Nonexistence of wave operators via strong propagation estimates for Schr\"{o}dinger operators with sub-quadratic repulsive potentials

Sub-quadratic repulsive potentials accelerate quantum particles and can relax the decay rate in the $x$ of the external potentials $V$ that guarantee the existence of the quantum wave operators. In the case where the sub-quadratic potential is $- |x|^{\alpha}$ with $0< \alpha < 2$ and the external potential satisfies $|V(x) | \leq C (1+|x|) ^{-(1- \alpha /2) - \varepsilon}$ with $\varepsilon>0$, Bony et. al [3] determined the existence and completeness of the wave operators, and Itakura [12, 13, 14] then obtained their results using stationary scattering theory for more generalized external potentials. Based on their results, we naturally expect the following. If the decay power of the external potential $V$ is less than ${ -(1- \alpha /2) }$, V is included in the short-range class. If the decay power is greater than or equal to ${ -(1- \alpha /2) }$, $V$ is included in the long-range class. In this study, we first prove the new propagation estimates for the time propagator that can be applied to scattering theory. Second, we prove that the wave operators do not exist if the power is greater than or equal to $-(1- \alpha /2)$ and that the threshold expectation of ${ -(1- \alpha /2) }$ is true using the new propagation estimates

Outcomes and Treatments of Mal Fractures Caused by the Split-Crest Technique in the Mandible

In this study, we investigated cases of mal fracture occurring during the split-crest procedure. In all subjects (six patients), the free cortical bone segment caused by the mal fracture was carefully maintained in the lateral position without fixation using a titanium plate or screw. On pre- and postoperative multiplanar reconstruction CT, the average total alveolar increase was 5.0 mm in the lower portion 1 mm from the top of the alveolar ridge, and the average total alveolar increase in the lower portion 11 mm from the top of the alveolar ridge was 2.2 mm. A total of 11 dental implants were placed immediately at the same time as the split-crest procedure, while three dental implants were placed after a waiting period of 4-11 months from bone augmentation. During an average follow-up of 27.8 months, there were no complications or cases of failed implants. Consequently, among the patients who experienced mal fracture during the split-crest technique, a sufficient volume of alveolar bone was obtained without the need for rigid fixation of the free bone segment, and the dental implants placed within the area of the mal fracture showed a good prognosis

Efficient ultrashort-pulse generation of Yb:YAG laser overcoming the fluorescence spectrum limit by using nonlinear medium

One-hundred-ten-fs and 72-fs pulse-widths were obtained directly from a mode-locked Yb:YAG laser with SESAM and without SESAM, respectively. The laser-spectrum-center and the fluorescence-center were almost same. The oscillation-spectra were much broader than the fluorescence

Stent-graft detachment from aortic wall after stent-graft repair of acute aortic dissection

AbstractPreliminary studies have demonstrated that stent-graft repair is an attractive method for treatment of aortic dissection. However, few reports have described long-term results. A 72-year-old woman with acute type B aortic dissection underwent stent-graft repair. The entry was closed, and the false lumen disappeared completely. The postoperative course was uneventful for 4 years 5 months, when detachment of the stent graft from the aortic wall was noted. Because the device appears to be stable, follow-up is on an outpatient basis