arXiv subjects
Publications and source records attributed to Ihyeok Seo.
We obtain a reverse Hölder inequality for the eigenfuctions of the Schrödinger operator with slowly decaying potentials. The class of potentials includes singular potentials which decay like $|x|^{-α}$ with $0<α<2$, in particular the Coulomb potential.
In this paper we obtain some new Strichartz estimates for the wave propagator $e^{it\sqrt{-Δ}}$ in the context of Wiener amalgam spaces. While it is well understood for the Schrödinger case, nothing is known about the wave propagator. This is because there is no such thing as an explicit formula for the integral kernel of the propagator unlike the Schrödinger case. To overcome this lack, we instead approach the kernel by rephrasing it as an oscillatory integral involving Bessel functions and then by carefully making use of cancellation in such integrals based on the asymptotic expansion of Bessel functions. Our approach can be applied to the Schrödinger case as well. We also obtain some corresponding retarded estimates to give applications to nonlinear wave equations where Wiener amalgam spaces as solution spaces can lead to a finer analysis of the local and global behavior of the solution.
In this paper we study the Cauchy problem for the energy-critical inhomogeneous nonlinear Schrödinger equation $i\partial_{t}u+Δu=λ|x|^{-α}|u|^βu$ in $H^1$. The well-posedness theory in $H^1$ has been intensively studied in recent years, but the currently known approaches do not work for the critical case $β=(4-2α)/(n-2)$. It is still an open problem. The main contribution of this paper is to develop the theory in this case.
In this paper, we consider the Lamé operator $-Δ^\ast$ and study resolvent estimate, uniform Sobolev estimate, and Carleman estimate for $-Δ^\ast$. First, we obtain sharp $L^p$--$L^q$ resolvent estimates for $-Δ^\ast$ for admissible $p,q$. This extends the particular case $q=\frac p{p-1}$ due to Barceló et al. \cite{BFPRV} and Cossetti \cite{Co19}. Secondly, we show failure of uniform Sobolev estimate and Carleman estimate for $-Δ^\ast$. For the purpose we directly analyze the Fourier multiplier of the resolvent. This allows us to prove not only the upper bound but also the lower bound on the resolvent, so we get the sharp $L^p$--$L^q$ bounds for the resolvent of $-Δ^\ast$. Strikingly, the relevant uniform Sobolev and Carleman estimates turn out to be false for the Lamé operator $-Δ^\ast$ even though the uniform resolvent estimates for $-Δ^\ast$ are valid for certain range of $p, q$. This contrasts with the classical result regarding the Laplacian $Δ$ due to Kenig, Ruiz, and Sogge \cite{KRS87} in which the uniform resolvent estimate plays crucial role in proving the uniform Sobolev and Carleman estimates for $Δ$. We also describe locations of the $L^q$-eigenvalues of $-Δ^\ast+V$ with complex potential $V$ by making use of the sharp $L^p$--$L^q$ resolvent estimates for $-Δ^\ast$.
The Liouville-von Neumann equation describes the change in the density matrix with time. Interestingly, this equation was recently regarded as a wave equation for wave functions but not a equation for density functions. This setting leads to an extended form of the Schrödinger wave equation governing the motion of a quantum particle. In this paper we obtain the integrability of the wave propagator arising from the Liouville-von Neumann equation in this setting.
There have been a lot of works concerning the Strichartz estimates for the perturbed Schrödinger equation by potential. These can be basically carried out adopting the well-known procedure for obtaining the Strichartz estimates from the weighted $L^2$ resolvent estimates for the Laplacian. In this paper we handle the Strichartz estimates without relying on the resolvent estimates. This enables us to consider various potential classes such as the Morrey-Campanato classes.
In this paper we study the well-posedness for the inhomogeneous nonlinear Schrödinger equation $i\partial_{t}u+Δu=λ|x|^{-α}|u|^βu$ in Sobolev spaces $H^s$, $s\geq0$. The well-posedness theory for this model has been intensively studied in recent years, but much less is understood compared to the classical NLS model where $α=0$. The conventional approach does not work particularly for the critical cases $β=\frac{4-2α}{d-2s}$. It is still an open problem. The main contribution of this paper is to develop the well-posedness theory in this critical case (as well as non-critical cases). To this end, we approach to the matter in a new way based on a weighted $L^p$ setting which seems to be more suitable to perform a finer analysis for this model. This is because it makes it possible to handle the singularity $|x|^{-α}$ in the nonlinearity more effectively. This observation is a core of our approach that covers the critical case successfully.
In this paper we obtain lower bounds on the radius of spatial analyticity of solutions to the Kawahara equation $u_t + uu_x + αu_{xxx} + βu_{xxxxx} = 0$, $β\neq0$, given initial data which is analytic with a fixed radius. It is shown that the uniform radius of spatial analyticity of solutions at later time $t$ can decay no faster than $1/|t|$ as $|t|\rightarrow\infty$.
