A semiclassical approach to spectra of quantized torus Hamiltonians
Final Report Abstract
The main results of my project was a novel Gutzwiller trace formula for quantized torus Hamiltonians and a mathematical proof of the Bohr-Sommerfeld quantization condition for torus Hamiltonians. For Weyl-quantized symbols on the torus no comparable results exist. To obtain these results I have derived several intermediate results. The first step was to find a discrete stationary phase argument for finite sums. The second important step was to find a suitable definition of a discrete Fourier integral operator. The crucial idea was here to formulate the operator kernels on a proper subset of the covering space for the torus. The third step was to find an appropriate norm estimate of the error for the semiclassical approximation of the time evolution operator. A further essential step was to find the semiclassical asymptotics of the action of the quantized Hamiltonian on the constructed discrete Fourier integral operator. Using the discrete stationary phase argument the semiclassical time evolution operator yields also a Van Vleck formula at non caustic points of the classical Hamiltonian flow. The final result for the Gutzwiller trace formula for torus Hamiltonians has a surprising modification of well-known results for the standard case of Weyl-quantized Hamiltonians on a cotangent bundle. The action term of the phases in the periodic orbit expansion in the trace formula has to be modified by a term involving the winding number w.r.t. the momentum coordinates. This term vanishes if the corresponding path of the periodic orbit in the covering space is closed and becomes relevant if the path in covering space is not closed. Moreover, I was also able to obtain a modified Bohr-Sommerfeld quantisation condition for the eigenvalues of the Hamiltonian on the basis of the trace formula. Such a formula was assumed in the physical literature before but a mathematical proof was missing. This result was obtained only on the basis of the trace formula and not by means of eigenfunction representation in terms of quasi-modes. The results were used to examine the spectral properties of models systems. Examples are Hamiltonians depending only on position or momentum, the lattice Laplacian or some Harper models.
Publications
- A Gutzwiller trace formula for large Hermitian matrices
J. Bolte, S. Egger and S. Keppeler
(See online at https://doi.org/10.1142/S0129055X17500271) - A Gutzwiller trace formula for large Hermitian matrices, 22.-26.06.2015, workshop Quantum Correlated Matter and Chaos, Max Planck Institute for the Physics of Complex Systems, Dresden
Sebastian Egger
- Heat-kernel and resolvent asymptotics for Schrödinger operators on metric graphs. Appl. Math. Res. Express: 1 : 129 − 165, 2015
J. Bolte, S. Egger and R. Rueckriemen
(See online at https://doi.org/10.1093/amrx/abu009) - Zero modes for quantum graph Laplacians and an index theorem. Ann. H. Poincare, 16 : 1155 − 1189, 2015
J. Bolte, S. Egger and F. Steiner
(See online at https://doi.org/10.1007/s00023-014-0347-z)
