Physical systems are usually not closed and insight about their internal structure is experimentally derived by scattering. This is efficiently described by resonance eigenfunctions of non-Hermitian quantum systems with a corresponding classical dynamics that allows for the escape of particles. For the phase-space distribution of resonance eigenfunctions in chaotic systems with partial and full escape we obtain a universal description of their semiclassical limit in terms of classical conditional invariant measures with the same decay rate. For partial escape, we introduce a family of conditionally invariant measures with arbitrary decay rates based on the hyperbolic dynamics and the natural measures of forward and backward dynamics. These measures explain the multifractal phase-space structure of resonance eigenfunctions and their dependence on the decay rate. Additionally, for the nontrivial limit of full escape we motivate the hypothesis that resonance eigenfunctions are described by conditionally invariant measures that are uniformly distributed on sets with the same temporal distance to the quantum resolved chaotic saddle. Overall we confirm quantum-to-classical correspondence for the phase-space densities, for their fractal dimensions, and by evaluating their Jensen–Shannon distance in a generic chaotic map with partial and full escape, respectively. / Typische physikalische Systeme sind nicht geschlossen, sodass ihre innere Struktur mit Hilfe von Streuexperimenten untersucht werden kann. Diese werden mit Hilfe einer nicht-Hermiteschen Quantendynamik und deren Resonanzeigenzuständen beschrieben. Die dabei zugrunde liegende klassische Dynamik berücksichtigt den Verlust von Teilchen. Für die semiklassische Phasenraumverteilung solcher Resonanzeigenzustände in chaotischen Systemen mit partieller und voller Öffnung entwickeln wir eine universelle Beschreibung mittels bedingt invarianter Maße gleicher Zerfallsrate. Für partiellen Zerfall stellen wir eine Familie bedingt invarianter Maße mit beliebiger Zerfallsrate vor, welche auf der hyperbolischen Dynamik und den natürlichen Maßen der vorwärts gerichteten und der invertierten Dynamik aufbauen. Diese Maße erklären die multifraktale Phasenraumstruktur der Resonanzzustände und deren Abhängigkeit von der Zerfallsrate. Darüber hinaus motivieren wir für den nicht trivialen Grenzfall voll geöffneter Systeme die Hypothese, dass Resonanzeigenzustände durch ein bedingt invariantes Maß beschrieben werden, welches gleichverteilt auf solchen Mengen ist, die den gleichen zeitlichen Abstand zum quantenunscharfen chaotischen Sattel haben. Insgesamt bestätigen wir die quantenklassische Korrespondenz für die Phasenraumdichten, deren fraktale Dimensionen und durch Auswertung ihres Jensen–Shannon Abstandes in einer generischen chaotischen Abbildung sowohl für partielle als auch für volle Öffnung.

Diese Arbeit befasst sich mit der Entwicklung einer Theorie bedingter Mengen. Bedingte Mengenlehre ist reich genug um einen bedingten mathematischen Diskurs zu führen, dessen Möglichkeit wir durch die Konstruktion einer bedingten Topologielehre und bedingter reeller Analysis aufzeigen. Wir beweisen die bedingte Version folgender Sätze: Ultrafilterlemma, Tychonoff, Borel-Lebesgue, Heine-Borel, Bolzano-Weierstraß, und das Gaplemma von Debreu. Darüberhinaus beweisen wir die bedingte Version derjenigen Resultate der klassischen Mathematik, die in den Beweisen dieser Sätze benötigt werden, beginnend mit der Mengenlehre. Wir diskutieren die Verbindung von bedingter Mengenlehre zur Garben-, Topos- und L0-Theorie. / In this thesis, we develop a theory of conditional sets. Conditional set theory is sufficiently rich in order to allow for a conditional mathematical reasoning, the possibility of which we demonstrate by constructing a conditional general topology and a conditional real analysis. We prove the conditional version of the following theorems: Ultrafilter Lemma, Tychonoff, Borel-Lebesgue, Heine-Borel, Bolzano-Weierstraß, and Debreu’s Gap Lemma. Moreover, we prove the conditional version of those results in classical mathematics which are needed in the proofs of these theorems, starting from set theory. We discuss the connection of conditional set theory to sheaf, topos and L0-theory.

Classical partial transport barriers govern both classical and quantum dynamics of generic Hamiltonian systems. Chaotic eigenstates of quantum systems are known to localize on either side of a partial barrier if the flux connecting the two sides is not resolved by means of Heisenberg's uncertainty. Surprisingly, in open systems, in which orbits can escape, chaotic resonance states exhibit such a localization even if the flux across the partial barrier is quantum mechanically resolved. We explain this using the concept of conditionally invariant measures by introducing a new quantum mechanically relevant class of such fractal measures. We numerically find quantum-to-classical correspondence for localization transitions depending on the openness of the system and on the decay rate of resonance states. Moreover, we show that the number of long-lived chaotic resonance states that localize on one particular side of the partial barrier is described by an individual fractal Weyl law. For a generic phase space, this implies a hierarchy of fractal Weyl laws, one for each region of the hierarchical decomposition of phase space.

Classical partial transport barriers govern both classical and quantum dynamics of generic Hamiltonian systems. Chaotic eigenstates of quantum systems are known to localize on either side of a partial barrier if the flux connecting the two sides is not resolved by means of Heisenberg's uncertainty. Surprisingly, in open systems, in which orbits can escape, chaotic resonance states exhibit such a localization even if the flux across the partial barrier is quantum mechanically resolved. We explain this using the concept of conditionally invariant measures by introducing a new quantum mechanically relevant class of such fractal measures. We numerically find quantum-to-classical correspondence for localization transitions depending on the openness of the system and on the decay rate of resonance states. Moreover, we show that the number of long-lived chaotic resonance states that localize on one particular side of the partial barrier is described by an individual fractal Weyl law. For a generic phase space, this implies a hierarchy of fractal Weyl laws, one for each region of the hierarchical decomposition of phase space.