Browsing by Author "Kalimeris, Konstantinos"

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Open Access
Dispersion estimates for the boundary integral operator associated with the fourth order Schrödinger equation posed on the half line
(Element d.o.o., 2021-09) Özsarı, Türker; Alkan, Kıvılcım; Kalimeris, Konstantinos
In this paper, we prove dispersion estimates for the boundary integral operator associated with the fourth order Schr¨odinger equation posed on the half line. Proofs of such estimates for domains with boundaries are rare and generally require highly technical approaches, as opposed to our simple treatment which is based on constructing a boundary integral operator of oscillatory nature via the Fokas method. Our method is uniform and can be extended to other higher order partial differential equations where the main equation possibly involves more than one spatial derivatives.
Open Access
Existence of unattainable states for Schrödinger type flows on the half-line
(Oxford University Press, 2023-12-01) Özsarı, Türker; Kalimeris, Konstantinos
We prove that the solutions of the Schrödinger and biharmonic Schrödinger equations do not have the exact boundary controllability property on the half-line by showing that the associated adjoint models lack observability. We consider the framework of L2 boundary controls with data spaces H−1(R+) and H−2(R+) for the classical and biharmonic Schrödinger equations, respectively. The lack of controllability on the half-line contrasts with the corresponding dynamics on a finite interval for a similar regularity setting. Our proof is based on an argument that uses the sharp fractional time trace estimates for solutions of the adjoint models. We also make several remarks on the connection of controllability and temporal regularity of spatial traces.