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We consider the higher order Schr\"odinger operator $H=(-\Delta)^m+V(x)$ in $n$ dimensions with real-valued potential $V$ when $n>4m$, $m\in \mathbb N$. We adapt our recent results for $m>1$ to show that when $H$ has a threshold eigenvalue the wave operators are bounded on $L^p(\mathbb R^n)$ for the natural range $1\leq p<\frac{n}{2m}$ in both even and odd dimensions. The approach used works without distinguishing even and odd cases, and matches the range of boundedness in the classical case when $m=1$. The proof applies in the classical $m=1$ case as well and simplifies the argument., Comment: 23 pages. arXiv admin note: text overlap with arXiv:2207.14264

In this paper we study the effect of rotation on nonlinear wave phenomena in weakly dispersive media modeled by the Korteweg-de Vries equation on the real line. It is well known that smoothing in the case of the KdV equation with periodic boundary conditions is a result of the presence of high frequency waves that weaken the nonlinearity through time averaging, \cite{bit}, \cite{et1}. High frequency interactions through normal form transformations can be thought of as adding a fast rotating term in certain toy PDE models that arrest the development of singularities, \cite{bit}. It is crucial for this phenomena that the zero Fourier mode can be removed due to the conservation of the mean. On the real line this mechanism breaks down as the resonance sets close to zero frequency are sizable and normal form transformations are not useful, \cite{imt}, and hence smoothing fails. The model we study is a perturbation of the KdV equation on a rotating frame of reference. A combination of a suitable normal form transformation along with the restricted norm method of Bourgain \cite{Bou2} implies smoothing on the real line., Comment: 17 pages

We prove a family of dispersive estimates for the higher order Schr\"odinger equation $iu_t=(-\Delta)^mu +Vu$ for $m\in \mathbb N$ with $m>1$ and $2m

We utilize exponential sum techniques to obtain upper and lower bounds for the fractal dimension of the graph of solutions to the linear Schr\"odinger equation on $\mathbb{S}^d$ and $\mathbb{T}^d$. Specifically for $\mathbb S^d$, we provide dimension bounds using both $L^p$ estimates of Littlewood-Paley blocks, as well as assumptions on the Fourier coefficients. In the appendix, we present a slight improvement to the bilinear Strichartz estimate on $\mathbb{S}^2$ for functions supported on the zonal harmonics. We apply this to demonstrate an improved local well-posedness result for the zonal cubic NLS when $d=2$, and a nonlinear smoothing estimate when $d\geq 2$. As a corollary of the nonlinear smoothing for solutions to the zonal cubic NLS, we find dimension bounds generalizing the results of the first author and Tzirakis for solutions to the cubic NLS on $\mathbb{T}$. Additionally, we obtain several results on $\mathbb{T}^d$ generalizing the results of the $d=1$ case., Comment: 42 pages

We consider the higher order Schr\"odinger operator $H=(-\Delta)^m+V(x)$ in $n$ dimensions with real-valued potential $V$ when $n>2m$, $m\in \mathbb N$. We adapt our recent results for $m>1$ to show that the wave operators are bounded on $L^p(\mathbb R^n)$ for the full the range $1\leq p\leq \infty$ in both even and odd dimensions without assuming the potential is small. The approach used works without distinguishing even and odd cases, captures the endpoints $p=1,\infty$, and somehow simplifies the low energy argument even in the classical case of $m=1$., Comment: 16 pages, revised. This paper extends the authors' work in arXiv:2107.09620 to capture endpoints in even dimensions. Updated according to referee comments, to appear in the J. of Differential Equations

We consider the higher order Schr\"odinger operator $H=(-\Delta)^m+V(x)$ in $n$ dimensions with real-valued potential $V$ when $n>4m-1$, $m\in \mathbb N$. We show that for any $\frac{2n}{n-4m+1}3$ and $\frac{2n}{n-3}

