Spectral and Parametric Averaging for Integrable Systems
|
|
- Anissa Hopkins
- 5 years ago
- Views:
Transcription
1 Spectral and Parametric Averaging for Integrable Systems Tao a and R.A. Serota Department of Physics, University of Cincinnati, Cincinnati, OH 5- We analyze two theoretical approaches to ensemble averaging for integrable systems in quantum chaos - spectral averaging and parametric averaging. For spectral averaging, we introduce a new procedure - rescaled spectral averaging. Unlike traditional spectral averaging, it can describe the correlation function of spectral staircase and produce persistent oscillations of the interval level number variance. Parametric averaging, while not as accurate as rescaled spectral averaging for the correlation function of spectral staircase and interval level number variance, can also produce persistent oscillations of the global level number variance and better describes saturation level rigidity as a function of the running energy. Overall, it is the most reliable method for a wide range of statistics. I. INTRODUCTION The framework of quantum chaos is structured around the concept of ensemble averaging. Statistics, such as correlation function of level density [], interval level number variance (IV) [], global level number variance (GV) [], spectral rigidity (SR) [3] and nearest neighbor spacing distribution [] are defined through ensemble averaging. In the literature, two methods are employed to achieve ensemble averaging for integrable systems. Traditionally, ensemble averaging in semiclassical theories was understood in terms of spectral averaging (SA). [3, 5] A numerical simulation of IV using SA was performed in []. The oscillations of IV were found to decay, while the other ensemble averaging method for integrable systems parametric averaging (PA) correctly showed persistent oscillations. [] We explained that SA tends to suppress the non-decaying oscillatory behavior due to destructive interference of the running-energy-dependant non-coherent terms. [] oreover, this paper will show that when, in order to avoid such destructive interference, SA is performed over a short range of sampled energies, sampling is insu cient and sample-specific fluctuations are observed. uch as impurity averaging in disordered systems, PA corresponds to ensemble averaging for a fixed value of the running energy; specifically for rectangular billiards averaging is over an ensemble of rectangles of varying aspect ratios and fixed area. To our knowledge, PA for integrable system was first performed by Casati et. al. to prove the saturation of SR of integrable system. In their words, computed an average 3(L)... byaver- aging 3 over a number of di erent values of chosen at random in a given interval ( ensemble averaging ). [7] ( 3(L) denotes SR over the interval L and is the aspect ratio defined in [7].) PA with better implementations was used to reproduce saturation of SR [, ], produce persistent oscillations of IV [, ] and GV [] and prove level repulsion in integrable systems []. These studies lectronic address: serota@ucmail.uc.edu demonstrated that PA is a reliable and versatile method for numerical computation of statistics of integrable systems. (Note that PA can also be used as an experimental technique to study orbital magnetism. []) Here we undertake a detailed comparison of SA and PA previously unaddressed in literature. The central result of this work is that, unlike PA, traditional SA cannot produce persistent oscillations of IV and GV. ven with our newly proposed rescaled spectral averaging (RSA), one can only address IV oscillations. These results are argued theoretically and proved numerically. This paper is organized as follows. In Sec. II, we review the periodic orbit theory of level fluctuations and the semiclassical theory of IV, GV, SR, and correlation function of spectral staircase (CFSS). In Sec. III, we define SA and PA for IV, CFSS, and SR. From linear expansion of SA, we argue that SA suppresses the oscillations of IV when averaging over large intervals. To preserve persistent oscillations, we propose RSA. In Sec. IV, we present spectral fluctuations, IV, and GV computed from SA and PA, IV from RSA, CFSS from PA and RSA, and SR from SA, PA and RSA. In Sec. V, we discuss advantages and shortcomings of RSA and PA and outline their applicability. In this paper, we use uppercase letters to indicate ensemble averaged statistics and lowercase ones to indicate their corresponding sample-specific values. For instance, denotes IV and denote sample IV; g denotes GV and g denotes sample GV; 3 denotes SR and 3 denotes sample SR; K N denotes CFSS and k N denotes sample CFSS. The subscripts A and indicate numerical computation and theoretical calculation respectively. The superscript indicates the ensemble averaging method, that is SA, RSA or PA. II. STATISTICS A. Periodic orbit theory of level fluctuations We use rectangular billiards as a model system to illustrate our theory. For a particle of mass m in a rectangular billiard with sides a, b and aspect ratio a /b,
