On the Fourth order Schr\"odinger equation in four dimensions: dispersive estimates and zero energy resonances

We study the fourth order Schr\"odinger operator $H=(-\Delta)^2+V$ for a decaying potential $V$ in four dimensions. In particular, we show that the $t^{-1}$ decay rate holds in the $L^1\to L^\infty$ setting if zero energy is regular. Furthermore, if the threshold energies are regular then a faster decay rate of $t^{-1}(\log t)^{-2}$ is attained for large $t$, at the cost of logarithmic spatial weights. Zero is not regular for the free equation, hence the free evolution does not satisfy this bound due to the presence of a resonance at the zero energy. We provide a full classification of the different types of zero energy resonances and study the effect of each type on the time decay in the dispersive bounds.Comment: Revised according to referee suggestions. To appear in J. Differential Equation

Dispersive estimates for massive Dirac operators in dimension two

We study the massive two dimensional Dirac operator with an electric potential. In particular, we show that the $t^{-1}$ decay rate holds in the $L^1\to L^\infty$ setting if the threshold energies are regular. We also show these bounds hold in the presence of s-wave resonances at the threshold. We further show that, if the threshold energies are regular that a faster decay rate of $t^{-1}(\log t)^{-2}$ is attained for large $t$, at the cost of logarithmic spatial weights. The free Dirac equation does not satisfy this bound due to the s-wave resonances at the threshold energies.Comment: 40 page

Global dynamics of Schrodinger and Dirac equations

In this document, we study the linear Schr\"odinger operator and linear massive Dirac operator in the $L^1\to L^\infty$ settings. In Chapter~I, we focus on the two dimensional Schr\"odinger operator in the weighted $L^1(\R^2) \rightarrow L^{\infty}(\R^2)$ setting when there is a resonance of the first kind at zero energy. In particular, we show that if |V(x)|\les \la x \ra ^{-4-} and there is only s-wave resonance at zero of $H$, then \big\| w^{-1} \big( e^{itH}P_{ac} f - {\f 1 {\pi it} } F f \big) \big\| _{\infty} \leq \frac {C} {|t| (\log|t|)^2 } \|wf\|_1,\,\,\,\,\,\,|t|>2, with $w(x)=\log^2(2+|x|)$. Here Ff=-{\f 14} \psi\la \psi,f \ra, where $\psi$ is an s-wave resonance function. We also extend this result to matrix Schr\"odinger equations with potentials under similar conditions. In Chapter~II, we focus on the two and three dimensional massive Dirac equation with a potential. In two dimension, we show that the $t^{-1}$ decay rate holds if the threshold energies are regular or if there are s-wave resonances at the threshold. We further show that, if the threshold energies are regular then a faster decay rate of $t^{-1}(\log t)^{-2}$ is attained for large $t$, at the cost of logarithmic spatial weights, which is not the case for the free Dirac equation. In three dimension, we show that the solution operator is composed of a finite rank operator that decays at the rate $t^{-1/2}$ plus a term that decays at the rate $t^{-3/2}$

L1→L∞L^1\rightarrow L^\infty Dispersive estimates for Coulomb waves

We show the time decay of spherically symmetric Coulomb waves in $\R^{3}$ for the case of a repulsive charge. By means of a distorted Fourier transform adapted to $H=-\Delta+q\cdot |x|^{-1}$, with $q>0$, we explicitly compute the kernel of the evolution operator $e^{itH}$. A detailed analysis of the kernel is then used to prove that for large times, $e^{i t H}$ obeys an $L^1 \to L^\infty$ dispersive estimate with the natural decay rate t^{-\f32}.Comment: 60 page

Efficacy of injectable platelet-rich fibrin in the erosive oral lichen planus: a split-mouth, randomized, controlled clinical trial

Objective: Our study compared the effects of injectable platelet-rich fibrin (i-PRF) with those of corticosteroids in the treatment of erosive oral lichen planus (EOLP). Methodology: This split-mouth study included 24 individuals diagnosed histopathologically with bilateral EOLP. One bilateral lesion was injected with i-PRF, whereas the other was injected with methylprednisolone acetate in four sessions at 15-day intervals. Visual analog scale (VAS) for pain and satisfaction, oral health impact profile scale-14, and the lesion size were used. Results: The intragroup comparisons showed a significant decrease in VAS-pain and lesion size in both the i-PRF group (from 81.88±17.74 to 13.33±18.34, and from 4.79±0.41 to 1.88±1.08, respectively) and the corticosteroid group (from 80.21±17.35 to 23.33±26.81, and from 4.71±0.46 to 2.21±1.35, respectively) in the 6th month compared to baseline (p<0.001). Moreover, VAS-satisfaction increased significantly in both the i-PRF group (from 26.67±17.8 to 85.63±16.24) and the corticosteroid group (from 28.33±17.05 to 74.38±24.11) in the 6th month compared to baseline (p<0.001). However, no significant difference in any value occurred in the intergroup comparisons. Conclusion: In patients with EOLP, both methods decreased pain and lesion size similarly, and both increased satisfaction. Therefore, the use of i-PRF may be considered an option in cases refractory to topical corticosteroid therapy. Biochemical and histopathological studies are required to reveal the mechanism of i-PRF action in EOLP treatment