Astronomical sites occupying observing instruments have to be selected according to many factors. Among these factors, geographic location of the site and quality of atmosphere above the site play an important role in the decision process. The following factors were chosen to create layers at their geographic locations (observatories: 1905 northern, 235 southern) from the \href{https://www.astrogis.org}{astroGIS database}: CC (cloud coverage), PWV (precipitable water vapor), AOD (atmospheric optical depth), VWV (vertical wind velocity) and HWV (horizontal wind velocity). In order to estimate astronomical importance of geographic location of the sites and quality of airmass above the sites, DEM (digital elevation model) and LAT (latitude of observatory location) layers were also included. Two periodic variations have been produced from these factors: monthly and annual averages. In addition to the variations or trends a complete statistical analysis was carried out for all factors to investigate the potential correlations between the factors: There is a clear difference between northern and southern hemispheres. Exchange of meteorological seasons between hemispheres are also compliant within factors. The geographical locations of most of the observatories found to be "not suitable": On the average, DEM is low (550 m), CC is high (70\%) and PWV is high (14 mm). There seems to be no apparent long-term variations and/or patterns in all factors. We once again confirm the common expectation of astronomy: \textit{as DEM increases astronomical conditions get better} (CC, PWV and AOD gets lower values). All the results will be made available online through astroGIS database., Comment: 14 pages, 5 tables, 6 figures, Submitted to MNRAS

We consider the higher order Schr\"odinger operator $H=(-\Delta)^m+V(x)$ in $n$ dimensions with real-valued potential $V$ when $n>2m$, $m\in \mathbb N$, $m>1$. When $n$ is odd, we prove that the wave operators extend to bounded operators on $L^p(\mathbb R^n)$ for all $1\leq p\leq\infty$ under $n$ and $m$ dependent conditions on the potential analogous to the case when $m=1$. Further, if $V$ is small in certain norms, that depend $n$ and $m$, the wave operators are bounded on the same range for even $n$. We further show that if the smallness assumption is removed in even dimensions the wave operators remain bounded in the range $1

Artificial Light pollution (AL) in Turkey and in Turkish observatories between 2012--2020 have been studied using the archival data of Visible Infrared Imaging Radiometer Suite (VIIRS) instrument. The astroGIS database has been used in processing the data (https://www.astrogis.org) \cite{2020MNRAS.493.1204A}. The total energy released to space from Turkey increased by 80\% in 2019 with respect to 2012. In the span of the dataset, a steady and continuous increase has been observed throughout all cities of the country. On the other hand, Dark Sky Park locations, East and Southeast Anatolian regions and mostly rural areas around the cities kept their AL level constant. Four demographic parameters have been studied and they were found to be correlated very well with AL: Population ($R \simeq 0.90$); GDP ($R \simeq 0.87$); Total Power Consumption ($R \simeq 0.66$) and Outdoor Lightening ($R \simeq 0.67$). Contrary to countries acting to prevent AL increases, Turkey seems to be at the beginning of an era where AL will arithmetically increase throughout the country and enormous amount of energy will continuously escape to space and therefore will be wasted. Therefore, a preventive legislation, especially for invaluable astronomical site locations such as TURAG, TUG, DAG and \c{C}AAM where each is counted as a truly dark site due to their SQM values, has to be enacted in Turkey, in very near future., Comment: 15 pages, 7 figures and 4 tables, Accepted for publication in Astrophysics and Space Science

France issued a decree to restrict and prohibit mainly outdoor lighting effective from January 1st, 2019. Effectiveness of this legislation has been evaluated in this study using GIS data which was first used in \cite{2020MNRAS.493.1204A} (so called astroGIS database - \url{astrogis.org}). A subset of Artificial Light layer of astroGIS database has been adapted for years between January 2012 and December 2019. During 2019, radiance of $1.9 \times 10^{9}$ W cm$^{-2}$ sr$^{-1}$ has been released into space. Annual light pollution in France decreased by 6\% after the enactment of artificial light legislation. France continue to have potential Dark Sky Park locations for example cities like Indre, Lot, Nievre and Creuse having the lowest light pollution values. A strong correlation between population and light pollution ($R\simeq 0.83$) has been observed. A similar but a weak correlation can also be observed for GDP ($R\simeq 0.28$). However, it is still too early to justify whether the improvements observed in the dataset are due to the enactment of the legislation or not., Comment: 10 pages, 4 figures, 2 tables, Submitted to Astrophysics and Space Science

Let $\mu$ be an $\alpha$-dimensional probability measure. We prove new upper and lower bounds on the decay rate of hyperbolic averages of the Fourier transform $\widehat{\mu}$. More precisely, if $\mathbb{H}$ is a truncated hyperbolic paraboloid in $\mathbb{R}^d$ we study the optimal $\beta$ for which $$\int_{\mathbb{H}} |\hat{\mu}(R\xi)|^2 \, d \sigma (\xi)\leq C(\alpha, \mu) R^{-\beta}$$ for all $R > 1$. Our estimates for $\beta$ depend on the minimum between the number of positive and negative principal curvatures of $\mathbb{H}$; if this number is as large as possible our estimates are sharp in all dimensions., Comment: Final version, to appear in Transactions of the AMS

A global site selection for astronomy was performed with 1 km spatial resolution ($\sim$ 1 Giga pixel in size) using long term and up-to-date datasets to classify the entire terrestrial surface on the Earth. Satellite instruments are used to get the following datasets of Geographical Information System (GIS) layers: Cloud Coverage, Digital Elevation Model, Artificial Light, Precipitable Water Vapor, Aerosol Optical Depth, Wind Speed and Land Use -- Land Cover. A Multi Criteria Decision Analysis (MCDA) technique is applied to these datasets creating four different series where each layer will have a specific weight. We introduce for the first time a ``Suitability Index for Astronomical Sites'' namely, SIAS. This index can be used to find suitable locations and to compare different sites or observatories. Mid-western Andes in South America and Tibetan Plateau in west China were found to be the best in all SIAS Series. Considering all the series, less than 3 \% of all terrestrial surfaces are found to be the best regions to establish an astronomical observatory. In addition to this, only approximately 10 \% of all current observatories are located in good locations in all SIAS series. Amateurs, institutions or countries aiming to construct an observatory could create a short-list of potential site locations using layout of SIAS values for each country without spending time and budget.The outcomes and datasets of this study has been made available through a web site, namely ``Astro GIS Database'' on \texttt{\url{www.astrogis.org}}., Comment: 19 Pages, 4 Figures, 7 tables, Accepted for publication in MNRAS

We prove Strichartz estimates for the Schr\"odinger equation in $\mathbb R^n$, $n\geq 3$, with a Hamiltonian $H = -\Delta + \mu$. The perturbation $\mu$ is a compactly supported measure in $\mathbb R^n$ with dimension $\alpha > n-(1+\frac{1}{n-1})$. The main intermediate step is a local decay estimate in $L^2(\mu)$ for both the free and perturbed Schr\"odinger evolution., Comment: 16 pages, submitted

In this paper we study the fifth order Kadomtsev--Petviashvili II (KP--II) equation on the upper half-plane $U=\{(x,y)\in \R^2: y>0\}$. In particular we obtain low regularity local well-posedness using the restricted norm method of Bourgain and the Fourier-Laplace method of solving initial and boundary value problems. Moreover we prove that the nonlinear part of the solution is in a smoother space than the initial data., Comment: 39 pages

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 pages

We study the initial-boundary value problem for the derivative nonlinear Schr\"odinger (DNLS) equation. More precisely we study the wellposedness theory and the regularity properties of the DNLS equation on the half line. We prove almost sharp local wellposedness, nonlinear smoothing, and small data global wellposedness in the energy space. One of the obstructions is that the crucial gauge transformation we use replaces the boundary condition with a nonlocal one. We resolve this issue by running an additional fixed point argument. Our method also implies almost sharp local and small energy global wellposedness, and an improved smoothing estimate for the quintic Schr\"odinger equation on the half line. In the last part of the paper we consider the DNLS equation on $\R$ and prove smoothing estimates by combining the restricted norm method with a normal form transformation., Comment: 36 pages