2 the eigenenergy with quantum numbers n, is given by " n,n = ~ n m a + n b. () The spectral staircase is defined as N (") X n,n (" " n,n ), () where is unit step function. According to Weyl s formula, the ensemble-averaged spectral staircase is given by [,, ] hn (")i = " S " / p + (3) A where = ~ /ma; A and S are the rectangular billiard area and perimeter respectively; and hi denotes ensemble averaging. The second and third terms are usually called perimeter correction and corner correction respectively. In previous works [, 9], we only considered the perimeter correction when unfolding the spectrum. In the present paper, we account for both terms. After unfolding the spectrum by (3), the mean level spacing becomes unity and [] hn (")i = ", () which would be correct for a perfect ensemble averaging method and is approximately correct for SA and PA as will be shown in Sec. IV A. From the periodic orbit theory,[, 3] the fluctuation of level density is given by (") (") h (")i = X apple S (") ~ µ+ A (") cos ~ and the fluctuation of spectral staircase by N (") N(") hn(")i = X A (") ~ µ T (") apple S (") sin ~ (5). () Here µ =( )/, is the dimensionality of phase space and the period, action, and amplitude of the periodic orbit are given respectively by [3] T (") =[m( a + b )/"] / S (") ="T A (") =m a b / 3 "T, (7) with =(, ) and non-negative, as winding numbers. Above >< = = = / if only one of, is zero () >: otherwise. Compared with [3], in (5) and (), we have an extra factor / from a quantum mechanical calculation [9]. In Sec. IV A, we show that this factor matters. B. Interval and global level number variance IV is defined as where N N(" ) (", ) h[n hni] i = h[n ] i, (9) N(" )with and ". GV is defined as [, 5] " = " / () " = " + / () g (") h[n (") hn(")] i = h[n (") "] i. () In (9) and (), we used (). We term sample IV and (", ) [N hni] =[N ] (3) g(") [N (") hn(")i] =[N (") "] () sample GV. mploying the diagonal approximation [, 9], theoretical sample IV is expressed as [] (", ) = ~ µ X A (") T (") sin T (") ~. (5) Substituting (7) and unfolding the spectrum, the above equation can be written as [] r " X appler (", ) = 5 R 3 sin " R, () where R = p / + /. Numerical sample IV, A(", ), is a jagged line as a function of, while theoretical sample IV, (", ), is a smooth line by (). ensemble averaging is able to bridge this di erence. SR is defined as 3(", ) min (A,B) C. Spectral rigidity Z " which has the explicit form [3] * " Z / N (" +!)d! / Z / / " [N (x) A Bx] dx, (7)!N (" +!)d! " # + Z / /. N (" +!)d! # ()
3 3 Sample SR is defined as 3(", ) min (A,B) Z " " [N (x) A Bx] dx, (9) which is computed from () without ensemble averaging (that is from the expression inside hi). The saturation SR 3 (") and its sample value 3 (") are numerically computed as 3(", ) p and 3 (", ) respectivelywithsufficiently large ". For the saturation SR, the minimization fit A+B" is approximately given by ". Hence [] 3(", ) Z " " g(x)dx, () where we used (). Based on DA, we have the sample value of saturation SR [] r " X ( 3 ) (") = 5 R 3, () where we used (7) and unfolded the spectrum. D. Correlation function of spectral staircase CFSS is defined as [, 5] K N (", ) h N (" ) N (" )i. () The sample CFSS is defined as Using DA, we have [] k N (", ) = 3 (") k N (", ) N (" ) N (" ). (3) 3 (") The ensemble averaged form is (", ), for p ". () where A(x, ) and (x, ) implicitly depend on the aspect ratio and f SA (x) is the density of sampled energies and is chosen as equally spaced points in a range centered at ". In other words, x [" /,"+ /] with uniform density. In PA, the numerical and theoretical values of IV are respectively defined as PA A (", ) R A(", )f PA ( )d () PA (", ) (",, ), (9) where A (", ) and (", ) implicitly depend on and f PA ( ) is a Gaussian distribution with mean and standard deviation. Numerical computation of IV from SA and PA can be understood as numerical integration of () and () respectively. Similarly we can define SA and PA for CFSS and SR. [] B. Linear expansion of spectral averaging Using (), a representative term in (7) reads Z r appler x sin x R f SA (x)dx r " 5 R 3 Z "+ 5 R 3 " sin appler x R f SA (x)dx. (3) When is far larger than the period of the sine term, the integrand can be replaced by / and one will not observe persistent oscillations of IV. The first-order derivative of the argument of sine is given by d p x R dx whereof we find that when x=" = p R " 3/, (3) > p " 3/ R "3/, (3) oscillations of IV will decay. Notice that oscillations are observed when >" / [, 9] and that the decay of IV oscillations becomes faster with larger. [] K N (", ) = 3 (") (", ). (5) C. Rescaled spectral averaging III. THORY OF SPCTRAL AND PARATRIC AVRAGING A. Spectral and parametric averaging In SA, the numerical and theoretical values of IV can be respectively defined by the following integrals: SA A (", ) R A(x, )f SA (x)dx () SA (", ) R (x, )f SA (x)dx, (7) We just saw that traditional SA su ers from an inherent flaw due to destructive interference of oscillating terms. In order to observe persistent oscillations with larger and larger interval width, one needs to sample su ciently large energy range centered on " to achieve proper ensemble averaging. Yet, q. (3) sets the limit to how large such range can be in order to avoid destructive interference and observe persistent oscillation of IV. Furthermore, the limit deceases with the increase of. A possible workaround would be to sample various parts of spectrum, not necessarily around ". However,
