Skip to main content Accessibility help
×
Home

On the asymptotic decay of L2-solutions of one-body Schrödinger equations in unbounded domains*

  • M. Hoffmann-Ostenhof (a1)

Synopsis

The asymptotic decay of L2-solutions of Schrödinger equations (-Δ+V)ψ=0 in ΔR= {x εRn∣∣x∣=r>R} is investigated, where V(x) = V1(r) + V2(x) with V1→ ∞ for r↑∞ and with some ε > 0 for large r. Under additional assumptions on the decay of V1, pointwise upper bounds to |ψ |and lower bounds to the spherical average of ψ are given showing the same asymptotics for r→ ∞. For the case V→ const. > 0 for r→ ℝ (investigated in [8] a simplified treatment is given.

Copyright

References

Hide All
1Agmon, S.. On the asymptotic behaviour of solutions of Schrodinger type equations in unbounded domains. In Analyse Mathématique et Applications, pp. 122. (Paris: Gauthier-Villars, 1988).
2Amrein, W. O., Berthier, A. M. and Georgescu, V.. Lower bounds for zero energy eigenfunctions of Schrödinger operators. Helv. Phys. Acta 57 (1984), 301306.
3Bardos, C. and Merigot, M.. Asymptotic decay of solutions of a second order elliptic equation in an unbounded domain. Proc. Roy. Soc. Edinburgh Sect. A 76 (1977), 323344.
4Davies, B. and Simon, B.. Ultracontractivity and the Heat kernel for Schrödinger operators and Dirichlet Laplacians. J. Funct. 59 (1984), 335395.
5Froese, R., Herbst, I., Hoffmann-Ostenhof, M. and Hoffmann-Ostenhof, T.. L 2-lower bounds to solutions of one-body Schrödinger equations. Proc. Roy. Soc. Edinburgh Sect. A 95 (1983), 2538.
6Froese, R. and Herbst, I.. Exponential lower bounds to solutions of the Schrödinger equation: lower bounds for the spherical average. Comm. Math. Phys. 92 (1983), 7180.
7Gilbarg, D. and Trudinger, N. S.. Elliptic Partial Differential Equations of Second Order (Berlin: Springer, 1983).
8Hoffmann-Ostenhof, M., Hoffmann-Ostenhof, T. and Swetina, J.. Pointwise bounds on the asymptotics of spherically averaged L 2-solutions of one-body Schrödinger equations. Ann. Inst. H Poincaré Anal. Physique théorique 42 (1985), 341361.
9Hoffmann-Ostenhof, M. and Hoffmann-Ostenhof, T.. Asymptotics and continuity properties near infinity of solutions of Schrodinger equations in exterior domains. Ann. Inst. H. Poincaré Anal. Physique théorique 46 (1987), 247280.
10Hoffmann-Ostenhof, M.. Asymptotics of the nodal lines of solutions of 2-dimensional Schrödinger equations. Math. Z. 198 (1988), 161179.
11Hoffmann-Ostenhof, M. and Hoffmann-Ostenhof, T.. On the asymptotics of nodes of L 3-solutions of Schrödinger equations in dimensions ≧ 3. Comm. Math. Phys. 117 (1988), 4977.
12Reed, M. and Simon, B.. Methods of mathematical physics II, Fourier analysis, selfadjointness (New York: Academic Press, 1975).
13Reed, M. and Simon, B.. Methods of mathematical physics IV, Analysis of operators (New York: Academic Press, 1978).
14Thoe, D. W.. Lower bounds for solutions of perturbed Helmholtz equations in exterior regions. J. Math. Anal. Appl. 102 (1984), 113122.
15Uchiyama, J.. Decay order of eigenfunctions of second order elliptic operators in an unbounded domain, and its application. Publ. RIMS. Kyoto Univ. 22 (1986), 10791104.

On the asymptotic decay of L2-solutions of one-body Schrödinger equations in unbounded domains*

  • M. Hoffmann-Ostenhof (a1)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed