arXiv subjects

Ebru Toprak

Publications and source records attributed to Ebru Toprak.

11 recordsLinked to original sources

arXiv2025

Global Solutions for 5D Quadratic Fourth-Order Schr\"{o}dinger Equations

We prove small data scattering for the fourth-order Schr\"odinger equation with quadratic nonlinearity \begin{equation*} i\partial_t u+\Delta^2 u+\alpha u^2 + \beta \bar{u}^2=0\qquad\text{in }\mathbb{R}^5 \end{equation*} for $\alpha, \beta \in \mathbb{R}$. We extend the space-time resonance method, originally introduced by Germain, Masmoudi, and Shatah, to the setting involving the bilaplacian. We show that under a smallness condition on the initial data measured in a suitable norm, the solution satisfies $\|u\|_{L^{\infty}_x }\lesssim t^{-\frac{5}{4}} $ and scatters to the solution to the free equation. Although our work builds upon an established method, the fourth-order nature of the equation presents substantial challenges, requiring different techniques to overcome them.

math.AP↗

arXiv2024

Pointwise decay for radial solutions of the Schrödinger equation with a repulsive Coulomb potential

We study the long-time behavior of solutions to the Schrödinger equation with a repulsive Coulomb potential on $\mathbb{R}^3$ for spherically symmetric initial data. Our approach involves computing the distorted Fourier transform of the action of the associated Hamiltonian $H=-Δ+\frac{q}{|x|}$ on radial data $f$, which allows us to explicitly write the evolution $e^{itH}f$. A comprehensive analysis of the kernel is then used to establish that, for large times, $\|e^{i t H}f\|_{L^{\infty}} \leq C t^{-\frac{3}{2}}\|f\|_{L^1}$. Our analysis of the distorted Fourier transform is expected to have applications to other long-range repulsive problems.

math.AP↗

arXiv2023

Bounds on the number of scattering poles of half-Laplacian in odd dimensions, $d\geq 3$

We study the scattering poles of $\sqrt{-Δ} + V$, where $V$ is a compactly supported, bounded and complex valued potential. We show that the resolvent operator $ χR_V χ$ has a meromorphic continuation to the whole Riemannian surface of $Λ$ of $ \log z $ as an operator $L^2 \to L^2 $. We then obtain the upper bound on the counting function $N(r,a)= \# \{ z_j \in Λ: 0 \leq |z_j| \leq r, |\arg z_j| \leq a \}$, $r >1$, $ |a| >1$ as $C \langle a \rangle ( \langle r \rangle^{d} + (\log \langle a \rangle)^d) $, where $z_j$ are the poles of $ χR_V χ$.

math.AP↗

arXiv2021

On the Dirac operator for a test electron in a Reissner--Weyl--Nordström black hole spacetime

The present paper studies the Dirac Hamiltonian of a test electron with a domain of bi-spinor wave functions supported on the static region inside the Cauchy horizon of the subextremal RWN black hole spacetime, respectively inside the event horizon of the extremal RWN black hole spacetime. It is found that this Dirac Hamiltonian is not essentially self-adjoint, yet has infinitely many self-adjoint extensions. Including a sufficiently large anomalous magnetic moment interaction in the Dirac Hamiltonian restores essential self-adjointness; the empirical value of the electron's anomalous magnetic moment is large enough. The spectrum of the subextremal self-adjoint Dirac operator with anomalous magnetic moment is purely absolutely continuous and consists of the whole real line; in particular, there are no eigenvalues. The same is true for the spectrum of any self-adjoint extension of the Dirac operator without anomalous magnetic moment interaction, in the subextremal black hole context. In the extremal black hole sector the point spectrum, if non-empty, consists of a single eigenvalue, which is identified.