4 since persistent oscillations strongly depend on " (the point of onset, the amplitude and the period [, 9]), such procedure, executed without a proper account for this "-dependence, would have an e ect similar to the destructive interference above a wash-out of persistent oscillations. Consequently, we introduce a modified procedure, RSA, that allows sampling of di erent parts of the spectrum. Our approach is based on a scaling transformation (c", p c) = p c (", ), which follows from (). Consequently, in RSA, when the running energy and the interval are scalled as "! c" and! p c respectively, A(c", p c) needs to be rescaled by a factor / p c before averaging. Numerically, IV is computed as RSA A (", ) n + nx i= p ci A (c i ", p c i ), (33) where c i is the ratio of the energy of a sampled spectral location to " and n+ is the number of sampled energies and theoretically, by design, it is given by RSA (", ) = (", ), (3) that is coincides with (9). We note due to the close relation between IV and CFSS in (5), RSA can be similarly defined for the latter and we have (K N ) RSA (", ) =(k N ) (", ), (35) where theoretical (K N ) RSA (", ) can be evaluated from (), (), and (). RSA of saturation SR is computed by nx p ( 3 ) A (c i "), (3) n + ci i= and its theoretical value is ( 3 ) RSA (") ( 3 ) ("). (37) ( 3 ) RSA A (") scales as p " for billiard systems, including elliptic billiards []. A SA A SA FIG. : Spectral averaging of the fluctuation of spectral staircase with di erent ranges of sampled energies. Left: the range is ;Right: 3. A PA, PA A PA, PA IV. NURICAL SIULATIONS Below, except in Fig., for SA and RSA, the aspect ratio is set to be ( p 5 )/.93 to avoid degeneracy; for PA, the distribution of is a Gaussian distribution with mean and standard deviation.. In the computation of IV and CFSS, we set " = 5. FIG. : Parametric averaging of the fluctuation of spectral staircase. Top: black dots, numerical hn (") "i PA A calculated by averaging over from a Gaussian distribution with the mean and the standard deviation.; magenta line, hn (") "i PA calculated from () and averaged over ; dashed blue line, hn (") "i PA calculated from () without the / factor and averaged over. Bottom: thesamemagentaline is shifted leftward by 5.. A. Fluctuations of spectral staircase We study SA and PA of the fluctuation of spectral staircase hn (") "i SA (3) hn (") "i PA. (39) In Figs. and, we present the results obtained with SA and PA respectively. For a large range of sampled energies, SA gives near zero result. PA produces regular oscillations about zero line. Theoretically, the oscillations are due to the sine term with the periodic orbit (,) in (), which does not vanish upon PA. [] In Fig., the theoretical result obtained from () with PA needs to be shifted leftward to be consistent with the numerical
5 5 S A RSA H,L, SA PA H,L, s QH,L FIG. : IV calculated with RSA and PA. Black dots: RSA calculated from an ensemble of sampled energies in [ 5, 5 ]. Green dashed line: PA. Blue solid line: theoretical result calculated from (). A SA, result. This shift is due to the perimeter correction and can be calculated as r = 5. from r / = 5, where 5 is the average energy in Fig.. We also observe that the factor / is critical for a good vertical fit. The deviation of hn (") "i from in PA reveals a shortcoming of PA. But its small magnitude indicates PA is su ciently proficient as an ensemble averaging method for spectral staircase. An advantage of PA is that the distribution f PA ( ) works for any energy scale, while the range of sampled energies needs to grow as " / in SA. FIG. 3: IV calculated from SA. Thin and thick lines: the ranges of sampled energies are [95, 5] and [75, 5] respectively. B. Interval level number variance In Fig. 3, we present IV computed from SA. Clearly, SA cannot properly produce the persistent oscillations of IV. If the range of sampled energies is small, SA produces close to sample specific oscillations, indicating insu cient sampling. If the range of sampled energies is large, SA suppresses IV oscillations when the interval grows. In Fig., we present IV computed from RSA and PA. We observe that RSA is in better agreement with the theoretical result, (3) (or, equivalently, (9)) and, than PA. It should be emphasized that while we are using rectangular billiars as a model system, RSA formalism readily A RSA,, RSA, FIG. 5: IV of circular billiard computed with RSA from an ensemble of sampled energies in [ 5, 5 ]. Black dots: numerical result. Cyan line: theoretical result.