In this paper we study the cubic fractional nonlinear Schrodinger equation (NLS) on the torus and on the real line. Combining the normal form and the restricted norm methods we prove that the nonlinear part of the solution is smoother than the initial data. Our method applies to both focusing and defocusing nonlinearities. In the case of full dispersion (NLS) and on the torus, the gain is a full derivative, while on the real line we get a derivative smoothing with an $\epsilon$ loss. Our result lowers the regularity requirement of a recent theorem of Kappeler et al. on the periodic defocusing cubic NLS, and extends it to the focusing case and to the real line. We also obtain estimates on the higher order Sobolev norms of the global smooth solutions in the defocusing case., Comment: 22 pages

The energy eigenvalues and the wave functions of an $\alpha$ particle in a Bohrium $270$ nucleus were calculated by solving Schr\"odinger equation for Generalized Symmetric Woods-Saxon potential. Using the energy spectrum by excluding and including the quasi-bound eigenvalues, entropy, internal energy, Helmholtz energy, and specific heat, as functions of reduced temperature were calculated. Stability and emission characteristics are interpreted in terms of the wave and thermodynamic functions. The kinetic energy of a decayed $\alpha$ particle was calculated using the quasi-bound states, which is found close to the experimental value., Comment: 11 pages, 6 figures, 2 Tables, SDUFBED 2017

We investigate dispersive estimates for the two dimensional Dirac equation with a potential. In particular, we show that the Dirac evolution satisfies a $t^{-1}$ decay rate as an operator from the Hardy space $H^1$ to $BMO$, the space of functions of bounded mean oscillation. This estimate, along with the $L^2$ conservation law allows one to deduce a family of Strichartz estimates. We classify the structure of threshold obstructions as being composed of s-wave resonances, p-wave resonances and eigenfunctions. We show that, as in the case of the Schr\"odinger evolution, the presence of a threshold s-wave resonance does not destroy the $t^{-1}$ decay rate. As a consequence of our analysis we obtain a limiting absorption principle in the neighborhood of the threshold, and show that there are only finitely many eigenvalues in the spectral gap., Comment: Revised according to referee suggestions. Added several references and expanded the discussion of the spectral theory of the perturbed Dirac operator. 40 pages

The exact analytical solution of the Schr\"odinger equation for a generalized symmetrical Woods-Saxon potential are examined for a nucleon in Fluorine 17 nucleus for bound and quasi-bound states in one dimension. The wave functions imply that the nucleon is completely confined within the nucleus, i.e., no decay probability for bound states, while tunneling probabilities arise for the quasi-bound state. We have calculated the temperature dependent Helmholtz free energies, the internal energies, the entropies and the specific heat capacities of the system. It is shown that, when the quasi-bound state is included, the internal energy and entropy increase, while the Helmholtz energy decreases at high temperatures. Very high excitation temperatures imply that the nucleus does not tend to release a nucleon. The calculated quasi bound state energy is in reasonable agreement with the experimental data on the cumulative fission energy issued by IAEA., Comment: 7 pages, 1 Table and 4 figures

In this paper we study the local and global regularity properties of the cubic nonlinear Schr\"odinger equation (NLS) on the half line with rough initial data. These properties include local and global wellposedness results, local and global smoothing results and the behavior of higher order Sobolev norms of the solutions. In particular, we prove that the nonlinear part of the cubic NLS on the half line is smoother than the initial data. The gain in regularity coincides with the gain that was observed for the periodic cubic NLS \cite{et2} and the cubic NLS on the line \cite{erin}. We also prove that in the defocusing case the norm of the solution grows at most polynomially-in-time while in the focusing case it grows exponentially-in-time. As a byproduct of our analysis we provide a different proof of an almost sharp local wellposedness in $H^s(\R^+)$. Sharp $L^2$ local wellposedness was obtained in \cite{holmer} and \cite{bonaetal}. Our methods simplify some ideas in the wellposedness theory of initial and boundary value problems that were developed in \cite{collianderkenig, holmer,holmer1,bonaetal}., Comment: 30 pages. Minor revisions. To appear in Journal of Functional Analysis

