On the asymptotics of solutions to the second initial boundary value problem for Schrödinger systems in domains with conical points.
The paper deals with the second initial boundary value problem for a general Schrödinger system of order \(2m\) in \(\Omega \times (0,\infty )\) where the boundary \(\partial \Omega \) has some conical point, say \(x=0\). While existence and uniqueness of weak solutions was proved in a previous paper (see [\textit{N. M. Hung} and \textit{N. T. K. Son}, Electron. J. Differ. Equ. 2008, Paper No. 35, 11 p. (2008; Zbl 1170.35360)]) the authors now study the asymptotic behavior of generalized solutions in a neighbourhood of the conical point. To this end they derive an elliptic eigenvalue problem depending on the parameter \(t\) which is defined on a part \(G\) of the sphere \(S^{n-1}\) generated by the cone. If this eigenvalue problem in a given strip has semisimple eigenvalues with invariant multiplicity for all \(t\in [0,\infty ]\) there is proven a representation of a weak solution in the cone in terms of powers of \(r=| x| \) and vector polynomials of \(\ln r\). The proof of this result is the main part of the paper. In the final section, as an example the Neumann problem for the classical Schrödinger equation in quantum mechanics is discussed for the cases of space dimensions \(n=2\), \(n=3\) and \(n\geq 4\), respectively.
scientific article; zbMATH DE number 6162813