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Eric Soccorsi

Publications and source records attributed to Eric Soccorsi.

At least 19 recordsLinked to original sources
arXiv2025

Determination of the Schr\"odinger-Robin operator by incomplete or asymptotic spectral boundary data

This article deals with the inverse problem of determining the unbounded real-valued electric potential of the Robin Laplacian on a bounded domain of dimension 3 or greater, by incomplete knowledge of its boundary spectral data. Namely, the main result establishes that the unknown potential can be H\"older stably retrieved from the asymptotic behavior of the eigenvalues and the sequence of the boundary measurements of the corresponding eigenfunctions where finitely many terms are missing.

math.AP↗
arXiv2024

Inverse Coefficient Problem for One-Dimensional Subdiffusion with Data on Disjoint Sets in Time

In this work we investigate an inverse coefficient problem for the one-dimensional subdiffusion model, which involves a Caputo fractional derivative in time. The inverse problem is to determine two coefficients and multiple parameters (the order, and length of the interval) from one pair of lateral Cauchy data. The lateral Cauchy data are given on disjoint sets in time with a single excitation and the measurement is made on a time sequence located outside the support of the excitation. We prove two uniqueness results for different lateral Cauchy data. The analysis is based on the solution representation, analyticity of the observation and a refined version of inverse Sturm-Liouville theory due to Sini [35]. Our results heavily exploit the memory effect of fractional diffusion for the unique recovery of the coefficients in the model. Several numerical experiments are also presented to complement the analysis.

math.AP↗
arXiv2024

Asymptotic analysis of time-fractional quantum diffusion

We study the large-time asymptotics of the mean-square displacement for the time-fractional Schrodinger equation in $\mathbb{R}^d$. We define the time-fractional derivative by the Caputo derivative and we consider the initial-value problem for the free evolution of wave packets in $\mathbb{R}^d$ governed by the time-fractional Schrodinger equation $ i^\beta \partial_t^\alpha u = - \Delta u, ~~~~u(t=0) = u_0$, parameterized by two indices $\alpha, \beta \in (0,1]$. We show distinctly different long-time evolution of the mean square displacement according to the relation between $\alpha$ and $\beta$. In particular, asymptotically ballistic motion occurs only for $\alpha=\beta$.

math.AP↗
arXiv2024

Edge currents for the time-fractional, half-plane, Schrodinger equation with constant magnetic field

We study the large-time asymptotics of the edge current for a family of time-fractional Schrodinger equations with a constant, transverse magnetic field on a half-plane $(x,y) \in \mathbb{R}_x^+ \times \mathbb{R}_y$. The TFSE is parameterized by two constants $(\alpha, \beta)$ in $(0,1]$, where $\alpha$ is the fractional order of the time derivative, and $\beta$ is the power of $i$ in the Schrodinger equation. We prove that for fixed $\alpha$, there is a transition in the transport properties as $\beta$ varies in $(0,1]$: For $0 < \beta < \alpha$, the edge current grows exponentially in time, for $\alpha = \beta$, the edge current is asymptotically constant, and for $\beta > \alpha$, the edge current decays in time. We prove that the mean square displacement in the $y\in \mathbb{R}$-direction undergoes a similar transport transition. These results provide quantitative support for the comments of Laskin \cite{laskin2000_1} that the latter two cases, $\alpha = \beta$ and $\alpha < \beta$, are the physically relevant ones.

math-ph↗
arXiv2023

Determining an Iwatsuka Hamiltonian by knowledge of its first band function

We investigate the inverse problem of retrieving the magnetic potential of an Iwatsuka Hamiltonian through knowledge of its first band function. We prove that two magnetic potentials sharing the same first band function not just between them, but between all fields within the family linearly interpolating between them and accumulating at one end point, coincide.

math.AP↗
arXiv2022

Solving time-fractional diffusion equations with singular source term

This article deals with time-fractional diffusion equations with time-dependent singular source term. Whenever the order of the time-fractional derivative is either multi-term, distributed or space-dependent, we prove that the system admits a unique weak solution enjoying a Duhamel representation, provided that the time-dependence of the source term is a distribution.

math.AP↗
arXiv2021

Logarithmic stable recovery of the source and the initial state of time fractional diffusion equations