6 PA HKNL A H,L RSA HKNL A H,L HkNLQH,L FIG. : CFSS with RSA and PA. Black dots: numerical result calculated from the definition of K N in () and averaged through RSA. Green dashed line: PA. Blue solid line: theoretical result calculated from the first q. (). Insert shows small behavior - close to linear, according to the second q. (). extends to any system with continuous dependence on ", even if the latter is non-monotonic as, for instance, in Fig. 9 below for the saturation spectral rigidity (which has the same "-dependence as IV). In Fig. 5, we see that IV computed with RSA for a circular billiard [, ] is in excellent agreement with theory. C. Correlation function of spectral staircase In Fig. we plot K N (", ) computed with RSA and PA. Again, we observe that RSA is in a better agreement with theoretical result than PA. 3 A RSA, 3 A PA A, 3 A SA, 3 A PA 3 FIG. 7: Numerical result of saturation SR computed from RSA and PA. agenta line: PA. Dashed blue line: RSA. For ( 3 ) RSA A ("), the range of sampled energies [", "]. D. Saturation spectral rigidity In Fig. 7, we present saturation SR computed from RSA and PA. Clearly, PA yields a better result since RSA shows small oscillations, while by theory () saturation SR should be a smooth function of ". In Fig., we present saturation SR computed with PA and SA and sample saturation SR (computed with (9)). The latter shows large-range oscillations, which is absent in the PA result. If the range of sampled energies is su ciently large, SA gives a result close to PA; otherwise, SA gives behaves similarly to sample specific SR. FIG. : Numerical results of saturation SR computed from SA and PA. agenta line without any oscillations: PA result. Thin and thick black lines: SA with averaging range and 5 respectively. Jagged cyan line: sample saturation SR computed with = 3 for "<= and =5 3 for ">.. Global level number variance GV is discussed in detail in []. In Fig. (9), we present numerical results, computed using PA, for four di erent integrable systems: rectangular billiard [9], modified Kepler problem [9] and circular and elliptical billiards [, ]. In each case, GV exhibits large oscillations around SR []. With the exception of rectangular billiards, SA/RSA is incapable of reproducing the complex behavior of SR
7 7 # PA!!g "PA A!!",!"3 " A!!" Rectangular billiard 5 not find a rescaled form of SA for GV. A simple definition of SA for GV is n 5 g (") = 5! # PA!!g "PA A!!",!"3 " A!!" odified Kepler problem ! Quarter circular billiard # PA!!g "PA A!!",!"3 " A!!" 5 5 5! Quarter elliptic billiard # PA!!g "PA A!!",!"3 " A!!" 5 CONCLUSIONS We introduced a new SA procedure RSA to cure some of the intrinsic problems of SA. For rectangular billiards, we found that SA cannot produce persistent oscillations of IV and has some difficulties with SR. Any spectral averaging is unsuitable for GV oscillations. RSA is best suited for oscillations of IV and CFSS and generally works for SR, while PA is best suited for SR, GV and generally works for IV and CFSS. Relative RSA success for SR in rectangular billiards does not carry over to more complex system, such as odified Kepler Problem [9] and elliptic billiards [], where SR exhibits non-trivial dependence on the running energy (spectral position) that RSA is incapable of capturing. To summarize our findings: PA always works numerically, RSA may be occasionally more accurate while traditional SA is almost always inadequate. Alternatively, we can also cast our findings as follows: For quantities depending on the interval, such as IV and CFSS, RSA works slightly better than PA, while standard SA fails. 5 For fluctuations of spectral staircase, any SA produces zero, while PA yields unphysical persistent oscillations, albeit very small. () For quantities depending on ", such as saturation spectral rigidity and GV, PA should be used. 5 "i ], where the sampled energy "i is equally distributed in the range ["!/, " +!/]. This is the definition of sample SR: 3 (",!). The integration in () (after we change into!) P can be approximated by the numerical integration n as n+!/ + i!/n), which becomes () if i= g (" "i = "!/ + i!/n. We arrive at the conclusion that no variant of SA is capable of reproducing large oscillations of GV around SR. V X [N ("i ) n + i=! FIG. 9: Comparison between GV and saturation SR of rectangular billiard, modified Kepler problem, quarter circular billiard, and quarter elliptic billiard Jagged black line: g (") directly computed () through PA. Smooth cyan line: 3 ("). Inserts show a shorter range of g (") and 3 ("). observed in Fig. (9). Also, unlike IV and CFSS, we can We also have good agreement between theory and numerical results. The latter includes the fact that, with the exception of GV, diagonal approximation yields sufficiently accurate predictions. RSA should find its use in fixed area circular and square billiards, for which no proper PA procedure exists. On the other hand, PA may also find application in chaotic billiard and potential systems. For instance, PA of a Sinai billiard a circular hole in a rectangular billiard can be achieved through varying the aspect ratios of the sides of the billiard. [7]
