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Ketzmerick R, Clauß K, Fritzsch F, Bäcker A. Chaotic Resonance Modes in Dielectric Cavities: Product of Conditionally Invariant Measure and Universal Fluctuations. PHYSICAL REVIEW LETTERS 2022; 129:193901. [PMID: 36399737 DOI: 10.1103/physrevlett.129.193901] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/16/2022] [Revised: 07/19/2022] [Accepted: 09/28/2022] [Indexed: 06/16/2023]
Abstract
We conjecture that chaotic resonance modes in scattering systems are a product of a conditionally invariant measure from classical dynamics and universal exponentially distributed fluctuations. The multifractal structure of the first factor depends strongly on the lifetime of the mode and describes the average of modes with similar lifetime. The conjecture is supported for a dielectric cavity with chaotic ray dynamics at small wavelengths, in particular for experimentally relevant modes with longest lifetime. We explain scarring of the vast majority of modes along segments of rays based on multifractality and universal fluctuations, which is conceptually different from periodic-orbit scarring.
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Affiliation(s)
- Roland Ketzmerick
- Technische Universität Dresden, Institut für Theoretische Physik and Center for Dynamics, 01062 Dresden, Germany
| | - Konstantin Clauß
- Technische Universität Dresden, Institut für Theoretische Physik and Center for Dynamics, 01062 Dresden, Germany
- Department of Mathematics, Technical University of Munich, Boltzmannstrasse 3, 85748 Garching, Germany
| | - Felix Fritzsch
- Technische Universität Dresden, Institut für Theoretische Physik and Center for Dynamics, 01062 Dresden, Germany
- Physics Department, Faculty of Mathematics and Physics, University of Ljubljana, 1000 Ljubljana, Slovenia
| | - Arnd Bäcker
- Technische Universität Dresden, Institut für Theoretische Physik and Center for Dynamics, 01062 Dresden, Germany
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2
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Clauß K, Kunzmann F, Bäcker A, Ketzmerick R. Universal intensity statistics of multifractal resonance states. Phys Rev E 2021; 103:042204. [PMID: 34005999 DOI: 10.1103/physreve.103.042204] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/07/2020] [Accepted: 03/12/2021] [Indexed: 11/06/2022]
Abstract
We conjecture that in chaotic quantum systems with escape, the intensity statistics for resonance states universally follows an exponential distribution. This requires a scaling by the multifractal mean intensity, which depends on the system and the decay rate of the resonance state. We numerically support the conjecture by studying the phase-space Husimi function and the position representation of resonance states of the chaotic standard map, the baker map, and a random matrix model, each with partial escape.
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Affiliation(s)
- Konstantin Clauß
- Technische Universität Dresden, Institut für Theoretische Physik and Center for Dynamics, 01062 Dresden, Germany
| | - Felix Kunzmann
- Technische Universität Dresden, Institut für Theoretische Physik and Center for Dynamics, 01062 Dresden, Germany
| | - Arnd Bäcker
- Technische Universität Dresden, Institut für Theoretische Physik and Center for Dynamics, 01062 Dresden, Germany.,Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Straße 38, 01187 Dresden, Germany
| | - Roland Ketzmerick
- Technische Universität Dresden, Institut für Theoretische Physik and Center for Dynamics, 01062 Dresden, Germany.,Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Straße 38, 01187 Dresden, Germany
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3
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Carlo GG, Borondo F. Lagrangian descriptors for open maps. Phys Rev E 2020; 101:022208. [PMID: 32168688 DOI: 10.1103/physreve.101.022208] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/23/2019] [Accepted: 01/27/2020] [Indexed: 11/07/2022]
Abstract
We adapt the concept of Lagrangian descriptors, which have been recently introduced as efficient indicators of phase space structures in chaotic systems, to unveil the key features of open maps. We apply them to the open tribaker map, a paradigmatic example not only in classical but also in quantum chaos. Our definition allows us to identify in a very simple way the inner structure of the chaotic repeller, which is the fundamental invariant set that governs the dynamics of this system. The homoclinic tangles of periodic orbits (POs) that belong to this set are clearly found. This could also have important consequences for chaotic scattering and in the development of the semiclassical theory of short POs for open systems.
