arXiv subjects
Publications and source records attributed to Jochen Schmid.
We investigate the behavior of infinite-time admissibility under compact perturbations. We show, by means of two completely different examples, that infinite-time admissibility is not preserved under compact perturbations $Q$ of the underlying semigroup generator $A$, even if $A$ and $A+Q$ both generate strongly stable semigroups.
We establish adiabatic theorems with and without spectral gap condition for general -- typically dissipative -- linear operators $A(t): D(A(t)) \subset X \to X$ with time-dependent domains $D(A(t))$ in some Banach space $X$. In these theorems, we do not require the considered spectral values $λ(t)$ of $A(t)$ to be (weakly) semisimple. We then apply our general theorems to the special case of skew-adjoint operators $A(t) = 1/i A_{a(t)}$ defined by symmetric sesquilinear forms $a(t)$ and thus generalize, in a very simple way, the only adiabatic theorem for operators with time-dependent domains known so far.
We establish adiabatic theorems with and without spectral gap condition for general -- typically dissipative -- linear operators $A(t): D(A(t)) \subset X \to X$ with time-independent domains $D(A(t)) = D$ in some Banach space $X$. Compared to the previously known adiabatic theorems -- especially those without spectral gap condition -- we do not require the considered spectral values $λ(t)$ of $A(t)$ to be (weakly) semisimple. We also impose only fairly weak regularity conditions. Applications are given to slowly time-varying open quantum systems and to adiabatic switching processes.
In this paper, we are concerned with the stabilization of linear port-Hamiltonian systems of arbitrary order $N \in \mathbb{N}$ on a bounded $1$-dimensional spatial domain $(a,b)$. In order to achieve stabilization, we couple the system to a dynamic boundary controller, that is, a controller that acts on the system only via the boundary points $a,b$ of the spatial domain. We use a nonlinear controller in order to capture the nonlinear behavior that realistic actuators often exhibit and, moreover, we allow the output of the controller to be corrupted by actuator disturbances before it is fed back into the system. What we show here is that the resulting nonlinear closed-loop system is input-to-state stable w.r.t.~square-integrable disturbance inputs. In particular, we obtain uniform input-to-state stability for systems of order $N=1$ and a special class of nonlinear controllers, and weak input-to-state stability for systems of arbitrary order $N \in \mathbb{N}$ and a more general class of nonlinear controllers. Also, in both cases, we obtain convergence to $0$ of all solutions as $t \to \infty$. Applications are given to vibrating strings and beams.
We establish an exponential stabilization result for linear port-Hamiltonian systems of first order with quite general, not necessarily continuous, energy densities. In fact, we have only to require the energy density of the system to be of bounded variation. In particular, and in contrast to the previously known stabilization results, our result applies to vibrating strings or beams with jumps in their mass density and modulus of elasticity.
We establish characterizations of weak input-to-state stability for abstract dynamical systems with inputs, which are similar to characterizations of uniform and of strong input-to-state stability established in a recent paper by A. Mironchenko and F. Wirth. We also answer, by means of suitable counterexamples, two open questions concerning weak input-to-state stability (and its relation to other common stability concepts) raised in the aforementioned paper.
This paper addresses the problem of wellposedness of non-autonomous linear evolution equations $\dot x = A(t)x$ in uniformly convex Banach spaces. We assume that $A(t):D \subset X\to X$, for each $t$ is the generator of a quasi-contractive $C_0$-group where the domain $D$ and the growth exponent are independent of $t$. Well-posedness holds provided that $t\mapsto A(t)y$ is Lipschitz for all $y\in D$. Hölder continuity of degree $α<1$ is not sufficient and the assumption of uniform convexity cannot be dropped.
We consider the dynamics of the mean-field polaron in the weak-coupling limit of vanishing electron-phonon interaction, $\varepsilon \to 0$. This is a singular limit formally leading to a Schrödinger--Poisson system that is equivalent to the nonlinear Choquard equation. By establishing estimates between the approximation obtained via the Choquard equation and true solutions of the original system we show that the Choquard equation makes correct predictions about the dynamics of the polaron mean-field model for small values of $\varepsilon > 0$.