8 [] J..A.S.P. Wickramasinghe, B. Goodman, and R. A. Serota, Phys. Rev. 7, 59(5). [] T. a and R.A. Serota, arxiv:3. (). [3]. Berry, Proc. R. Soc. London, Ser. A, 9(95). [] T. a and R.A. Serota, Int. J. od. Phys. B 595 (). [5]. Berry, Springer Lecture Notes in Physics, No. (lsevier-health Sciences Division, 9), p [] C. Grosche, merging Applications of Number Theory p. 9-9 (Springer-Verlag, New York, 999). [7] G. Casati, B. Chirikov, and I. Guarneri, Phys. Rev. Lett. 5, 35 (95). [] J..A.S.P. Wickramasinghe, B. Goodman, and R.A. Serota, Phys. Rev. 77, 5(). [9] T. a and R. Serota, Phys. Rev. 5, 3(). [] T. a and R. Serota, arxiv:3.7 (). [] L.P. Lévy, D. Reich, L. Pfei er, and K. West, Physica B: Condensed atter 9, (993). [].C. Gutzwiller, Chaos in Classical and Quantum echanics (Springer, 99);. [3] H-J Stöckmann Quantum Chaos: An Introduction (Cambridge University Press, Cambridge, UK, ). While in a very di erent fashion, in chaotic systems the periodic orbit theory now accounts for the key features of the spectrum (. Sieber and K. Richter, Phys. Scr. T9, (); S. üller, S. Heusler, P. Braun, F. Haake and A. Altland, Phys. Rev. Lett ()) and scarring of the wave function (L. Kaplan and. J. Heller, in NATO ASI series volume Supersymmetry and Trace Formulae: Chaos and Disorder, ed. by I. V. Lerner, J. P. Keating, and D.. Khmelnitskii (Plenum, 999)). [] Here and below circular billiards are near circular elliptical billiards with the aspect ratio picked from a distribution centered at and elliptical billiards have aspect ratio distribution centered at /. [5] R.A. Serota, arxiv:.3 (). [] Alternatively, one could use a di erent definition of theoretical PA, [, 9] namely PA (", ) R (", )f PA ( )d, () With this definition, an argument similar to Sec. III B, using (), shows that oscillations due to (, ) terms with = would decay to the average value when becomes large. But for (,) terms, this argument fails as the first order derivative over vanishes (approximately) for and the oscillations persist even when becomes large. Then the persistent oscillations of PA (", ) are solely due to periodic orbit (,). Replacing the sine terms in () and () with /, except for =, we have for large r " PA " (", ) 5 + X = = X = = R 3 R p sin R " # f PA ( )d. R 3 () The implication of the above is that DA breaks down in such procedure. For IV, () describes numerical results fairly well, even for large intervals. In order to obtain good agreement with numerical results for GV, however, the use of f PA ( ) in theoretical averaging necessitates taking non-diagonal terms into account. For instance, the theoretical sample GV containing both diagonal and nondiagonal terms is given by [] ( g) (") " = X ~ µ A (") T (") S (") sin ~ #. (3) One of the reasons to use theoretical f PA ( ) averaging in the first place is to obtain agreement with numerical results for spectral staircase, as shown in Fig.. [7] Averaging for two common classes of chaotic systems strongly interacting systems. such as large nuclei, and disordered systems, such as metals with impurities naturally falls, respectively, into SA and disorder averaging (analogue of PA). The spectral properties of such systems are now well understood and are broadly covered by the Random atrix Theory: L. P. Gorkov and G.. liashberg, Sov. Phys. JTP ; 9 (95), K. fetov, Supersymmetry in Disorder and Chaos (Cambridge University Press, Cambridge, UK, 997); T. A. Brody, J. Flores, J. B. French, P. A. ello, A. Pandey, and S. S.. Wong, Rev. od. Phys. 53, 35(9)and.L. ehta, Random atrices (Academic Press, New York, 99); A. V. Andreev, O. Agam, B. D. Simons, and B. L. Altshuler, Phys. Rev. Lett 7, 395(99). Unlike persistent oscillations, discontinuities and non-monotonic behavior displayed by integrable systems, averaged spectral properties of chaotic systems are generally continuos and monotonic. However, chaotic billiard systems, such as Sinai billiard, may exhibit some residual properties of integrability.