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Affiliation(s)
- Gabriel G Carlo
- Comisión Nacional de Energía Atómica, CONICET, Departamento de Física, Av. del Libertador 8250, 1429 Buenos Aires, Argentina
| | - F Borondo
- Departamento de Química, Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid, Spain.,Instituto de Ciencias Matemáticas (ICMAT), Cantoblanco, 28049 Madrid, Spain
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4
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Bilen AM, García-Mata I, Georgeot B, Giraud O. Multifractality of open quantum systems. Phys Rev E 2019; 100:032223. [PMID: 31639907 DOI: 10.1103/physreve.100.032223] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/12/2019] [Indexed: 11/07/2022]
Abstract
We study the eigenstates of open maps whose classical dynamics is pseudointegrable and for which the corresponding closed quantum system has multifractal properties. Adapting the existing general framework developed for open chaotic quantum maps, we specify the relationship between the eigenstates and the classical structures, and we quantify their multifractality at different scales. Based on this study, we conjecture that quantum states in such systems are distributed according to a hierarchy of classical structures, but these states are multifractal instead of ergodic at each level of the hierarchy. This is visible for sufficiently long-lived resonance states at scales smaller than the classical structures. Our results can guide experimentalists in order to observe multifractal behavior in open systems.
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Affiliation(s)
- Agustín M Bilen
- Instituto de Investigaciones Físicas de Mar del Plata (IFIMAR), Facultad de Ciencias Exactas y Naturales, Universidad Nacional de Mar del Plata, CONICET, 7600 Mar del Plata, Argentina
| | - Ignacio García-Mata
- Instituto de Investigaciones Físicas de Mar del Plata (IFIMAR), Facultad de Ciencias Exactas y Naturales, Universidad Nacional de Mar del Plata, CONICET, 7600 Mar del Plata, Argentina
| | - Bertrand Georgeot
- Laboratoire de Physique Théorique, IRSAMC, Université de Toulouse, CNRS, UPS, France
| | - Olivier Giraud
- LPTMS, CNRS, Univ. Paris-Sud, Université Paris-Saclay - 91405 Orsay, France
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5
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Prado CA, Carlo GG, Benito RM, Borondo F. Role of short periodic orbits in quantum maps with continuous openings. Phys Rev E 2018; 97:042211. [PMID: 29758733 DOI: 10.1103/physreve.97.042211] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/08/2017] [Indexed: 11/07/2022]
Abstract
We apply a recently developed semiclassical theory of short periodic orbits to the continuously open quantum tribaker map. In this paradigmatic system the trajectories are partially bounced back according to continuous reflectivity functions. This is relevant in many situations that include optical microresonators and more complicated boundary conditions. In a perturbative regime, the shortest periodic orbits belonging to the classical repeller of the open map-a cantor set given by a region of exactly zero reflectivity-prove to be extremely robust in supporting a set of long-lived resonances of the continuously open quantum maps. Moreover, for steplike functions a significant reduction in the number needed is obtained, similarly to the completely open situation. This happens despite a strong change in the spectral properties when compared to the discontinuous reflectivity case. In order to give a more realistic interpretation of these results we compare with a Fresnel-type reflectivity function.
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Affiliation(s)
- Carlos A Prado
- Comisión Nacional de Energía Atómica, Departamento de Física, Av. del Libertador 8250, 1429 Buenos Aires, Argentina.,Departamento de Física, FCEyN, Universidad de Buenos Aires, Ciudad Universitaria, 1428 Buenos Aires, Argentina
| | - Gabriel G Carlo
- Comisión Nacional de Energía Atómica, CONICET, Departamento de Física, Av. del Libertador 8250, 1429 Buenos Aires, Argentina
| | - R M Benito
- Grupo de Sistemas Complejos and Departamento de Física, Escuela Técnica Superior de Ingenieros Agrónomos, Universidad Politécnica de Madrid, 28040 Madrid, Spain
| | - F Borondo
- Departamento de Química, and Instituto de Ciencias Matemáticas (ICMAT), Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid, Spain
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6
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Mertig N, Shudo A. Open quantum maps from complex scaling of kicked scattering systems. Phys Rev E 2018; 97:042216. [PMID: 29758738 DOI: 10.1103/physreve.97.042216] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/05/2017] [Indexed: 11/07/2022]
Abstract
We derive open quantum maps from periodically kicked scattering systems and discuss the computation of their resonance spectra in terms of theoretically grounded methods, such as complex scaling and sufficiently weak absorbing potentials. In contrast, we also show that current implementations of open quantum maps, based on strong absorptive or even projective openings, fail to produce the resonance spectra of kicked scattering systems. This comparison pinpoints flaws in current implementations of open quantum maps, namely, the inability to separate resonance eigenvalues from the continuum as well as the presence of diffraction effects due to strong absorption. The reported deviations from the true resonance spectra appear, even if the openings do not affect the classical trapped set, and become appreciable for shorter-lived resonances, e.g., those associated with chaotic orbits. This makes the open quantum maps, which we derive in this paper, a valuable alternative for future explorations of quantum-chaotic scattering systems, for example, in the context of the fractal Weyl law. The results are illustrated for a quantum map model whose classical dynamics exhibits key features of ionization and a trapped set which is organized by a topological horseshoe.
