Calculation of spectral determinants

doi:10.1098/rspa.1994.0148

Calculation of spectral determinants

Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences ◽

10.1098/rspa.1994.0148 ◽

1994 ◽

Vol 447 (1930) ◽

pp. 413-437 ◽

Cited By ~ 23

Keyword(s):

Periodic Orbit ◽

Finite Number ◽

Dirichlet Series ◽

Energy Levels ◽

Euler Product ◽

Quantum Energy ◽

Type Formula

A method for regularizing spectral determinants is developed which facilitates their computation from a finite number of eigenvalues. This is used to calculate the determinant ∆ for the hyperbola billiard over a range which includes 46 quantum energy levels. The result is compared with semiclassical periodic orbit evaluations of ∆ using the Dirichlet series, Euler product, and a Riemann-Siegel-type formula. It is found that the Riemann-Siegel-type expansion, which uses the least number of orbits, gives the closest approximation. This provides explicit numerical support for recent conjectures concerning the analytic properties of semiclassical formulae, and in particular for the existence of resummation relations connecting long and short pseudo-orbits.

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Periodic orbit resummation and the quantization of chaos

Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences ◽

10.1098/rspa.1992.0007 ◽

1992 ◽

Vol 436 (1896) ◽

pp. 99-108 ◽

Cited By ~ 45

Keyword(s):

Periodic Orbit ◽

Periodic Orbits ◽

Energy Levels ◽

Convergence Properties ◽

Classical Motion ◽

Quantum Energy ◽

Formal Derivation ◽

Spectral Determinant

We study the spectral determinant ∆ ( E ), which has, by construction, zeros at the quantum energy levels of a given system. If the classical motion of the system in question is chaotic then ∆ ( E ) has a semiclassical representation as a sum over combinations of periodic orbits. There are, however, a number of fundamental problems associated with its convergence properties. Imposing upon the sum the condition that, like ∆ ( E ) itself, it is real for real E , we obtain formal resummation equations relating the contributions from asymptotically long orbits to those of the short orbits. These then lead to a formal derivation of the previously conjectured ‘Riemann-Siegel lookalike’ formula, which involves only a finite orbit sum and thus represents, in principle, a semiclassical rule for quantizing chaos.

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QUANTUM EFFECTS OF A DOMAIN WALL IN THE FERROMAGNETIC SYSTEM

Modern Physics Letters B ◽

10.1142/s0217984911025808 ◽

2011 ◽

Vol 25 (06) ◽

pp. 413-418

Author(s):

JI-SUO WANG ◽

KE-ZHU YAN ◽

BAO-LONG LIANG

Keyword(s):

Domain Wall ◽

Unitary Transformation ◽

Energy Levels ◽

Canonical Quantization ◽

Classical Equation ◽

Quantization Method ◽

Damping Term ◽

Quantum Energy ◽

Moving Wall ◽

Ferromagnetic System

Starting from the classical equation of the motion of a domain wall in the ferromagnetic systems, the quantum energy levels of the wall and the corresponding eigenfunctions in the case of considering damping term are given by using the canonical quantization method and unitary transformation. The quantum fluctuations of displacement and momentum of the moving wall has also been given as well as the uncertain relation.

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Energy spectrum and wavefunction of electrons in hybrid superconducting nanowires

International Journal of Modern Physics B ◽

10.1142/s021797921642008x ◽

2016 ◽

Vol 30 (13) ◽

pp. 1642008 ◽

Cited By ~ 2

Author(s):

S. P. Kruchinin

Keyword(s):

Energy Spectrum ◽

Energy Levels ◽

Superconducting Nanowires ◽

Quantum Energy ◽

Close Proximity ◽

Superconducting Nanowire

Recent experiments have fabricated structured arrays. We study hybrid nanowires, in which normal and superconducting regions are in close proximity, by using the Bogoliubov–de Gennes equations for superconductivity in a cylindrical nanowire. We succeed to obtain the quantum energy levels and wavefunctions of a superconducting nanowire. The obtained spectra of electrons remind Hofstadter’s butterfly.

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Geometric Models of the Relativistic Harmonic Oscillator

International Journal of Modern Physics A ◽

10.1142/s0217751x97001833 ◽

1997 ◽

Vol 12 (20) ◽

pp. 3545-3550 ◽

Cited By ~ 11

Author(s):

Ion I. Cotăescu

Keyword(s):

Harmonic Oscillator ◽

Finite Number ◽

Continuous Spectrum ◽

Energy Levels ◽

Nonrelativistic Limit ◽

Energy Spectra ◽

De Sitter ◽

Geometric Models ◽

Relativistic Harmonic Oscillator ◽

Quantum Models

A family of relativistic geometric models is defined as a generalization of the actual anti-de Sitter (1 + 1) model of the relativistic harmonic oscillator. It is shown that all these models lead to the usual harmonic oscillator in the nonrelativistic limit, even though their relativistic behavior is quite different. Among quantum models we find a set of models with countable energy spectra, and another one having only a finite number of energy levels and in addition a continuous spectrum.

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Lyapunov optimization for non-generic one-dimensional expanding Markov maps

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2019.6 ◽

2019 ◽

Vol 40 (9) ◽

pp. 2571-2592 ◽

Cited By ~ 1

Author(s):

MAO SHINODA ◽

HIROKI TAKAHASI

Keyword(s):

Periodic Orbit ◽

Finite Number ◽

Dense Subset ◽

Equilibrium States ◽

Positive Entropy ◽

Hölder Continuous ◽

Perturbation Theorem ◽

One Dimensional ◽

Lyapunov Optimization ◽

Shift Map

For a non-generic, yet dense subset of$C^{1}$expanding Markov maps of the interval we prove the existence of uncountably many Lyapunov optimizing measures which are ergodic, fully supported and have positive entropy. These measures are equilibrium states for some Hölder continuous potentials. We also prove the existence of another non-generic dense subset for which the optimizing measure is unique and supported on a periodic orbit. A key ingredient is a new$C^{1}$perturbation theorem which allows us to interpolate between expanding Markov maps and the shift map on a finite number of symbols.

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Lower bounds for quantum energy levels through statistical mechanics methods

Journal of Mathematical Physics ◽

10.1063/1.524463 ◽

1980 ◽

Vol 21 (4) ◽

pp. 834-839 ◽

Cited By ~ 5

Author(s):

L. Nitti ◽

M. Pellicoro ◽

M. Villani

Keyword(s):

Statistical Mechanics ◽

Lower Bounds ◽

Energy Levels ◽

Quantum Energy

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Quantum states in a nanotube quantum dot

International Journal of Modern Physics B ◽

10.1142/s0217979217450230 ◽

2017 ◽

Vol 31 (25) ◽

pp. 1745023

Author(s):

J. T. Wang ◽

J. D. Fan

Keyword(s):

Carbon Nanotube ◽

High Efficiency ◽

Energy Levels ◽

Quantum States ◽

Semiconductor Surface ◽

Energy Structure ◽

Two Dimensional ◽

Quantum Energy ◽

Quantum Bit ◽

Computer Chips

In this paper, we carry out a theoretical calculation of quantum state and quantum energy structure in carbon nanotube embedded semiconductor surface. In this theoretical model, the electrons in the carbon nanotube are considered as in a two-dimensional cylindrical surface. Their motion, therefore, can be described by the Dirac equation. We solve the equation and find that the energy levels are quantized and are linearly dependent on the wave vectors along the [Formula: see text]-direction that is along the direction of the nanotube. This type of energy structure may have potential application for fabricating high efficiency solar cell or quantum bit in computer chips.

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