arXiv subjects

Thanasis Zacharopoulos

Publications and source records attributed to Thanasis Zacharopoulos.

6 recordsLinked to original sources

arXiv2026

Stability estimates for the initial-to-final-state inverse problem

The initial-to-final-state inverse problem for the Schr\"odinger equation consists in determining uniquely the Hamiltonian that generates the evolution, assuming the knowledge of the initial-to-final-state map. This functional maps each initial state $f\in L^2(\mathbb{R}^n)$ to the corresponding final state at a fixed time $T$. The problem was formulated by Caro and Ruiz in the case of Hamiltonians arising from time-dependent bounded electric potentials that exhibit super-exponential decay at infinity. Caro, Parissis and the authors of this article established that uniqueness also holds for time-independent bounded potentials with super-linear decay at infinity. In this paper, we quantify the above uniqueness results establishing that the potentials are stable under small changes of the initial-to-final-state maps. In the case of time-dependent potentials we get a stability of logarithmic type. A notable improvement is achieved when the potentials are time-independent, where under this assumption we prove H\"older stability estimates.

math.AP↗

arXiv2026

Reconstruction of non-self-adjoint anisotropic and complex inclusions in the Calder\'on problem

We generalize recent results on the monotonicity method, for inclusion detection in the partial data anisotropic Calder\'on problem, to very general non-self-adjoint perturbations. This involves a forward model that accounts for both the anisotropic real conductivity and the anisotropic permittivity, and the results hold in any spatial dimension $d \geq 2$. We assume that the inclusion boundaries can be reached from the domain boundary via a set on which the background conductivity is self-adjoint, and that a definiteness condition holds near the inclusion boundaries. Away from the inclusion boundaries we allow general $L^\infty$ non-self-adjoint perturbations. We only require unique continuation based on the self-adjoint part of the background conductivity, thus making the methods compatible with generic unique continuation results.

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arXiv2026

The initial-to-final-state inverse problem with critically-singular potentials

The Schr\"odinger equation in high dimensions describes the evolution of a quantum system. Assume that we are given the evolution map sending each initial state $f\in L^2(\mathbb{R}^n)$ of the system to the corresponding final state at a fixed time $T$. The main question we address in this paper is whether this initial-to-final-state map uniquely determines the Hamiltonian $-\Delta+V$ that generates the evolution. We restrict attention to time-independent potentials $V$ and show that uniqueness holds provided $V \in L^1(\mathbb{R}^n)\cap L^q(\mathbb{R}^n)$, with $q>1$ if $n=2$ or $q\geq n/2$ if $n\geq 3$. This should be compared with the results of Caro and Ruiz, who proved that in the time-dependent case, uniqueness holds under the stronger assumption that the potential exhibits super-exponential decay at infinity, for both bounded and unbounded potentials. This paper extends earlier work of the same authors, where uniqueness was obtained for bounded time-independent potentials with polynomial decay at infinity. Here we only require $L^1$-type decay at infinity and allow for $L^q$-type singularities. We reach this improvement by providing a refinement of the Kenig-Ruiz-Sogge resolvent estimate, which replaces the classical Agmon-H\"ormander estimates used previously. Crucially, the time-independent setting allows us to avoid the use of complex geometrical optics solutions and thereby dispense with strong decay assumptions at infinity.

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arXiv2025

Direct reconstruction of anisotropic self-adjoint inclusions in the Calder\'on problem

We extend the monotonicity method for direct exact reconstruction of inclusions in the partial data Calder\'on problem, to the case of general anisotropic conductivities in any spatial dimension $d\geq 2$. From a local Neumann-to-Dirichlet map, we give reconstruction methods of inclusions based on unknown anisotropic self-adjoint perturbations to a known anisotropic conductivity coefficient. This additionally provides new insights into the non-uniqueness issues of the anisotropic Calder\'on problem. The main assumption is a definiteness condition for the perturbations near the outer inclusion boundaries. Beyond this condition, they are $L^\infty$-perturbations that may be indefinite away from the outer inclusion boundaries, and with no boundary regularity requirement for the inclusions. Alternatively, we allow extreme parts that are perfectly insulating or perfectly conducting, in which case we require Lipschitz regularity of the outer inclusion boundaries.

