xxx-|u|^p-1u in R×R_+, p > 1, and to its higher-order counterparts are presented. The goal is, by using various techniques, to show that there exists a countable sequence of critical exponents p_l=1+3/(l+1), l=0,1,2,... such that, at each p= p_l , a p-branch of VSSs bifurcates from the corresponding eigenfunction ψ_l of the linear operator B above. , number=3, journal=Journal of Partial Differential Equations, year=2011, month=Aug., pages=207–263