We obtain weighted $L^2$ estimates for the elastic wave equation perturbed by singular potentials including the inverse-square potential. We then deduce the Strichartz estimates under the sole ellipticity condition for the Lamé operator $-Δ^\ast$. This improves upon the previous result in \cite{BFRVV} which relies on a stronger condition to guarantee the self-adjointness of $-Δ^\ast$. Furthermore, by establishing local energy estimates for the elastic wave equation we also prove that the solution has local regularity.
We obtain some new Morawetz estimates for the Klein-Gordon flow of the form \begin{equation*} \big\||\nabla|^σ e^{it \sqrt{1-Δ}}f \big\|_{L^2_{x,t}(|(x,t)|^{-α})} \lesssim \|f\|_{H^s} \end{equation*} where $σ,s\geq0$ and $α>0$. The conventional approaches to Morawetz estimates with $|x|^{-α}$ are no longer available in the case of time-dependent weights $|(x,t)|^{-α}$. Here we instead apply the Littlewood-Paley theory with Muckenhoupt $A_2$ weights to frequency localized estimates thereof that are obtained by making use of the bilinear interpolation between their bilinear form estimates which need to carefully analyze some relevant oscillatory integrals according to the different scaling of $\sqrt{1-Δ}$ for low and high frequencies.
We obtain Strichartz inequalities for the wave equation with potentials which behave like the inverse square potential $|x|^{-2}$ but might be not a radially symmetric function.
In this paper we study spatial analyticity of solutions to the defocusing nonlinear Schrödinger equations $iu_t + Δu = |u|^{p-1}u$, given initial data which is analytic with fixed radius. It is shown that the uniform radius of spatial analyticity of solutions at later time $t$ cannot decay faster than $1/|t|$ as $|t|\rightarrow\infty$. This extends the previous work of Tesfahun for the cubic case $p=3$ to the cases where $p$ is any odd integer greater than $3$.
We prove weighted $L^2$ estimates for the Klein-Gordon equation perturbed with singular potentials such as the inverse-square potential. We then deduce the well-posedness of the Cauchy problem for this equation with small perturbations, and go on to discuss local smoothing and Strichartz estimates which improve previously known ones.
The primary objective in this paper is to give an answer to an open question posed by J. A. Barceló, J. M. Bennett, A. Carbery, A. Ruiz and M. C. Vilela concerning the problem of determining the optimal range on $s\geq0$ and $p\geq1$ for which the following Strichartz estimate with time-dependent weights $w$ in Morrey-Campanato type classes $\mathfrak{L}^{2s+2,p}_2$ holds: \begin{equation}\label{absset} \|e^{itΔ}f\|_{L_{x,t}^2(w(x,t))}\leq C\|w\|_{\mathfrak{L}^{2s+2,p}_2}^{1/2}\|f\|_{\dot{H}^s}. \end{equation} Beyond the case $s\geq0$, we further ask how much regularity we can expect on this setting. But interestingly, it turns out that this estimate is false whenever $s<0$, which shows that the smoothing effect cannot occur in this time-dependent setting and the dispersion in the Schrödinger flow $e^{itΔ}$ is not strong enough to have the effect. This naturally leads us to consider the possibility of having the effect at best in higher-order versions of this estimate with $e^{-it(-Δ)^{γ/2}}$ ($γ>2$) whose dispersion is more strong. We do obtain a smoothing effect exactly for these higher-order versions. In fact, we will obtain the estimates where $γ\geq1$ in a unified manner and also their corresponding inhomogeneous estimates to give applications to the global well-posedness for Schrödinger and wave equations with time-dependent perturbations. This is our secondary objective in this paper.
In this paper we study the Strichartz estimates for the Schrödinger propagator in the context of Wiener amalgam spaces which, unlike the Lebesgue spaces, control the local regularity of a function and its decay at infinity separately. This separability makes it possible to perform a finer analysis of the local and global behavior of the propagator. Our results improve some of the classical ones in the case of large time.
The Kato-Yajima smoothing estimate is a smoothing weighted $L^2$ estimate with a singular power weight for the Schrödinger propagator. The weight has been generalized relatively recently to Morrey-Campanato weights. In this paper we make this generalization more sharp in terms of the so-called Kerman-Sawyer weights. Our result is based on a more sharpened Fourier restriction estimate in a weighted $L^2$ space. Obtained results are also extended to the fractional Schrödinger propagator.
We obtain a new bound on the location of eigenvalues for a non-self-adjoint Schrödinger operator with complex-valued potentials by obtaining a weighted $L^2$ estimate for the resolvent of the Laplacian.
We obtain a unique continuation result for the differential inequality $| (i\partial_t +Δ)u | \leq |Vu| + | W\cdot\nabla u |$ by establishing $L^2$ Carleman estimates. Here, $V$ is a scalar function and $W$ is a vector function, which may be time-dependent or time-independent. As a consequence, we give a similar result for the magnetic Schrödinger equation.