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M. Burak Erdogan

Publications and source records attributed to M. Burak Erdogan.

At least 19 recordsLinked to original sources
arXiv2026

Dispersive Estimates for Dirac Operators with General Domain Walls: One and Two Dimensions

We establish dispersive decay estimates for two-dimensional Dirac equations with bounded and unbounded domain walls, together with estimates for the analogous one-dimensional problem. These are paradigmatic models of bulk-edge correspondence in topological insulators, yet their dispersive dynamics have not been previously studied. We show that topologically equivalent models may exhibit qualitatively different dynamics. Our new approach allows us to analyze models which are beyond the usual framework of localized perturbations of constant-coefficient Dirac operators.

math.AP↗
arXiv2026

Dispersive estimates for fractional order Schrödinger operators

We prove dispersive bounds for fractional Schrödinger operators on $\mathbb R^n$ of the form $H=(-Δ)^α+V$ with $V$ a real-valued, decaying potential and $α\notin\mathbb N$. We derive pointwise bounds on the resolvent operators for all $0<α<\frac{n}{2}$, a quantitative limiting absorption principle for $\frac12<α<\frac{n}{2}$, and establish global dispersive estimates in dimension $n\geq 2$ for the range $\frac{n+1}{4}\leq α<\frac{n}2$.

math.AP↗
arXiv2026

$L^p$ boundedness of wave operators for higher order schrödinger operators with threshold eigenvalues

We consider the higher order Schrödinger operator $H=(-Δ)^m+V(x)$ in $n$ dimensions with real-valued potential $V$ when $n>2m$, $m\in \mathbb N$ when $H$ has a threshold eigenvalue. We adapt our recent results for $m\geq 1$ when $n>4m$ to lower dimensions $2m 2m$. The range is $p\in [1,\infty)$ and $p\in[1,\infty]$ when $k_0=2m$ and $k_0>2m$ respectively. The proofs apply in the classical $m=1$ case as well and streamlines existing arguments in the eigenvalue only case, in particular the $L^\infty(\mathbb R^n)$ boundedness is new when $n>3$.

math.AP↗
arXiv2026

The $L^p$-continuity of wave operators for fractional order Schrödinger operators

We consider fractional Schrödinger operators $H=(-Δ)^α+V(x)$ in $n$ dimensions with real-valued potential $V$ when $n>2α$, $α>1$. We show that the wave operators extend to bounded operators on $L^p(\mathbb R^n)$ for all $1\leq p\leq\infty$ under conditions on the potential that depend on $n$ and $α$ analogously to the case when $α\in \mathbb N$. As a consequence, we deduce a family of dispersive and Strichartz estimates for the perturbed fractional Schrödinger operator.

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arXiv2025

Dispersive estimates for higher order Schrödinger operators with scaling-critical potentials

We prove a family of dispersive estimates for the higher order Schrödinger equation $iu_t=(-Δ)^mu +Vu$ for $m\in \mathbb N$ with $m>1$ and $2m<n<4m$. Here $V$ is a real-valued potential belonging to the closure of $C_0$ functions with respect to the generalized Kato norm, which has critical scaling. Under standard assumptions on the spectrum, we show that $e^{-itH}P_{ac}(H)$ satisfies a $|t|^{-\frac{n}{2m}}$ bound mapping $L^1$ to $L^\infty$ by adapting a Wiener inversion theorem. We further show the lack of positive resonances for the operator $(-Δ)^m +V$ and a family of dispersive estimates for operators of the form $|H|^{β-\frac{n}{2m}}e^{-itH}P_{ac}(H)$ for $0<β\leq \frac{n}{2}$. The results apply in both even and odd dimensions in the allowed range.

math.AP↗
arXiv2025

$L^p$-continuity of wave operators for higher order Schrödinger operators with threshold eigenvalues in high dimensions

We consider the higher order Schrödinger operator $H=(-Δ)^m+V(x)$ 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.

math.AP↗
arXiv2023

Talbot Effect on the Sphere and Torus for $d\geq 2$

We utilize exponential sum techniques to obtain upper and lower bounds for the fractal dimension of the graph of solutions to the linear Schrödinger 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.

math.AP↗
arXiv2023

A note on endpoint $L^p$-continuity of wave operators for classical and higher order Schrödinger operators

We consider the higher order Schrödinger operator $H=(-Δ)^m+V(x)$ 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 $m=1$.

math.AP↗
arXiv2022

The $L^p$-continuity of wave operators for higher order Schrödinger operators

We consider the higher order Schrödinger operator $H=(-Δ)^m+V(x)$ 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 $1<p<\infty$.

math.AP↗
arXiv2020

Fourier decay of fractal measures on hyperboloids

Let $μ$ be an $α$-dimensional probability measure. We prove new upper and lower bounds on the decay rate of hyperbolic averages of the Fourier transform $\widehatμ$. More precisely, if $\mathbb{H}$ is a truncated hyperbolic paraboloid in $\mathbb{R}^d$ we study the optimal $β$ for which $$\int_{\mathbb{H}} |\hatμ(Rξ)|^2 \, d σ(ξ)\leq C(α, μ) R^{-β}$$ for all $R > 1$. Our estimates for $β$ 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.

math.CA↗
arXiv2019

Strichartz Estimates for the Schrödinger Equation with a Measure-Valued Potential

We prove Strichartz estimates for the Schrödinger equation in $\mathbb R^n$, $n\geq 3$, with a Hamiltonian $H = -Δ+ μ$. The perturbation $μ$ is a compactly supported measure in $\mathbb R^n$ with dimension $α> n-(1+\frac{1}{n-1})$. The main intermediate step is a local decay estimate in $L^2(μ)$ for both the free and perturbed Schrödinger evolution.

math.AP↗
arXiv2016

The Dirac equation in two dimensions: Dispersive estimates and classification of threshold obstructions

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 Schrödinger 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.

math.AP↗
arXiv2016

Regularity properties of the cubic nonlinear Schrödinger equation on the half line

In this paper we study the local and global regularity properties of the cubic nonlinear Schrödinger 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}.

math.AP↗
arXiv2015

The Structure of Global Attractors for Dissipative Zakharov Systems with Forcing on the Torus

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.

math.AP↗
arXiv2013

Dispersive estimates for four dimensional Schrödinger and wave equations with obstructions at zero energy

We investigate $L^1(\mathbb R^4)\to L^\infty(\mathbb R^4)$ dispersive estimates for the Schrödinger operator $H=-Δ+V$ 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.

math.AP↗
arXiv2012

Dispersive estimates for Schrödinger operators in dimension two with obstructions at zero energy

We investigate $L^1(\R^2)\to L^\infty(\R^2)$ dispersive estimates for the Schrödinger operator $H=-Δ+V$ when there are obstructions, resonances or an eigenvalue, at zero energy. In particular, we show that the existence of an s-wave resonance at zero energy does not destroy the $t^{-1}$ decay rate. We also show that if there is a p-wave resonance or an eigenvalue at zero energy then there is a time dependent operator $F_t$ satisfying $\|F_t\|_{L^1\to L^\infty} \lesssim 1$ such that $$\|e^{itH}P_{ac}-F_t\|_{L^1\to L^\infty} \lesssim |t|^{-1}, \text{for} |t|>1.$$ We also establish a weighted dispersive estimate with $t^{-1}$ decay rate in the case when there is an eigenvalue at zero energy but no resonances.

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