arXiv subjects
Publications and source records attributed to Tomoya Tagawa.
We discuss the low energy resolvent estimates for the Schr\"odinger operator with slowly decaying attractive potential. The main results are Rellich's theorem, the limiting absorption principle and Sommerfeld's uniqueness theorem. For the proofs we employ an elementary commutator method due to Ito--Skibsted, for which neither of microlocal or functional-analytic techniques is required.
We discuss propagation of space-time singularities for the quantum harmonic oscillator with time-dependent metric and potential perturbations. Reformulating the quasi-homogeneous wave front set according to Lascar (1977) in a semiclassical manner, we obtain a characterization of its appearance in comparison with the unperturbed system. The idea of our proof is based on the argument of Nakamura (2009), which was originally devised for the analysis of spatial singularities of the Schr\"odinger equation, however, the application is non-trivial since the time is no more a parameter, but takes a part in the base variables.
For the generalized oscillator, we prove a Rellich type theorem, or characterize the order of growth of eigenfunctions. The proofs are given by an extensive use of commutator arguments invented recently by Ito and Skibsted. These arguments are simple and elementary and do not employ energy cut-offs or microlocal analysis.