A site selection of potential observatory locations in Turkey have been carried out by using Multi-Criteria Decision Analysis (MCDA) coupled with Geographical Information Systems (GIS) and satellite imagery which in turn reduced cost and time and increased the accuracy of the final outcome. The layers of cloud cover, digital elevation model, artificial lights, precipitable water vapor, aerosol optical thickness and wind speed were studied in the GIS system. In conclusion of MCDA, the most suitable regions were found to be located in a strip crossing from southwest to northeast including also a diverted region in southeast of Turkey. These regions are thus our prime candidate locations for future on-site testing. In addition to this major outcome, this study has also been applied to locations of major observatories sites. Since no goal is set for \textit{the best}, the results of this study is limited with a list of positions. Therefore, the list has to be further confirmed with on-site tests. A national funding has been awarded to produce a prototype of an on-site test unit (to measure both astronomical and meteorological parameters) which might be used in this list of locations., Comment: 17 pages, 10 figures, accepted by Experimental Astronomy

In this paper we study fractal solutions of linear and nonlinear dispersive PDE on the torus. In the first part we answer some open questions on the fractal solutions of linear Schr\"odinger equation and equations with higher order dispersion. We also discuss applications to their nonlinear counterparts like the cubic Schr\"odinger equation (NLS) and the Korteweg-de Vries equation (KdV). In the second part, we study fractal solutions of the vortex filament equation and the associated Schr\"odinger map equation (SM). In particular, we construct global strong solutions of the SM in $H^s$ for $s>\frac32$ for which the evolution of the curvature is given by a periodic nonlinear Schr\"odinger evolution. We also construct unique weak solutions in the energy level. Our analysis follows the frame construction of Chang {\em et al.} \cite{csu} and Nahmod {\em et al.} \cite{nsvz}., Comment: 28 pages

We study the Cauchy problem for the $1$-d periodic fractional Schr\"odinger equation with cubic nonlinearity. In particular we prove local well-posedness in Sobolev spaces, for solutions evolving from rough initial data. In addition we show the existence of global-in-time infinite energy solutions. Our tools include a new Strichartz estimate on the torus along with ideas that Bourgain developed in studying the periodic cubic NLS., Comment: 19 pages

We investigate $L^1(\mathbb R^4)\to L^\infty(\mathbb R^4)$ dispersive estimates for the Schr\"odinger operator $H=-\Delta+V$ when there are obstructions, a resonance or an eigenvalue, at zero energy. In particular, we show that if there is a resonance or an eigenvalue at zero energy then there is a time dependent, finite rank operator $F_t$ satisfying $\|F_t\|_{L^1\to L^\infty} \lesssim 1/\log t$ for $t>2$ such that $$\|e^{itH}P_{ac}-F_t\|_{L^1\to L^\infty} \lesssim t^{-1},\,\,\,\,\,\text{for} t>2.$$ We also show that the operator $F_t=0$ if there is an eigenvalue but no resonance at zero energy. We then develop analogous dispersive estimates for the solution operator to the four dimensional wave equation with potential., Comment: 32 pages

The Zakharov system was originally proposed to study the propagation of Langmuir waves in an ionized plasma. In this paper, motivated by earlier work of the first and third authors, we numerically and analytically investigate the dynamics of the dissipative Zakharov system on the torus in 1 dimension. We find an interesting family of stable periodic orbits and fixed points, and explore bifurcations of those points as we take weaker and weaker dissipation., Comment: 16 pages, 7 figures