LEVEL REPULSION IN INTEGRABLE SYSTEMS
LEVEL REPULSION IN INTEGRABLE SYSTEMS Tao Ma and R. A. Serota Department of Physics University of Cincinnati Cincinnati, OH 45244-0011 serota@ucmail.uc.edu Abstract Contrary to conventional wisdom, level
More informationarxiv: v1 [quant-ph] 7 Mar 2012
Global Level Number Variance in Integrable Systems Tao Ma, R.A. Serota Department of Physics University of Cincinnati Cincinnati, OH 5-11 (Dated: October, 13) arxiv:3.1v1 [quant-ph] 7 Mar 1 We study previously
More informationExperimental and theoretical aspects of quantum chaos
Experimental and theoretical aspects of quantum chaos A SOCRATES Lecture Course at CAMTP, University of Maribor, Slovenia Hans-Jürgen Stöckmann Fachbereich Physik, Philipps-Universität Marburg, D-35032
More informationUniversality. Why? (Bohigas, Giannoni, Schmit 84; see also Casati, Vals-Gris, Guarneri; Berry, Tabor)
Universality Many quantum properties of chaotic systems are universal and agree with predictions from random matrix theory, in particular the statistics of energy levels. (Bohigas, Giannoni, Schmit 84;
More informationConstructing a sigma model from semiclassics
Constructing a sigma model from semiclassics In collaboration with: Alexander Altland, Petr Braun, Fritz Haake, Stefan Heusler Sebastian Müller Approaches to spectral statistics Semiclassics Approaches
More informationMisleading signatures of quantum chaos
Misleading signatures of quantum chaos J. M. G. Gómez, R. A. Molina,* A. Relaño, and J. Retamosa Departamento de Física Atómica, Molecular y Nuclear, Universidad Complutense de Madrid, E-28040 Madrid,
More informationRecent results in quantum chaos and its applications to nuclei and particles
Recent results in quantum chaos and its applications to nuclei and particles J. M. G. Gómez, L. Muñoz, J. Retamosa Universidad Complutense de Madrid R. A. Molina, A. Relaño Instituto de Estructura de la
More informationSpectral Fluctuations in A=32 Nuclei Using the Framework of the Nuclear Shell Model
American Journal of Physics and Applications 2017; 5(): 5-40 http://www.sciencepublishinggroup.com/j/ajpa doi: 10.11648/j.ajpa.2017050.11 ISSN: 20-4286 (Print); ISSN: 20-408 (Online) Spectral Fluctuations
More informationIntroduction to Theory of Mesoscopic Systems
Introduction to Theory of Mesoscopic Systems Boris Altshuler Princeton University, Columbia University & NEC Laboratories America Lecture 3 Beforehand Weak Localization and Mesoscopic Fluctuations Today
More informationSize Effect of Diagonal Random Matrices
Abstract Size Effect of Diagonal Random Matrices A.A. Abul-Magd and A.Y. Abul-Magd Faculty of Engineering Science, Sinai University, El-Arish, Egypt The statistical distribution of levels of an integrable
More informationarxiv:cond-mat/ v1 [cond-mat.stat-mech] 13 Mar 2003
arxiv:cond-mat/0303262v1 [cond-mat.stat-mech] 13 Mar 2003 Quantum fluctuations and random matrix theory Maciej M. Duras Institute of Physics, Cracow University of Technology, ulica Podchor ażych 1, PL-30084
More informationDepartment of Electrical and Electronic Engineering, Ege University, Bornova 3500, Izmir, Turkey
The effect of anisotropy on the absorption spectrum and the density of states of two-dimensional Frenkel exciton systems with Gaussian diagonal disorder I. Avgin a and D. L. Huber b,* a Department of Electrical
More informationAutocorrelation function of level velocities for ray-splitting billiards
PHYSICAL REVIEW E VOLUME 61, NUMBER 1 JANUARY 2000 Autocorrelation function of level velocities for ray-splitting billiards Y. Hlushchuk, 1,2 A. Kohler, 3 Sz. Bauch, 1 L. Sirko, 1,2 R. Blümel, 4 M. Barth,
More informationErgodicity of quantum eigenfunctions in classically chaotic systems
Ergodicity of quantum eigenfunctions in classically chaotic systems Mar 1, 24 Alex Barnett barnett@cims.nyu.edu Courant Institute work in collaboration with Peter Sarnak, Courant/Princeton p.1 Classical
More informationarxiv:chao-dyn/ v1 3 Jul 1995
Chaotic Spectra of Classically Integrable Systems arxiv:chao-dyn/9506014v1 3 Jul 1995 P. Crehan Dept. of Mathematical Physics, University College Dublin, Belfield, Dublin 2, Ireland PCREH89@OLLAMH.UCD.IE
More informationORIGINS. E.P. Wigner, Conference on Neutron Physics by Time of Flight, November 1956
ORIGINS E.P. Wigner, Conference on Neutron Physics by Time of Flight, November 1956 P.W. Anderson, Absence of Diffusion in Certain Random Lattices ; Phys.Rev., 1958, v.109, p.1492 L.D. Landau, Fermi-Liquid
More informationEmergence of chaotic scattering in ultracold lanthanides.