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Affiliation(s)
- Normann Mertig
- Department of Physics, Tokyo Metropolitan University, Minami-Osawa, Hachioji 192-0397, Japan
| | - Akira Shudo
- Department of Physics, Tokyo Metropolitan University, Minami-Osawa, Hachioji 192-0397, Japan
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7
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Revuelta F, Vergini E, Benito RM, Borondo F. Semiclassical basis sets for the computation of molecular vibrational states. J Chem Phys 2017; 146:014107. [PMID: 28063420 DOI: 10.1063/1.4973376] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/31/2022] Open
Abstract
In this paper, we extend a method recently reported [F. Revuelta et al., Phys. Rev. E 87, 042921 (2013)] for the calculation of the eigenstates of classically highly chaotic systems to cases of mixed dynamics, i.e., those presenting regular and irregular motions at the same energy. The efficiency of the method, which is based on the use of a semiclassical basis set of localized wave functions, is demonstrated by applying it to the determination of the vibrational states of a realistic molecular system, namely, the LiCN molecule.
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Affiliation(s)
- F Revuelta
- Grupo de Sistemas Complejos, Escuela Técnica Superior de Ingeniería Agronómica, Alimentaria y de Biosistemas, Universidad Politécnica de Madrid, 28040 Madrid, Spain
| | - E Vergini
- Departamento de Física, Comisión Nacional de Energía Atómica, Av. del Libertador 8250, 1429 Buenos Aires, Argentina
| | - R M Benito
- Grupo de Sistemas Complejos, Escuela Técnica Superior de Ingeniería Agronómica, Alimentaria y de Biosistemas, Universidad Politécnica de Madrid, 28040 Madrid, Spain
| | - F Borondo
- Instituto de Ciencias Matemáticas (ICMAT), 28049 Cantoblanco, Madrid, Spain
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8
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Carlo GG, Benito RM, Borondo F. Theory of short periodic orbits for partially open quantum maps. Phys Rev E 2016; 94:012222. [PMID: 27575138 DOI: 10.1103/physreve.94.012222] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/06/2016] [Indexed: 11/07/2022]
Abstract
We extend the semiclassical theory of short periodic orbits [M. Novaes et al., Phys. Rev. E 80, 035202(R) (2009)PLEEE81539-375510.1103/PhysRevE.80.035202] to partially open quantum maps, which correspond to classical maps where the trajectories are partially bounced back due to a finite reflectivity R. These maps are representative of a class that has many experimental applications. The open scar functions are conveniently redefined, providing a suitable tool for the investigation of this kind of system. Our theory is applied to the paradigmatic partially open tribaker map. We find that the set of periodic orbits that belongs to the classical repeller of the open map (R=0) is able to support the set of long-lived resonances of the partially open quantum map in a perturbative regime. By including the most relevant trajectories outside of this set, the validity of the approximation is extended to a broad range of R values. Finally, we identify the details of the transition from qualitatively open to qualitatively closed behavior, providing an explanation in terms of short periodic orbits.
