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We consider the problem of maximal regularity for the semilinear non-autonomous evolution equations $$\begin{aligned} u'(t)+A(t)u(t)=F(t,u),\, t \text {-a.e.}, \, u(0)=u_0. \end{aligned}$$ u ′ ( t ) + A ( t ) u ( t ) = F ( t , u ) , t -a.e. , u ( 0 ) = u 0 . Here, the time-dependent operators A(t) are associated with (time dependent) sesquilinear forms on a Hilbert space $$\mathcal {H}.$$ H . We prove the maximal regularity result in temporally weighted $$L^2$$ L 2 -spaces and other regularity properties for the solution of the previous problem under minimal regularity assumptions on the forms, the initial value $$u_0$$ u 0 and the inhomogeneous term F. Our results are motivated by boundary value problems.

We consider the inverse problem of determining an electromagnetic potential appearing in an infinite cylindrical domain from boundary measurements. More precisely, we prove the stable recovery of some general class of magnetic field and electric potential from boundary measurements. Assuming some knowledge of the unknown coefficients close to the boundary, we obtain also some results of stable recovery with measurements restricted to some portion of the boundary. Our approach combines construction of complex geometric optics solutions and Carleman estimates suitably designed for our stability results stated in an unbounded domain.

International audience; Abstract We study the well-posedness for initial boundary value problems associated with time fractional diffusion equations with non-homogenous boundary and initial values. We consider both weak and strong solutions for the problems. For weak solutions, we introduce a definition of solutions which allows to prove the existence of solution to the initial boundary value problems with non-zero initial and boundary values and non-homogeneous source terms lying in some negative-order Sobolev spaces. For strong solutions, we introduce an optimal compatibility condition and prove the existence of the solutions. We introduce also some sharp conditions guaranteeing the existence of solutions with more regularity in time and space.

This paper is a continuation of a previous work about the study of the survival probability modelizing the molecular predissociation in the Born-Oppenheimer framework. Here we consider the critical case where the reference energy corresponds to the value of a crossing of two electronic levels, one of these two levels being confining while the second dissociates. We show that the survival probability associated to a certain initial state is a sum of the usual time-dependent exponential contribution, and a reminder term that is jointly polynomially small with respect to the time and the semiclassical parameter. We also compute explicitly the main contribution of the remainder., 37 pages, 1 figure

In this paper, we prove Lipschitz stable determination of the complex-valued electric potential and the direction of the magnetic field appearing in the dynamic Schrodinger equation with static coefficients, by finitely many partial boundary measurements of the solution. This is by means of the Bukhgeim–Klibanov method, based on an appropriate Carleman estimate. Since the time symmetrization of the static magnetic Schrodinger equation around t = 0 is not possible, we preliminarily establish a Carleman inequality specifically designed for this problem.

Given ( M , g ) , a compact connected Riemannian manifold of dimension d ⩾ 2 , with boundary ∂M, we consider an initial boundary value problem for a fractional diffusion equation on ( 0 , T ) × M , T > 0 , with time-fractional Caputo derivative of order α ∈ ( 0 , 1 ) ∪ ( 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 solutions on a subset of ∂M at fixed time. In the “flat” case where M is a compact subset of R d , two out the three coefficients ρ (density), a (conductivity) and q (potential) appearing in the equation ρ ∂ t α u − div ( a ∇ u ) + q u = 0 on ( 0 , T ) × M are recovered simultaneously.

The dynamic reflection probability of Davies and Simon (Commun Math Phys 63(3):277–301, 1978) and the spectral reflection probability of Gesztesy et al. (Diff Integral Eqs 10(3):521–546, 1997) and Gesztesy and Simon (Helv Phys Acta 70:66–71, 1997) for a one-dimensional Schrodinger operator \(H = - \Delta + V\) are characterized in terms of the scattering theory of the pair \((H, H_\infty )\) where \(H_\infty \) is the operator obtained by decoupling the left and right half-lines \(\mathbb {R}_{\le 0}\) and \(\mathbb {R}_{\ge 0}\). An immediate consequence is that these reflection probabilities are in fact the same, thus providing a short and transparent proof of the main result of Breuer et al. (Commun Math Phys 295(2):531–550, 2010). This approach is inspired by recent developments in non-equilibrium statistical mechanics of the electronic black-box model and follows a strategy parallel to Jaksic (Commun Math Phys 332:827–838, 2014).