We prove the well-posedness of non-autonomous linear evolution equations for generators $A(t): D(A(t)) \subset X \to X$ whose pairwise commutators are complex scalars and, in addition, we establish an explicit representation formula for the evolution. We also prove well-posedness in the more general case where instead of the $1$-fold commutators only the $p$-fold commutators of the operators $A(t)$ are complex scalars. All these results are furnished with rather mild stability and regularity assumptions: indeed, stability in $X$ and strong continuity conditions are sufficient. Additionally, we improve a well-posedness result of Kato for group generators $A(t)$ by showing that the original norm continuity condition can be relaxed to strong continuity. Applications include Segal field operators and Schrödinger operators for particles in external electric fields.
This paper is devoted to a comparison of early works of Kato and Yosida on the integration of non-autonomous linear evolution equations $\dot{x} = A(t)x$ in Banach space, where the domain $D$ of $A(t)$ is independent of $t$. Our focus is on the regularity assumed of $t\mapsto A(t)$ and our main objective is to clarify the meaning of the rather involved set of assumptions given in Yosida's classic and highly influential \emph{Functional Analysis}. We prove Yosida's assumptions to be equivalent to Kato's condition that $t\mapsto A(t)x$ is continuously differentiable for each $x\in D$.
We establish adiabatic theorems with and without spectral gap condition for general operators $A(t): D(A(t)) \subset X \to X$ with possibly time-dependent domains in a Banach space $X$. We first prove adiabatic theorems with uniform and non-uniform spectral gap condition (including a slightly extended adiabatic theorem of higher order). In these adiabatic theorems the considered spectral subsets $σ(t)$ have only to be compact -- in particular, they need not consist of eigenvalues. We then prove an adiabatic theorem without spectral gap condition for not necessarily (weakly) semisimple eigenvalues: in essence, it is only required there that the considered spectral subsets $σ(t) = \{ λ(t) \}$ consist of eigenvalues $λ(t) \in \partial σ(A(t))$ and that there exist projections $P(t)$ reducing $A(t)$ such that $A(t)|_{P(t)D(A(t))}-λ(t)$ is nilpotent and $A(t)|_{(1-P(t))D(A(t))}-λ(t)$ is injective with dense range in $(1-P(t))X$ for almost every~$t$. In all these theorems, the regularity conditions imposed on $t \mapsto A(t)$, $σ(t)$, $P(t)$ are fairly mild. We explore the strength of the presented adiabatic theorems in numerous examples. And finally, we apply the adiabatic theorems for time-dependent domains to obtain -- in a very simple way -- adiabatic theorems for operators $A(t)$ defined by symmetric sesquilinear forms.
In this work we generalize some of the previously known adiabatic theorems to situations with non-unitary evolutions in Banach spaces. We prove adiabatic theorems with uniform gap condition (generalizing a theorem of Abou Salem), adiabatic theorems with non-uniform gap condition (generalizing a theorem of Kato) and qualitative as well as quantitative adiabatic theorems without gap condition (generalizing theorems of Avron and Elgart, and Teufel). Additionally, we give a generalized version of an adiabatic theorem of higher order due to Nenciu. In all these adiabatic theorems the considered spectral values need not lie on the imaginary axis and in the adiabatic theorems with spectral gap condition and the adiabatic theorem of higher order compact subsets of the spectrum are sufficient (in particular, these subsets need not consist of eigenvalues). We explore the strength of the presented adiabatic theorems in numerous examples. In particular, we show that the theorems of the present work are more general than the previously known theorems. This work was finished in October 2010 and handed in as a diploma thesis at the mathematics department of the University of Stuttgart. The more recent results of Avron, Fraas, Graf und Grech which appeared in the meantime (in June 2011) are therefore not taken into consideration here. We point out, however, that the theorems of the present work are more general than the corresponding theorems of Avron, Fraas, Graf und Grech - with the exception of the adiabatic theorem of higher order which is in no logical relation to the adiabatic theorem of higher order of Avron, Fraas, Graf and Grech. We are about to gather the most important theorems of the present work in an article and will upload it to arXiv soon.