Emergence of chaotic scattering in ultracold lanthanides. Phys. Rev. X 5, 041029 arxiv preprint 1506.05221 A. Frisch, S. Baier, K. Aikawa, L. Chomaz, M. J. Mark, F. Ferlaino in collaboration with : Dy
More informationInteraction Matrix Element Fluctuations
Interaction Matrix Element Fluctuations in Quantum Dots Lev Kaplan Tulane University and Yoram Alhassid Yale University Interaction Matrix Element Fluctuations p. 1/37 Outline Motivation: ballistic quantum
More informationQuantum coherent transport in Meso- and Nanoscopic Systems
Quantum coherent transport in Meso- and Nanoscopic Systems Philippe Jacquod pjacquod@physics.arizona.edu U of Arizona http://www.physics.arizona.edu/~pjacquod/ Quantum coherent transport Outline Quantum
More informationarxiv: v2 [cond-mat.stat-mech] 30 Mar 2012
Quantum chaos: An introduction via chains of interacting spins 1/2 Aviva Gubin and Lea F. Santos Department of Physics, Yeshiva University, 245 Lexington Avenue, New York, NY 10016, USA arxiv:1106.5557v2
More informationIs Quantum Mechanics Chaotic? Steven Anlage
Is Quantum Mechanics Chaotic? Steven Anlage Physics 40 0.5 Simple Chaos 1-Dimensional Iterated Maps The Logistic Map: x = 4 x (1 x ) n+ 1 μ n n Parameter: μ Initial condition: 0 = 0.5 μ 0.8 x 0 = 0.100
More informationChapter 29. Quantum Chaos
Chapter 29 Quantum Chaos What happens to a Hamiltonian system that for classical mechanics is chaotic when we include a nonzero h? There is no problem in principle to answering this question: given a classical
More informationString-like theory of many-particle Quantum Chaos
String-like theory of many-particle Quantum Chaos Boris Gutkin University of Duisburg-ssen Joint work with V. Al. Osipov Luchon, France March 05 p. Basic Question Quantum Chaos Theory: Standard semiclassical
More informationChaos, Quantum Mechanics and Number Theory
Chaos, Quantum Mechanics and Number Theory Peter Sarnak Mahler Lectures 2011 Hamiltonian Mechanics (x, ξ) generalized coordinates: x space coordinate, ξ phase coordinate. H(x, ξ), Hamiltonian Hamilton
More informationPersistent current flux correlations calculated by quantum chaology
J. Phys. A, Math. Gen. 27 (1994) 61674116. Printed in the UK Persistent current flux correlations calculated by quantum chaology M V Berry+ and J P Keatingf t H H Wills Physics Laboratory, TyndaU Avenue,
More informationPeriodic orbit quantization of chaotic systems with strong pruning
6 May 2002 Physics Letters A 297 (2002) 87 91 www.elsevier.com/locate/pla Periodic orbit quantization of chaotic systems with strong pruning Kirsten Weibert, Jörg Main, Günter Wunner Institut für Theoretische
More informationBreit-Wigner to Gaussian transition in strength functions
Breit-Wigner to Gaussian transition in strength functions V.K.B. Kota a and R. Sahu a,b a Physical Research Laboratory, Ahmedabad 380 009, India b Physics Department, Berhampur University, Berhampur 760
More informationInteraction Matrix Element Fluctuations
Interaction Matrix Element Fluctuations in Quantum Dots Lev Kaplan Tulane University and Yoram Alhassid Yale University Interaction Matrix Element Fluctuations p. 1/29 Outline Motivation: ballistic quantum
More informationThe concept of free electromagnetic field in quantum domain
Tr. J. of Physics 23 (1999), 839 845. c TÜBİTAK The concept of free electromagnetic field in quantum domain Alexander SHUMOVSKY and Özgür MÜSTECAPLIOĞLU Physics Department, Bilkent University 06533 Ankara-TURKEY
More informationarxiv:cond-mat/ v1 29 Dec 1996
Chaotic enhancement of hydrogen atoms excitation in magnetic and microwave fields Giuliano Benenti, Giulio Casati Università di Milano, sede di Como, Via Lucini 3, 22100 Como, Italy arxiv:cond-mat/9612238v1
More informationExperimental evidence of wave chaos signature in a microwave cavity with several singular perturbations
Chaotic Modeling and Simulation (CMSIM) 2: 205-214, 2018 Experimental evidence of wave chaos signature in a microwave cavity with several singular perturbations El M. Ganapolskii, Zoya E. Eremenko O.Ya.
More informationUniversal theory of complex SYK models and extremal charged black holes
HARVARD Universal theory of complex SYK models and extremal charged black holes Subir Sachdev Chaos and Order: from Strongly Correlated Systems to Black Holes, Kavli Institute for Theoretical Physics University
More informationEigenvalues and eigenfunctions of a clover plate
Eur. Phys. J. B 23, 365 372 (21) THE EUROPEAN PHYSICAL JOURNAL B c EDP Sciences Società Italiana di Fisica Springer-Verlag 21 Eigenvalues and eigenfunctions of a clover plate O. Brodier a,t.neicu b, and
More information4 Evolution of density perturbations
Spring term 2014: Dark Matter lecture 3/9 Torsten Bringmann (torsten.bringmann@fys.uio.no) reading: Weinberg, chapters 5-8 4 Evolution of density perturbations 4.1 Statistical description The cosmological
More informationClosing the Debates on Quantum Locality and Reality: EPR Theorem, Bell's Theorem, and Quantum Information from the Brown-Twiss Vantage