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Affiliation(s)
- Gabriel G Carlo
- Departamento de Física, Comisión Nacional de Energía Atómica, CONICET, Avenida del Libertador 8250, 1429 Buenos Aires, Argentina
| | - R M Benito
- Grupo de Sistemas Complejos and Departamento de Física, Escuela Técnica Superior de Ingenieros Agrónomos, Universidad Politécnica de Madrid, 28040 Madrid, Spain
| | - F Borondo
- Departamento de Química and Instituto de Ciencias Matemáticas, Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid, Spain
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9
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Revuelta F, Vergini E, Benito RM, Borondo F. Scar Functions, Barriers for Chemical Reactivity, and Vibrational Basis Sets. J Phys Chem A 2016; 120:4928-38. [DOI: 10.1021/acs.jpca.5b12541] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Affiliation(s)
- F. Revuelta
- Grupo
de Sistemas Complejos, Escuela Técnica Superior de Ingeniería
Agronómica, Agroambiental y de Biosistemas, Universidad Politécnica de Madrid, 28040 Madrid, Spain
- Instituto
de Ciencias Matemáticas (ICMAT), Cantoblanco, 28049 Madrid, Spain
| | - E. Vergini
- Departamento
de Física, Comisión Nacional de la Energía Atómica, Avenida del Libertador 8250, 1429 Buenos Aires, Argentina
| | - R. M. Benito
- Grupo
de Sistemas Complejos, Escuela Técnica Superior de Ingeniería
Agronómica, Agroambiental y de Biosistemas, Universidad Politécnica de Madrid, 28040 Madrid, Spain
| | - F. Borondo
- Instituto
de Ciencias Matemáticas (ICMAT), Cantoblanco, 28049 Madrid, Spain
- Departamento
de Química, Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid, Spain
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10
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Revuelta F, Benito RM, Borondo F, Vergini E. Using basis sets of scar functions. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 87:042921. [PMID: 23679503 DOI: 10.1103/physreve.87.042921] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/26/2012] [Revised: 03/12/2013] [Indexed: 06/02/2023]
Abstract
We present a method to efficiently compute the eigenfunctions of classically chaotic systems. The key point is the definition of a modified Gram-Schmidt procedure which selects the most suitable elements from a basis set of scar functions localized along the shortest periodic orbits of the system. In this way, one benefits from the semiclassical dynamical properties of such functions. The performance of the method is assessed by presenting an application to a quartic two-dimensional oscillator whose classical dynamics are highly chaotic. We have been able to compute the eigenfunctions of the system using a small basis set. An estimate of the basis size is obtained from the mean participation ratio. A thorough analysis of the results using different indicators, such as eigenstate reconstruction in the local representation, scar intensities, participation ratios, and error bounds, is also presented.
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Affiliation(s)
- F Revuelta
- Grupo de Sistemas Complejos and Departamento de Física, Escuela Técnica Superior de Ingenieros Agrónomos, Universidad Politécnica de Madrid, 28040 Madrid, Spain.
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11
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Carlo GG, Wisniacki DA, Ermann L, Benito RM, Borondo F. Classical transients and the support of open quantum maps. Phys Rev E 2013; 87:012909. [PMID: 23410406 DOI: 10.1103/physreve.87.012909] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/24/2012] [Revised: 10/09/2012] [Indexed: 11/07/2022]
Abstract
The basic ingredients in a semiclassical theory are the classical invariant objects serving as a support for quantization. Recent studies, mainly obtained on quantum maps, have led to the commonly accepted belief that the classical repeller-the set of nonescaping orbits in the future and past evolution-is the object that suitably plays this role in open scattering systems. In this paper we present numerical evidence warning that this may not always be the case. For this purpose we study recently introduced families of tribaker maps [L. Ermann, G. G. Carlo, J. M. Pedrosa, and M. Saraceno, Phys. Rev. E 85, 066204 (2012)], which share the same asymptotic properties but differ in their short-time behavior. We have found that although the eigenvalue distribution of the evolution operator of these maps follows the fractal Weyl law prediction, the theory of short periodic orbits for open maps fails to describe the resonance eigenfunctions of some of them. This is a strong indication that new elements must be included in the semiclassical description of open quantum systems. We provide an interpretation of the results in order to have hints about them.
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Affiliation(s)
- Gabriel G Carlo
- Departamento de Física, Comisión Nacional de Energía Atómica, Avenida del Libertador 8250, 1429 Buenos Aires, Argentina.
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Ermann L, Carlo GG, Pedrosa JM, Saraceno M. Transient features of quantum open maps. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 85:066204. [PMID: 23005193 DOI: 10.1103/physreve.85.066204] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/14/2012] [Indexed: 06/01/2023]
Abstract
We study families of open chaotic maps that classically share the same asymptotic properties--forward and backward trapped sets, repeller dimensions, and escape rate--but differ in their short time behavior. When these maps are quantized we find that the fine details of the distribution of resonances and the corresponding eigenfunctions are sensitive to the initial shape and size of the openings. We study phase space localization of the resonances with respect to the repeller and find strong delocalization effects when the area of the openings is smaller than ℏ.
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Affiliation(s)
- Leonardo Ermann
- Departamento de Física Teórica, GIyA, Comisión Nacional de Energía Atómica, Buenos Aires, Argentina.
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13
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Ishii A, Akaishi A, Shudo A, Schomerus H. Weyl law for open systems with sharply divided mixed phase space. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 85:046203. [PMID: 22680552 DOI: 10.1103/physreve.85.046203] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/03/2012] [Indexed: 06/01/2023]
Abstract
A generalization of the Weyl law to systems with a sharply divided mixed phase space is proposed. The ansatz is composed of the usual Weyl term which counts the number of states in regular islands and a term associated with sticky regions in phase space. For a piecewise linear map, we numerically check the validity of our hypothesis, and find good agreement not only for the case with a sharply divided phase space but also for the case where tiny island chains surround the main regular island. For the latter case, a nontrivial power law exponent appears in the survival probability of classical escaping orbits, which may provide a clue to develop the Weyl law for more generic mixed systems.