math-ph↗

arXiv2020

On general-relativistic hydrogen and hydrogenic ions

This paper studies how the static non-linear electromagnetic-vacuum spacetime of a point nucleus with negative bare mass affects the self-adjointness of the general-relativistic Dirac Hamiltonian for a test electron, without and with an anomalous magnetic moment. The study interpolates between the previously studied extreme cases of a test electron in (a) the Reissner--Weyl--Nordström spacetime (Maxwell's electromagnetic vacuum), which supports a very strong curvature singularity with negative infinite bare mass, and (b) the Hoffmann spacetime (Born or Born--Infeld's electromagnetic vacuum) with vanishing bare mass, which features the mildest possible curvature singularity. The main conclusion reached is: {on electrostatic spacetimes of a point nucleus with a strictly negative bare mass} (which may be $-\infty$) essential self-adjointness fails unless the radial electric field diverges sufficiently fast at the nucleus and the anomalous magnetic moment of the electron is taken into account. Thus on the Hoffmann spacetime with (strictly) negative bare mass the Dirac Hamiltonian of a test electron, with or without anomalous magnetic moment, is not essentially self-adjoint. All these operators have self-adjoint extensions, though, with the usual essential spectrum $(-\infty,-\mEL c^2]\cup[\mEL c^2,\infty)$ and an infinite discrete spectrum located in the gap $(-\mEL c^2,\mEL c^2)$

math-ph↗

arXiv2019

On the Fourth order Schrödinger equation in three dimensions: dispersive estimates and zero energy resonances

We study the fourth order Schrödinger operator $H=(-Δ)^2+V$ for a short range potential in three space dimensions. We provide a full classification of zero energy resonances and study the dynamic effect of each on the $L^1\to L^\infty$ dispersive bounds. In all cases, we show that the natural $|t|^{-\frac34}$ decay rate may be attained, though for some resonances this requires subtracting off a finite rank term, which we construct and analyze. The classification of these resonances, as well as their dynamical consequences differ from the Schrödinger operator $-Δ+V$.

math.AP↗

arXiv2019

On the Fourth order Schrödinger equation in four dimensions: dispersive estimates and zero energy resonances

We study the fourth order Schrödinger operator $H=(-Δ)^2+V$ 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.

math.AP↗

arXiv2017

Dispersive estimates for Dirac Operators in dimension three with obstructions at threshold energies

We investigate $L^1\to L^\infty$ dispersive estimates for the three dimensional Dirac equation with a potential. We also classify the structure of obstructions at the thresholds of the essential spectrum as being composed of a two dimensional space of resonances and finitely many eigenfunctions. We show that, as in the case of the Schrödinger evolution, the presence of a threshold obstruction generically leads to a loss of the natural $t^{-\frac32}$ decay rate. In this case we show that the solution operator is composed of a finite rank operator that decays at the rate $t^{-\frac12}$ plus a term that decays at the rate $t^{-\frac32}$.

math.AP↗

arXiv2017

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.

math.AP↗

arXiv2016

Decay estimates for four dimensional Schrödinger, Klein-Gordon and wave equations with obstructions at zero energy

We investigate dispersive estimates for the Schrödinger operator $H=-Δ+V$ with $V$ is a real-valued decaying potential when there are zero energy resonances and eigenvalues in four spatial dimensions. If there is a zero energy obstruction, we establish the low-energy expansion $$ e^{itH}χ(H) P_{ac}(H)=O(1/(\log t)) A_0+ O(1/t)A_1+O((t\log t)^{-1})A_2+ O(t^{-1}(\log t)^{-2})A_3. $$ Here $A_0,A_1:L^1(\mathbb R^n)\to L^\infty (\mathbb R^n)$, while $A_2,A_3$ are operators between logarithmically weighted spaces, with $A_0,A_1,A_2$ finite rank operators, further the operators are independent of time. We show that similar expansions are valid for the solution operators to Klein-Gordon and wave equations. Finally, we show that under certain orthogonality conditions, if there is a zero energy eigenvalue one can recover the $|t|^{-2}$ bound as an operator from $L^1\to L^\infty$. Hence, recovering the same dispersive bound as the free evolution in spite of the zero energy eigenvalue.

math.AP↗

arXiv2015

A weighted estimate for two dimensional Schrodinger, matrix schrodinger and wave equations with resonance of first kind at zero energy

We study the two dimensional Schrödinger operator, $H=-Δ+V$, 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 ^{-3-} and there is only s-wave resonance at zero of H, then \big\| w^{-1} \big( e^{itH}P_{ac} f - {\f 1 t } 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=c ψ\la f,ψ\ra, where ψis an s-wave resonance function. We also extend this result to matrix Schrödinger and wave equations with potentials under similar conditions.

math.AP↗