We present some results on the monotonicity of some traces involving functions of self-adjoint operators with respect to the natural ordering of their associated quadratic forms. The relation between these results and Löwner’s Theorem is discussed. We also apply these results to complete a proof of the Wegner estimate for continuum models of random Schrödinger operators as given in a 1994 paper by Combes and Hislop.

We consider two-dimensional massless Dirac operators in a radially symmetric electromagnetic field. In this case the fields may be described by one-dimensional electric and magnetic potentials $V$ and $A$. We show dynamical localization in the regime when $\displaystyle\lim_{r\to\infty}|V|/|A

Let Ω : = ω × R where ω ⊂ R 2 is a bounded domain, and let V : Ω → R be a bounded potential which is 2π-periodic in the variable x 3 ∈ R . We study the inverse problem consisting in the determination of V, through the boundary spectral data of the operator u ↦ A u : = − Δ u + V u , acting on L 2 ( ω × ( 0 , 2 π ) ) , with quasi-periodic and Dirichlet boundary conditions. More precisely we show that if for j = 1 , 2 two potentials V j are given so that ‖ V j ‖ ∞ ≤ R , and if we denote by ( λ j , k ) k the eigenvalues of the operators A j (that is the operator A with V : = V j ), then for a constant c > 0 , depending on ω and R > 0 , we have ‖ F ( ( V 1 − V 2 ) 1 ω × ( 0 , 2 π ) ) ‖ ∞ ≤ c lim sup k → ∞ | λ 1 , k − λ 2 , k | , provided that ∑ k ≥ 1 ‖ ψ 1 , k − ψ 2 , k ‖ L 2 ( ∂ ω × [ 0 , 2 π ] ) 2 ∞ , where ψ j , k : = ∂ ϕ j , k / ∂ n (here F denotes the Fourier transform). The arguments developed here may be applied to other spectral inverse problems, and similar results can be obtained.

International audience; Let $\Omega$ be a $C^\infty$-smooth bounded domain of $\mathbb{R}^n$, $n \geq 1$, and let the matrix ${\bf a} \in C^\infty (\overline{\Omega};\R^{n^2})$ be symmetric and uniformly elliptic. We consider the $L^2(\Omega)$-realization $A$ of the operator $-\mydiv ( {\bf a} \nabla \cdot)$ with Dirichlet boundary conditions. We perturb $A$ by some real valued potential $V \in C_0^\infty (\Omega)$ and note $A_V=A+V$. We compute the asymptotic expansion of $\mbox{tr}\left( e^{-t A_V}-e^{-t A}\right)$ as $t \downarrow 0$ for any matrix ${\bf a}$ whose coefficients are homogeneous of degree $0$. In the particular case where $A$ is the Dirichlet Laplacian in $\Omega$, that is when ${\bf a}$ is the identity of $\R^{n^2}$, we make the four main terms appearing in the asymptotic expansion formula explicit and prove that $L^\infty$-bounded sets of isospectral potentials of $A$ are $H^s$-compact for $s

In this paper, we determine the exact expression for the hydrogen binding energy in the Pauli-Fierz model up to the order $O(\alpha^5\log\alpha^{-1})$, where $\alpha$ denotes the finestructure constant, and prove rigorous bounds on the remainder term of the order $o(\alpha^5\log\alpha^{-1})$. As a consequence, we prove that the binding energy is not a real analytic function of $\alpha$, and verify the existence of logarithmic corrections to the expansion of the ground state energy in powers of $\alpha$, as conjectured in the recent literature.

The inequalities between the convergents of the Stieltjes continued fractions (CF) allowing to prove their convergence to the Stieltjes functions were first introduced in the famous work of Stieltjes (1894) [26]. Of course, the name of Stieltjes was attributed to these objects more later, such as the identification of the convergents of CF with Pade approximants (PA). The present review relates the evolution of PA errors inequalities for the Stieltjes functions up to now. Alphonse Magnus and the author (Gilewicz and Magnus (1979) [13]) remarked that the original Stieltjes inequalities are not optimal, not order equilibrated. From this time up to now we have established the optimal inequalities for the errors of PA for the Stieltjes functions in the cases of the classical one-point PA, of two-point (0 and ~) PA and of N-point (N>2) PA. The last case is presented for the first time in the present review.

International audience; Some examples of application of Padé approximant techniques to approximate the integrals in question and comparison with previous results, essentially the recent Panayi-Schock and Green-wood results, show the efficiency of this simple natural approximation.