Bogomolny's Transfer Operator Research Articles

Scattering Phase Correction for Semiclassical Quantization Rules in Multi-Dimensional Quantum Systems

While the scattering phase for several one-dimensional potentials can be exactly derived, less is known in multi-dimensional quantum systems. This work provides a method to extend the one-dimensional phase knowledge to multi-dimensional quantization rules. The extension is illustrated in the example of Bogomolny's transfer operator method applied in two quantum wells bounded by step potentials of different heights. This generalized semiclassical method accurately determines the energy spectrum of the systems, which indicates the substantial role of the proposed phase correction. Theoretically, the result can be extended to other semiclassical methods, such as Gutzwiller trace formula, dynamical zeta functions, and semiclassical Landauer–Büttiker formula. In practice, this recipe enhances the applicability of semiclassical methods to multi-dimensional quantum systems bounded by general soft potentials.

Transfer operator approach on three-dimensional quantum billiards with SO(2) symmetry.

This work demonstrates the application of Bogomolny's transfer operator method on three-dimensional dynamics. Motivated by experimental observations of lenslike metal clusters, the quantum billiards bounded by a flat bottom and an upper surface with SO(2) symmetry are studied. A precise determination of the energies with error less than 0.05% and exact predicted degeneracies in the special case of the half-sphere billiard confirm the efficiency of this method. Furthermore, the spectra and degeneracies of lens billiards with varying heights are explicitly determined.

Quantization conditions in Bogomolny's transfer operator method.

Bogomolny's transfer operator method plays a significant role in the study of quantum chaos, along with other well known methods like Gutzwiller's trace formula and the dynamical zeta function, which generalize the Einstein-Brillouin-Keller quantization rule from integrable systems to chaotic systems. According to the theory, the Fredholm determinant of the transfer operator, defined on a Poincaré section of a classical physical system, provides a quantization condition to the energy spectrum of the corresponding quantum system. This study presents two factorization formulas, which relate different quantization conditions defined on different classical trajectory segments. These explicit relations answer the question of why all these classical quantization conditions determine exactly the same energy spectrum of the corresponding quantum systems. As an example, these formulas are illustrated in the equilateral triangular billiard.

Trace formulas and Bogomolny's transfer operator

The trace formulas commonly used in discussing spectral properties of quantum or wave systems are derived simply and directly from the Bogomolny transfer operator. Special cases are the Gutzwiller formula, the Berry–Tabor formula, and the perturbed Berry–Tabor formula. A comment is made about interpolation formulas proposed in the latter case.

Application of Bogomolny's transfer operator to a circular harmonic oscillator plus potential

Bogomolny's transfer operator has been used to find an analytical solution for the semiclassical energy eigenvalues of a simple two-dimensional integrable system. The system studied consists of a particle moving in an isotropic harmonic oscillator potential plus a potential. The classical trajectories are used to construct the transfer matrix, and an expression is derived for the eigenvalues of this matrix as a function of the energy. These eigenvalue curves yield the semiclassical energy eigenvalues for the quantum system, which turn out to be exactly the same as the results obtained by solving the Schrodinger equation. Some insight into this unexpected agreement is provided by considering an exact transfer operator. We show that when this operator is expanded in powers of Planck's constant, the leading term in the expansion is Bogomolny's transfer operator.

Semiclassical approximations to diffractive effects in the annulus billiard

The semiclassical theory formulated by Bogomolny [E. B. Bogomolny, Nonlinearity 5, 805 (1992)] employs a transfer operator constructed from classical trajectories that connect points on a suitably chosen Poincar\'esurface of section. In this paper we study the two-dimensional annulus billiard, and modify Bogomolny's transfer operator to include diffractive paths. The penumbra contributions [H. Primack et al., Phys. Rev. Lett. 76, 1615 (1996)], which correspond to diffractive paths passing close to the inner circle of the annulus, are found to account for most of the difference between the exact transfer operator and Bogomolny's semiclassical transfer operator. When these diffractive effects are included, the semiclassical energy eigenvalues are brought much closer to the exact quantum energy eigenvalues.

Quasiclassical Random Matrix Theory.

We directly combine ideas of the quasiclassical approximation with random matrix theory and apply them to the study of the spectrum, in particular, to the two-level correlator. Bogomolny's transfer operator T, quasiclassically an $N\ifmmode\times\else\texttimes\fi{}N$ unitary matrix, is considered to be a random matrix. Rather than rejecting all knowledge of the system, except for its symmetry (as with Dyson's circular unitary ensemble), we choose an ensemble which incorporates the knowledge of the shortest periodic orbits, the prime quasiclassical information bearing on the spectrum. The results largely agree with expectations but contain novel features differing from other recent theories.

Energy spectra of the hydrogen atom and the harmonic oscillator in two dimensions from Bogomolny's semiclassical transfer operator.

The semiclassical quantization scheme formulated by Bogomolny [E.B. Bogomolny, Nonlinearity 5, 805 (1992); Chaos 2, 5 (1992)], employing a suitably chosen Poincar\'e surface of section, has been used to calculate the energy eigenvalues of the hydrogen atom in one and two dimensions and the anisotropic harmonic oscillator in two dimensions. For the two-dimensional systems it was found to be advantageous to decompose Bogomolny's transfer operator into two "half-mapping" operators. This approach, developed by Haggerty [M.R. Haggerty, Ph.D. thesis, Massachusetts Institute of Technology, 1994 (unpublished); Phys. Rev. E 52, 389 (1995)], leads to an analytical solution for the energy eigenvalues of the hydrogen atom. However, the energies are found to depend on the quantum number $n$ as ${(n\ensuremath{-}\frac{1}{4})}^{\ensuremath{-}2}$, unlike the exact quantum energies, which go as ${(n\ensuremath{-}\frac{1}{2})}^{\ensuremath{-}2}$. An attempt to explain this one-quarter shift on the basis of the Langer-modified WKB approximation is only partly successful. For the two-dimensional harmonic oscillator, numerical calculations yield results close to the exact quantum energies.