In this paper we study the inverse problem of identifying a source or an initial state in a time-fractional diffusion equation from the knowledge of a single boundary measurement. We derive logarithmic stability estimates for both inversions. These results show that the ill-posedness increases exponentially when the fractional derivative order tends to zero, while it exponentially decreases when the regularity of the source or the initial state becomes larger. The stability estimate concerning the problem of recovering the initial state can be considered as a weak observability inequality in control theory. The analysis is mainly based on Laplace inversion techniques and a precise quantification of the unique continuation property for the resolvent of the time-fractional diffusion operator as a function of the frequency in the complex plane. We also determine a global time regularity for the time-fractional diffusion equation which is of interest itself.

math.AP↗
arXiv2021

Determination of source and initial values for acoustic equations with a time-fractional attenuation

We consider the inverse problem of determining the initial states or the source term of a hyperbolic equation damped by some non-local time-fractional derivative. This framework is relevant to medical imaging such as thermoacoustic or photoacoustic tomography. We prove a stability estimate for each of these two problems, with the aid of a Carleman estimate specifically designed for the governing equation.

math.AP↗
arXiv2021

Identification of time-varying source term in time-fractional diffusion equations

This paper is concerned with the inverse problem of determining the time and space dependent source term of diffusion equations with constant-order time-fractional derivative in $(0,2)$. We examine two different cases. In the first one, the source is the product of two spatial and temporal terms, and we prove that both of them can be retrieved by knowledge of one arbitrary internal measurement of the solution for all times. In the second case, we assume that the first term of the product varies with one fixed space variable, while the second one is a function of all the remaining space variables and the time variable, and we show that both terms are uniquely determined by two arbitrary lateral measurements of the solution over the entire time span. These two source identification results boil down to a weak unique continuation principle in the first case and a unique continuation principle for Cauchy data in the second one, that are preliminarily established. Finally, numerical reconstruction of spatial term of source terms in the form of the product of two spatial and temporal terms, is carried out through an iterative algorithm based on the Tikhonov regularization method.

math.AP↗
arXiv2018

Eigenvalue Asymptotics in a Twisted Waveguide

We consider a twisted quantum wave guide, and are interested in the spectral analysis of the associated Dirichlet Laplacian H. We show that if the derivative of rotation angle decays slowly enough at infinity, then there is an infinite sequence of discrete eigenvalues lying below the infimum of the essential spectrum of H, and obtain the main asymptotic term of this sequence.

math.SP↗
arXiv2018

Carleman estimate for the Schrödinger equation and application to magnetic inverse problems

We prove that the stationary magnetic potential vector and the electrostatic potential entering the dynamic magnetic Schrödinger equation can be Lipschitz stably retrieved through finitely many local boundary measurements of the solution. The proof is by means of a specific global Carleman estimate for the Schrödinger equation, established in the first part of the paper.

math.AP↗
arXiv2017

Stable reconstruction of the volatility in a regime-switching local volatility model

Prices of European call options in a regime-switching local volatility model can be computed by solving a parabolic system which generalises the classical Black and Scholes equation, giving these prices as functionals of the local volatilities. We prove Lipschitz stability for the inverse problem of determining the local volatilities from quoted call option prices for a range of strikes, if the calls are indexed by the different states of the continuous Markov chain which governs the regime switches.

math.AP↗
arXiv2017

Initial-boundary value problem for distributed order time-fractional diffusion equations

We examine initial-boundary value problems for diffusion equations with distributed order time-fractional derivatives. We prove existence and uniqueness results for the weak solution to these systems, together with its continuous dependency on initial value and source term. Moreover, under suitable assumption on the source term, we establish that the solution is analytic in time.

math.AP↗
arXiv2017

Hölder stably determining the time-dependent electromagnetic potential of the Schrödinger equation

We consider the inverse problem of determining the time and space dependent electromagnetic potential of the Schrödinger equation in a bounded domain of $\mathbb R^n$, $n\geq 2$, by boundary observation of the solution over the entire time span. Assuming that the divergence of the magnetic potential is fixed, we prove that the electric potential and the magnetic potential can be Hölder stably retrieved from these data, whereas stability estimates for inverse time-dependent coefficients problems of evolution partial differential equations are usually of logarithmic type.

math.AP↗
arXiv2017

On time-fractional diffusion equations with space-dependent variable order

We investigate diffusion equations with time-fractional derivatives of space-dependent variable order. We examine the well-posedness issue and prove that the space-dependent variable order coefficient is uniquely determined among other coefficients of these equations, by the knowledge of a suitable time-sequence of partial Dirichlet-to-Neumann maps.

math.AP↗