Rellich’s theorem for spherically symmetric repulsive Hamiltonians
- 74 Downloads
For spherically symmetric repulsive Hamiltonians we prove Rellich’s theorem, or identify the largest weighted space of Agmon–Hörmander type where the generalized eigenfunctions are absent. The proof is intensively dependent on commutator arguments. Our novelty here is a use of conjugate operator associated with some radial flow, not with dilations and not with translations. Our method is simple and elementary, and does not employ any advanced tools such as the operational calculus or the Fourier analysis.
KeywordsRepulsive Hamiltonians Rellich’s theorem
Mathematics Subject Classification81Q05 35J10 35P05
The author would like to thank Kenichi Ito and Erik Skibsted for informative advice regarding this work.
- 3.Froese, R., Herbst, I.: Exponential bounds and absence of positive eigenvalues for \(N\)-body Schrödinger operators. Comm. Math. Phys. 87(3), 429-447 (1982/83)Google Scholar
- 5.Hörmander, L.: The Analysis of Linear Partial Differential Operators. vol. II, Grundlehren der Mathematischen Wissenschaften. Springer, Berlin (1985)Google Scholar
- 8.Ito, K., Skibsted, E.: Stationary scattering theory on manifolds, I. Preprint, (2016)Google Scholar
- 10.Isozaki, H.: A uniqueness theorem for the \(N\)-body Schrödinger equation and its applications. Spectral and scattering theory (Sanda, 1992), pp. 63–84, Lecture Notes in Pure and Appl. Math., 161, Dekkaer, New York (1994)Google Scholar
- 11.Kreh, M.: Bessel functions. Lecture notes, Penn State-Göttingen Summer School on Number Theory (2012)Google Scholar
- 13.Reed, M., Simon, B.: Methods of modern mathematical physics II and IV, New York: Academic Press (1975 and 1978)Google Scholar