The eigenfrequencies of resonance cavities shaped as stadium or Sinai billiards are determined by microwave absorption. In the applied frequency range 0-18.74 GHz the used cavities can be considered as two dimensional.For this case quantum-mechanical and electromagnetic boundary conditions are equivalent, and the resonance spectrum of the cavity is, if properly normalized, identical with the quantum-mechanical eigenvalue spectrum. Spectra, containing up to a thousand eigenfrequencies, are obtained within minutes. Statistical properties of the spectra as well as their correlation with classical periodic orbits are discussed. PACS numbers: 05.45.+bThe statistical properties of the eigenvalue spectrum of a Hamiltonian of a quantum system change in a characteristic way if the corresponding classical system shows a transition from integrable to nonintegrable behavior.Whereas in integrable systems for the distribution of energy differences of successive eigenenergies Poisson statistics is observed, the spectra of nonintegrable systems obey Wigner statistics. Experimentally, Wigner statistics were already observed many years ago in the spectra of highly excited nuclei, ' a very recent example is given by the optical fluorescence spectra of N02 molecules. Intense theoretical studies exist for the periodically kicked top, " nonlinearly coupled oscillators, 6 and billiards of different shapes. ' Pechukas and Yukawa' showed that the strength of the nonintegrable part of the Hamiltonian may be interpreted as a pseudotime. As a function of this "time, " the eigenenergies move on the energy axis in a similar way as the particles of an interacting one-dimensional gas.An alternative approach to the understanding of the statistics of eigenvalues is opened by the semiclassical approximation.Gutzwiller'' showed that the density of eigenvalues p(E) can be decomposed into a monotonic and an oscillatory part, p(E) =po(E)+gp"cos[(I/h)S"-y"l (I) (see also Ref. 12 for a review). In the case of twodimensional billiards, the monotonic part po(E) is given b 13 po(E)-A 4x L 1 8n JE where A is the area and L is the circumference of the billiard (the units were chosen such that h /2m =1, i.e. , E k, where k is the de Broglie wave number of the particle). The oscillatory part of p(E) in Eq. (1) is a sum over all classical periodic orbits y. 5" is the classical action for the orbit. The prefactor p"and the phase p" can be calculated within the frame of the semiclassical approximation. If the action S" is proportional to a power of E, then the contributions from the different periodic orbits to p(E) can be obtained by a Fourier transformation of the spectrum as was demonstrated by Wintgen. ' The classical periodic orbits show further up in so-called "scars, " regions of extra high amplitudes of some eigenfunctions in the neighborhood of periodic or-15, 16Calculations of eigenvalues of nonintegrable Hamiltonians are extremely time consuming even on modern computers. Probably for this reason in all publications mentioned above the total number of e...

Applying the method of continuous unitary transformations to a class ofHubbard models, we reexamine the derivation of the t/U expansion for the strong-coupling case. The flow equations for the coupling parameters of the higher order effective interactions can be solved exactly, resulting in a systematic expansion of the Hamiltonian in powers of t/U, valid for any lattice in arbitrary dimension and for general band filling. The expansion ensures a correct treatment of the operator products generated by the transformation, and only involves the explicit recursive calculation of numerical coefficients. This scheme provides a unifying framework to study the strong-coupling expansion for the Hubbard model, which clarifies and circumvents several difficulties inherent to earlier approaches. Our results are compared with those of other methods, and it is shown that the freedom in the choice of the unitary transformation that eliminates interactions between different Hubbard bands can affect the effective Hamiltonian only at order t3/U 2 or higher.

Wave functions of a stadium billiard are determined in a microwave analog experiment. It is shown that Gutzwiller's semiclassical representation of the Green's function in terms of classical trajectories can account not only for eigenvalue spectra but also for eigenfunction patterns.PACS numbers: 05.45.+bThe perhaps most impressive way to demonstrate the qualitative difference between integrable and nonintegrable systems is the presentation of eigenfunction patterns of billiards. Whereas for a rectangular or circular billiard the nodal lines form a regular grid with a large number of crossings, for nonintegrable billiards the node lines perform meandric walks while avoiding any crossing (in the presence of degeneracies the crossings can be destroyed by suitable superpositions of eigenfunctions also in integrable billiards; for details see Chap. of Ref. [l]). This was first demonstrated by McDonald and Kaufman[2] who calculated wave functions in a stadium billiard (a more detailed account of this work can be found in Ref.[3]). A surprise was then the discovery by Heller that many wave functions are not distributed more or less uniformly over the whole billiard but form so-called scars, regions of extra high amplitudes near classical periodic orbits [4,5]. An approach to explain these scars was made by Bogomolny [6]. He used the fact that the Green's function,can be expressed semiclassically as a sum over classically allowed trajectories [7], G(q A ,qB,E)Him) 1/2 XlA, I 1/2 exp -rS r (q A ,q B ,E) -im rn l(2)In Eq. (1) E n and 3(^,<7*,£)/9?!|, M-U2,where the p A are the two components of the momentum at the starting point, written as functions of q A ,q B ,E y and ql are the two components of the end position. Classically, A r is proportional to the density of trajectories at point qs starting isotropically from point q A (a very readable account of these questions can be found in Ref. [8]). Taking q A =q B =<7 in Eq.(2) and integrating over q, one gets the Gutzwiller trace formula establishing a correspondence between periodic orbits and the quantum-mechanical spectrum. This correspondence could be demonstrated experimentally by the present authors for electromagnetic eigenfrequency spectra of billiard-shaped microwave resonators (Ref.[9], hereafter...

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