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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.

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.

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.

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.