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arXiv2025

The initial-to-final-state inverse problem with time-independent potentials

The initial-to-final-state inverse problem consists in determining a quantum Hamiltonian assuming the knowledge of the state of the system at some fixed time, for every initial state. This problem was formulated by Caro and Ruiz and motivated by the data-driven prediction problem in quantum mechanics. Caro and Ruiz analysed the question of uniqueness for Hamiltonians of the form $-\Delta + V$ with an electric potential $V = V(\mathrm{t}, \mathrm{x})$ that depends on the time and space variables. In this context, they proved that uniqueness holds in dimension $n \geq 2$ whenever the potentials are bounded and have super-exponential decay at infinity. Although their result does not seem to be optimal, one would expect at least some degree of exponential decay to be necessary for the potentials. However, in this paper, we show that by restricting the analysis to Hamiltonians with time-independent electric potentials, namely $V = V(\mathrm{x})$, uniqueness can be established for bounded integrable potentials exhibiting only super-linear decay at infinity, in any dimension $n \geq 2$. This surprising improvement is possible because, unlike Caro and Ruiz's approach, our argument avoids the use of complex geometrical optics (CGO). Instead, we rely on the construction of stationary states at different energies -- this is possible because the potential does not depend on time. These states will have an explicit leading term, given by a Herglotz wave, plus a correction term that will vanish as the energy grows. Besides the significant relaxation of decay assumptions on the potential, the avoidance of CGO solutions is important in its own right, since such solutions are not readily available in more complicated geometric settings.

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arXiv2023

Varopoulos extensions in domains with Ahlfors-regular boundaries and applications to Boundary Value Problems for elliptic systems with $L^\infty$ coefficients

Let $\Omega \subset \mathbb{R}^{n+1}$, $n\geq 1$, be an open set with $s$-Ahlfors regular boundary $\partial \Omega$, for some $s \in(0,n]$, such that either $s=n$ and $\Omega$ is a corkscrew domain with the pointwise John condition, or $s<n$ and $\Omega= \mathbb{R}^{n+1} \setminus E$, for some $s$-Ahlfors regular set $E \subset \mathbb{R}^{n+1}$. In this paper we provide a unifying method to construct Varopoulos' type extensions of $BMO$ and $L^p$ boundary functions. In particular, we show that a) if $ f \in BMO(\partial \Omega)$, there exists $ F\in C^\infty(\Omega)$ such that $dist(x, \Omega^c)|\nabla F(x)|$ is uniformly bounded in $\Omega$ and the Carleson functional of $dist(x,\Omega^c)^{s-n}|\nabla F(x)|$ as well the sharp non-tangential maximal function of $ F$ are uniformly bounded on $\partial \Omega$ with norms controlled by the $BMO$-norm of $ f$, and $ F \to f$ in a certain non-tangential sense $\mathcal H^s|_{\partial \Omega}$-almost everywhere; b) if $\bar f \in L^p(\partial \Omega)$, $1 <p \leq \infty$, there exists $\bar F \in C^\infty(\Omega)$ such that the non-tangential maximal functions of $\bar F$ and $dist(\cdot, \Omega^c)|\nabla \bar F|$ as well as the Carleson functional of $dist(\cdot,\Omega^c)^{s-n}|\nabla \bar F|$ are in $L^p(\partial \Omega)$ with norms controlled by the $L^p$-norm of $\bar f$, and $\bar F \to \bar f$ in some non-tangential sense $\mathcal H^s|_{\partial \Omega}$-almost everywhere. If, in addition, the boundary function is Lipschitz with compact support, then both $F$ and $\bar F$ can be constructed so that they are also Lipschitz on $\bar\Omega$ and converge to the boundary data continuously. The latter results hold without the additional assumption of the pointwise John condition. Finally, we give some applications of the constructed extensions in the connection between Poisson problems and BVPs.

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