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Affiliation(s)
- Akihiro Ishii
- Department of Physics, Tokyo Metropolitan University, Minami-Osawa, Hachioji, Tokyo, Japan
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14
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Pedrosa JM, Wisniacki D, Carlo GG, Novaes M. Short periodic orbit approach to resonances and the fractal Weyl law. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 85:036203. [PMID: 22587157 DOI: 10.1103/physreve.85.036203] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/25/2011] [Indexed: 05/31/2023]
Abstract
We investigate the properties of the semiclassical short periodic orbit approach for the study of open quantum maps that was recently introduced [Novaes, Pedrosa, Wisniacki, Carlo, and Keating, Phys. Rev. E 80, 035202(R) (2009)]. We provide solid numerical evidence, for the paradigmatic systems of the open baker and cat maps, that by using this approach the dimensionality of the eigenvalue problem is reduced according to the fractal Weyl law. The method also reproduces the projectors |ψ(n)(R)><ψ(n)(L)|, which involves the right and left states associated with a given eigenvalue and is supported on the classical phase-space repeller.
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Affiliation(s)
- J M Pedrosa
- Departamento de Física, CNEA, Av. Libertador 8250, Buenos Aires C1429BNP, Argentina
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15
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Novaes M. Supersharp resonances in chaotic wave scattering. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 85:036202. [PMID: 22587156 DOI: 10.1103/physreve.85.036202] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/13/2012] [Indexed: 05/31/2023]
Abstract
Wave scattering in chaotic systems can be characterized by its spectrum of resonances, z(n)=E(n)-iΓ(n)/2, where E(n) is related to the energy and Γ(n) is the decay rate or width of the resonance. If the corresponding ray dynamics is chaotic, a gap is believed to develop in the large-energy limit: almost all Γ(n) become larger than some γ. However, rare cases with Γ<γ may be present and actually dominate scattering events. We consider the statistical properties of these supersharp resonances. We find that their number does not follow the fractal Weyl law conjectured for the bulk of the spectrum. We also test, for a simple model, the universal predictions of random matrix theory for density of states inside the gap and the hereby derived probability distribution of gap size.
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Affiliation(s)
- Marcel Novaes
- Departamento de Física, Universidade Federal de São Carlos, São Carlos, SP, 13565-905, Brazil
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16
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Revuelta F, Vergini EG, Benito RM, Borondo F. Computationally efficient method to construct scar functions. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 85:026214. [PMID: 22463306 DOI: 10.1103/physreve.85.026214] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/15/2011] [Revised: 01/25/2012] [Indexed: 05/31/2023]
Abstract
The performance of a simple method [E. L. Sibert III, E. Vergini, R. M. Benito, and F. Borondo, New J. Phys. 10, 053016 (2008)] to efficiently compute scar functions along unstable periodic orbits with complicated trajectories in configuration space is discussed, using a classically chaotic two-dimensional quartic oscillator as an illustration.
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Affiliation(s)
- F Revuelta
- Grupo de Sistemas Complejos and Departamento de Física, Escuela Técnica Superior de Ingenieros Agrónomos, Universidad Politécnica de Madrid, E-28040 Madrid, Spain
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17
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Vagov A, Schomerus H, Zalipaev VV. Asymptotic-boundary-layer method for unstable trajectories: semiclassical expansions for individual scar wave functions. Phys Rev E 2010; 80:056202. [PMID: 20365055 DOI: 10.1103/physreve.80.056202] [Citation(s) in RCA: 16] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/06/2009] [Indexed: 11/07/2022]
Abstract
We extend the asymptotic boundary layer (ABL) method, originally developed for stable resonator modes, to the description of individual wave functions localized around unstable periodic orbits. The formalism applies to the description of scar states in fully or partially chaotic quantum systems, and also allows for the presence of smooth and sharp potentials, as well as magnetic fields. We argue that the separatrix wave function provides the largest contribution to the scars on a single wave function. This agrees with earlier results on the wave-function asymptotics and on the quantization condition of the scar states. Predictions of the ABL formalism are compared with the exact numerical solution for a strip resonator with a parabolic confinement potential and a magnetic field.
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Affiliation(s)
- A Vagov
- Institut für Theoretische Physik III, Bayreuth Universität, Bayreuth 95440, Germany
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