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We consider the inverse coefficient problem of simultaneously determining the space dependent electromagnetic potential, the zero-th order coupling term and the first order coupling vector of a two-state Schr\"odinger equation in a bounded domain of $\mathbb{R}^d$, $d \ge 2$, from finitely many partial boundary measurements of the solution. We prove that these $3d+3$ unknown scalar coefficients can be H\"older stably retrieved by $(3d+2)$-times suitably changing the initial condition attached at the system.

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., Comment: 17 pages, 7 figures

This contribution considers the time-fractional subdiffusion with a time-dependent variable-order fractional operator of order $\beta(t)$. It is assumed that $\beta(t)$ is a piecewise constant function with a finite number of jumps. A proof technique based on the Fourier method and results from constant-order fractional subdiffusion equations has been designed. This novel approach results in the well-posedness of the problem.

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 $\alpha=\beta$.

We investigate the inverse problem of retrieving the magnetic potential of an Iwatsuka Hamiltonian by knowledge of the second component of the quantum velocity. We show that knowledge of the quantum currents carried by a suitable set of states with energy concentration within the first spectral band of the Schr\"odinger operator, uniquely determines the magnetic field.

This article deals with the uniqueness and stability issues in the inverse problem of determining the unbounded potential of the Schr\"odinger operator in a bounded domain of dimension 3 or greater, endowed with Robin boundary condition, from knowledge of its boundary spectral data. These data are defined by the pairs formed by the eigenvalues and either full or partial Dirichlet measurement of the eigenfunctions on the boundary of the domain.

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.

We consider the Dirichlet Laplacian $A_q=-\Delta+q$ in a bounded domain $\Omega \subset \mathbb{R}^d$, $d \ge 3$, with real-valued perturbation $q \in L^{\max(2 , 3 d / 5)}(\Omega)$. We examine the stability issue in the inverse problem of determining the electric potential $q$ from the asymptotic behavior of the eigenvalues of $A_q$. Assuming that the boundary measurement of the normal derivative of the eigenfunctions is a square summable sequence in $L^2(\partial \Omega)$, we prove that $q$ can be H\"older stably retrieved through knowledge of the asymptotics of the eigenvalues

We consider the Schr\"odinger operator with constant transverse magnetic field on a half-plane, endowed with Neumann boundary conditions. We study the low energy currents flowing along the boundary and we establish a Limiting Absorption Principle for the magnetic Neumann Laplacian under perturbation of an electric potential.

We consider the inverse coefficient problem of simultaneously determining the space dependent electric potential, the zero-th order coupling term and the first order coupling vector of a two-state Schr\"odinger equation in an infinite cylindrical domain of $\mathbb{R}^n$, $n \ge 2$, from finitely many partial boundary measurements of the solution. We prove that these $n+1$ unknown scalar coefficients can be H\"older stably retrieved by $(n+1)$-times suitably changing the initial condition attached at the system.

We prove that the real-valued electric potential $q \in L^{\max(2,3 n / 5)}(\Omega)$ of the Dirichlet Laplacian $-\Delta +q$ acting in a bounded domain $\Omega \subset \mathbb{R}^n$, $n \ge 3$, is uniquely determined by the asymptotics of the eigenpairs formed by the eigenvalues and the boundary observation of the normal derivative of the eigenfunctions.

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.

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.

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.

This text deals with multidimensional Borg-Levinson inverse theory. Its main purpose is to establish that the Dirichlet eigenvalues and Neumann boundary data of the Dirichlet Laplacian acting in a bounded domain of dimension 2 or greater, uniquely determine the real-valued bounded potential. We first address the case of incomplete spectral data, where finitely many boundary spectral eigen-pairs remain unknown. Under suitable summability condition on the Neumann data, we also consider the case where only the asymptotic behavior of the eigenvalues is known. Finally, we use the multidimensional Borg-Levinson theory for solving parabolic inverse coefficient problems.

We examine the short and long-time behaviors of time-fractional diffusion equations with variable space-dependent order. More precisely, we describe the time-evolution of the solution to these equations as the time parameter goes either to zero or to infinity.

We consider the inverse problem of determining the coupling coefficients in a two-state Schr\"odinger system. We prove a Lipschitz stability inequality for the zeroth and first order coupling terms by finitely many partial lateral measurements of the solution to the coupled Schr\"odinger equations.

We prove that the stationary magnetic potential vector and the electrostatic potential entering the dynamic magnetic Schr\"odinger 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 Schr\"odinger equation, established in the first part of the paper.

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.

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.

We consider the inverse problem of determining the time and space dependent electromagnetic potential of the Schr\"odinger 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 H\"older stably retrieved from these data, whereas stability estimates for inverse time-dependent coefficients problems of evolution partial differential equations are usually of logarithmic type.

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.

We prove logarithmic stability in the parabolic inverse problem of determining the space-varying factor in the source, by a single partial boundary measurement of the solution to the heat equation in an infinite closed waveguide, with homogeneous initial and Dirichlet data.

