Roman C V Schubert

Husimi functions at dielectric interfaces: Inside-outside duality for optical systems and beyond

We introduce generalized Husimi functions at the interfaces of dielectric systems. Four different functions can be defined, corresponding to the incident and departing wave on both sides of the interface. These functions allow to identify mechanisms of wave confinement and escape directions in optical microresonators, and give insight into the structure of resonance wave functions. Off resonance, where systematic interference can be neglected, the Husimi functions are related by Snells law and Fresnels coefficients.

Maximum norms of chaotic quantum eigenstates and random waves

Abstract The growth of the maximum norms of quantum eigenstates of classically chaotic systems with increasing energy is investigated. The maximum norms provide a measure for localization effects in eigenfunctions. An upper bound for the maxima of random superpositions of random functions is derived. For the random-wave model this gives the bound c ln E in the semiclassical limit E→∞. The growth of the maximum norms of random waves is compared with the growth of the maximum norms of the eigenstates of six quantum billiards which are classically chaotic. The maximum norms of these systems are numerically shown to be conform with the random-wave model. Furthermore, the distribution of the locations of the maximum norms is discussed.

ON THE NUMBER OF BOUNCING BALL MODES IN BILLIARDS

We study the number of bouncing ball modes in a class of two-dimensional quantized billiards with two parallel walls. Using an adiabatic approximation we show that asymptotically for , where depends on the shape of the billiard boundary. In particular for the class of two-dimensional Sinai billiards, which are chaotic, one can get arbitrarily close (from below) to , which corresponds to the leading term in Weyls law for the mean behaviour of the counting function of eigenstates. This result shows that one can come arbitrarily close to violating quantum ergodicity. We compare the theoretical results with the numerically determined counting function for the stadium billiard and the cosine billiard and find good agreement.

Upper Bounds on the Rate of Quantum Ergodicity

Abstract.We study the semiclassical behaviour of eigenfunctions of quantum systems with ergodic classical limit. By the quantum ergodicity theorem almost all of these eigenfunctions become equidistributed in a weak sense. We give a simple derivation of an upper bound of order

Autocorrelation function of eigenstates in chaotic and mixed systems

|{\rm ln{\hbar}}| ^{-1}

Amplitude distribution of eigenfunctions in mixed systems

on the rate of quantum ergodicity if the classical system is ergodic with a certain rate. In addition we obtain a similar bound on transition amplitudes if the classical system is weak mixing. Both results generalise previous ones by Zelditch.

How do wave packets spread? Time evolution on Ehrenfest time scales

The quantum evolution of the Wigner function for Gaussian wave packets generated by a non-Hermitian Hamiltonian is investigated. In the semiclassical limit ({h_bar}/2{pi}){yields}0 this yields the non-Hermitian analog of the Ehrenfest theorem for the dynamics of observable expectation values. The lack of Hermiticity reveals the importance of the complex structure on the classical phase space: The resulting equations of motion are coupled to an equation of motion for the phase-space metric - a phenomenon having no analog in Hermitian theories.

Classical and quantum dynamics in the (non-Hermitian) Swanson oscillator

We study the autocorrelation function of different types of eigenfunctions in quantum mechanical systems with either chaotic or mixed classical limits. We obtain an expansion of the autocorrelation function in terms of the correlation distance. For localized states in billiards, like bouncing ball modes or states living on tori, a simple model using only classical input gives good agreement with the exact result. In particular, a prediction for irregular eigenfunctions in mixed systems is derived and tested. For chaotic systems, the expansion of the autocorrelation function can be used to test quantum ergodicity on different length scales.