Time decay of solutions to the Schrödinger equation in exterior domains. I, II. (English) Zbl 0684.35019

Hayashi, Nakao

Ann. Inst. Henri Poincaré, Phys. Théor. 50, No. 1, 71-81, 83-93 (1989).

The time evolution given by the free particle Schrödinger operator on the complement, D, of a star-shaped bounded domain in \({\mathbb{R}}^ n\) is studied. Dirichlet boundary conditions are considered. Upper bounds for the decay rates of the \(L^ p(D)\)-norm of the evolved state, u(t), are obtained. The bounds obtained in the first part are improved in the second one. For orientation let us reproduce one particular result: for \(n\geq 5\), \(2\leq p\leq 2n/(n-4)\) \[ \| u(t)\|_ p\leq const. I^{1/2}(1+t)^{-n(1-2/p)/2}, \] where I is a functional of the initial value, u(0).

Reviewer: G.Nenciu

MSC:

35B40	Asymptotic behavior of solutions to PDEs
35Q99	Partial differential equations of mathematical physics and other areas of application
35K10	Second-order parabolic equations

Keywords:

exterior domains; Schrödinger operator; decay rates

References:

[1]	A. Friedman , Partial Differential Equations: Holt , Rinehart and Winston , New York , 1969 . MR 445088 | Zbl 0224.35002 · Zbl 0224.35002
[2]	Y. Jing-Qi , Comportement à l’infini des solutions d’une equation de Schrödinger non linéaire dans un domaine extérieur , C. R. Acad. Sc. Paris , t. 294 , 1982 , p. 163 - 166 . MR 651435 | Zbl 0511.35078 · Zbl 0511.35078
[3]	O.A. Ladyzhenskaya , The Mathematical Theory of Viscous Incompressible Flow , 2 nd English ed., Gordon and Breach , New York , 1969 . MR 254401 | Zbl 0184.52603 · Zbl 0184.52603
[4]	O.A. Ladyzhenskaya and N.N. Ural’ceva , Linear and Quasilinear Equations of Elliptic Type , Academic Press , New York , 1968 . · Zbl 0164.13002
[5]	Y. Tsutsumi , Global solution of the nonlinear Schrödinger equation in exterior domains , Comm. PDE , t. 8 , 1983 , p. 1337 - 1374 . MR 711442 | Zbl 0542.35027 · Zbl 0542.35027
[6]	K. Yosida , Functional Analysis , Springer-Verlag , Berlin , Heidelberg , New York , 1971 . Zbl 0217.16001 · Zbl 0217.16001
[7]	Y. Chen , Global existence of the solution of nonlinear Schrödinger equations in exterior domains , Acta Mathematicae Applicatae Sinica , t. 2 , ( 3 ), 1985 , p. 191 - 212 . Zbl 0641.35011 · Zbl 0641.35011