We study the inverse problem of determining the magnetic field and the electric potential entering the Schr\"odinger equation in an infinite 3D cylindrical domain, by Dirichlet-to-Neumann map. The cylindrical domain we consider is a closed waveguide in the sense that the cross section is a bounded domain of the plane. We prove that the knowledge of the Dirichlet-to-Neumann map determines uniquely, and even H\"older-stably, the magnetic field induced by the magnetic potential and the electric potential. Moreover, if the maximal strength of both the magnetic field and the electric potential, is attained in a fixed bounded subset of the domain, we extend the above results by taking finitely extended boundary observations of the solution, only.

We consider the Calder\`on problem in an infinite cylindrical domain, whose cross section is a bounded domain of the plane. We prove log-log stability in the determination of the isotropic periodic conductivity coefficient from partial Dirichlet data and partial Neumann boundary observations of the solution.

We study the inverse problem of identifying a periodic potential perturbation of the Dirichlet Laplacian acting in an infinite cylindrical domain, whose cross section is assumed to be bounded. We prove log-log stable determination of the potential with respect to the partial Dirichlet-to-Neumann map, where the Neumann data is taken on slightly more than half of the boundary of the domain.

Given $(M,g)$, a compact connected Riemannian manifold of dimension $d \geq 2$, with boundary $\partial M$, we consider an initial boundary value problem for a fractional diffusion equation on $(0,T) \times M$, $T>0$, with time-fractional Caputo derivative of order $\alpha \in (0,1) \cup (1,2)$. We prove uniqueness in the inverse problem of determining the smooth manifold $(M,g)$ (up to an isometry), and various time-independent smooth coefficients appearing in this equation, from measurements of the solution on a subset of $\partial M$ at fixed time. In the "flat" case where $M$ is a compact subset of $\mathbb R^d$, two out the three coefficients $\rho$ (weight), $a$ (conductivity) and $q$ (potential) appearing in the equation $\rho \partial_t^\alpha u-\textrm{div}(a \nabla u)+ q u=0$ on $(0,T)\times \Omega$ are recovered simultaneously.

In this paper we examine the inverse problem of determining the twisting function for the dynamicSchrodinger operator in a quantum waveguide from some suitable localized, either interior or boundary.

In this paper we investigate the inverse problem of determining the time independent scalar potential of the dynamic Schr\"odinger equation in an infinite cylindrical domain, from partial measurement of the solution on the boundary. Namely, if the potential is known in a neighborhood of the boundary of the spatial domain, we prove that it can be logarithmic stably determined in the whole waveguide from a single observation of the solution on any arbitrary strip-shaped subset of the boundary.

Let $\Omega =\omega\times\mathbb R$ where $\omega\subset \mathbb R^2$ be a bounded domain, and $V : \Omega \to\mathbb R$ a bounded potential which is $2\pi$-periodic in the variable $x_{3}\in \mathbb R$. We study the inverse problem consisting in the determination of $V$, through the boundary spectral data of the operator $u\mapsto Au := -\Delta u + Vu$, acting on $L^2(\omega\times(0,2\pi))$, with quasi-periodic and Dirichlet boundary conditions. More precisely we show that if for two potentials $V_{1}$ and $V_{2}$ we denote by $(\lambda_{1,k})_{k}$ and $(\lambda_{2,k})_{k}$ the eigenvalues associated to the operators $A_{1}$ and $A_{2}$ (that is the operator $A$ with $V := V_{1}$ or $V:=V_{2}$), then if $\lambda_{1,k} - \lambda_{2,k} \to 0$ as $k \to \infty$ we have that $V_{1} \equiv V_{2}$, provided one knows also that $\sum_{k\geq 1}\|\psi_{1,k} - \psi_{2,k}\|_{L^2(\partial\omega\times[0,2\pi])}^2 < \infty$, where $\psi_{m,k} := \partial\phi_{m,k}/\partial{\bf n}$. We establish also an optimal Lipschitz stability estimate. The arguments developed here may be applied to other spectral inverse problems, and similar results can be obtained.

We examine the stability issue in the inverse problem of determining a scalar potential appearing in the stationary Schr{\"o}dinger equation in a bounded domain, from a partial elliptic Dirichlet-to-Neumann map. Namely, the Dirichlet data is imposed on the shadowed face of the boundary of the domain and the Neumann data is measured on its illuminated face. We establish a log log stability estimate for the L2-norm (resp. the H minus 1-norm) of bounded (resp. L2) potentials whose difference is lying in any Sobolev space of order positive order.

We consider the multidimensional inverse problem of determining the conductivity coefficient of a hyperbolic equation in an infinite cylindrical domain, from a single boundary observation of the solution. We prove H{\"o}lder stability with the aid of a Carleman estimate specifically designed for hyperbolic waveguides.