Closing the Debates on Quantum Locality and Reality: EPR Theorem, Bell's Theorem, and Quantum Information from the Brown-Twiss Vantage C. S. Unnikrishnan Fundamental Interactions Laboratory Tata Institute
More information550 XU Hai-Bo, WANG Guang-Rui, and CHEN Shi-Gang Vol. 37 the denition of the domain. The map is a generalization of the standard map for which (J) = J
Commun. Theor. Phys. (Beijing, China) 37 (2002) pp 549{556 c International Academic Publishers Vol. 37, No. 5, May 15, 2002 Controlling Strong Chaos by an Aperiodic Perturbation in Area Preserving Maps
More informationBallistic quantum transport through nanostructures
Ballistic quantum transport through nanostructures S. Rotter, F. Libisch, F. Aigner, B. Weingartner, J. Feist, I. Březinová, and J. Burgdörfer Inst. for Theoretical Physics/E136 A major aim in ballistic
More informationAnderson Localization Looking Forward
Anderson Localization Looking Forward Boris Altshuler Physics Department, Columbia University Collaborations: Also Igor Aleiner Denis Basko, Gora Shlyapnikov, Vincent Michal, Vladimir Kravtsov, Lecture2
More informationMany-Body Localization. Geoffrey Ji
Many-Body Localization Geoffrey Ji Outline Aside: Quantum thermalization; ETH Single-particle (Anderson) localization Many-body localization Some phenomenology (l-bit model) Numerics & Experiments Thermalization
More information2.3 Band structure and lattice symmetries: example of diamond
2.2.9 Product of representaitons Besides the sums of representations, one can also define their products. Consider two groups G and H and their direct product G H. If we have two representations D 1 and
More informationQuantum Chaos. Dominique Delande. Laboratoire Kastler-Brossel Université Pierre et Marie Curie and Ecole Normale Supérieure Paris (European Union)
Quantum Chaos Dominique Delande Laboratoire Kastler-Brossel Université Pierre et Marie Curie and Ecole Normale Supérieure Paris (European Union) What is chaos? What is quantum chaos? Is it useful? Is quantum
More informationEigenvalue statistics and lattice points
Eigenvalue statistics and lattice points Zeév Rudnick Abstract. One of the more challenging problems in spectral theory and mathematical physics today is to understand the statistical distribution of eigenvalues
More information1 Supplementary Figure
Supplementary Figure Tunneling conductance ns.5..5..5 a n =... B = T B = T. - -5 - -5 5 Sample bias mv E n mev 5-5 - -5 5-5 - -5 4 n 8 4 8 nb / T / b T T 9T 8T 7T 6T 5T 4T Figure S: Landau-level spectra
More informationSpectral Degree of Coherence of a Random Three- Dimensional Electromagnetic Field
University of Miami Scholarly Repository Physics Articles and Papers Physics 1-1-004 Spectral Degree of Coherence of a Random Three- Dimensional Electromagnetic Field Olga Korotkova University of Miami,
More informationChaotic Transport in Antidot Lattices
Journal Chaotic of ELECTRONIC Transport MATERIALS, in Antidot Vol. Lattices 29, No. 5, 2000 557 Special Issue Paper Chaotic Transport in Antidot Lattices TSUNEYA ANDO and SEIJI URYU University of Tokyo,
More informationRelativistic integro-differential form of the Lorentz-Dirac equation in 3D without runaways.
Relativistic integro-differential form of the Lorentz-Dirac equation in 3D without runaways Michael Ibison, Harold E. Puthoff Institute for Advanced Studies at Austin 43 Braker Lane West, Suite 3 Austin,
More informationarxiv:nlin/ v2 [nlin.cd] 8 Feb 2005
Signatures of homoclinic motion in uantum chaos D. A. Wisniacki,,2 E. Vergini,, 3 R. M. Benito, 4 and F. Borondo, Departamento de Química C IX, Universidad Autónoma de Madrid, Cantoblanco, 2849 Madrid,
More informationEffect of Unfolding on the Spectral Statistics of Adjacency Matrices of Complex Networks
Effect of Unfolding on the Spectral Statistics of Adjacency Matrices of Complex Networks Sherif M. Abuelenin a,b, Adel Y. Abul-Magd b,c a Faculty of Engineering, Port Said University, Port Said, Egypt
More informationAbteilung Theoretische Physik Universitat Ulm Semiclassical approximations and periodic orbit expansions in chaotic and mixed systems Habilitationsschrift vorgelegt an der Fakultat fur Naturwissenschaften
More informationACCURATE FREE VIBRATION ANALYSIS OF POINT SUPPORTED MINDLIN PLATES BY THE SUPERPOSITION METHOD
Journal of Sound and Vibration (1999) 219(2), 265 277 Article No. jsvi.1998.1874, available online at http://www.idealibrary.com.on ACCURATE FREE VIBRATION ANALYSIS OF POINT SUPPORTED MINDLIN PLATES BY
More informationSpacing distributions for rhombus billiards
J. Phys. A: Math. Gen. 31 (1998) L637 L643. Printed in the UK PII: S0305-4470(98)93336-4 LETTER TO THE EDITOR Spacing distributions for rhombus billiards Benoît Grémaud and Sudhir R Jain Laboratoire Kastler
More informationSpin Peierls Effect in Spin Polarization of Fractional Quantum Hall States. Surface Science (2) P.1040-P.1046
Title Author(s) Spin Peierls Effect in Spin of Fractional Quantum Hall States Sasaki, Shosuke Citation Surface Science. 566-568(2) P.1040-P.1046 Issue Date 2004-09-20 Text Version author URL http://hdl.handle.net/11094/27149
More informationSymmetries in Quantum Transport : From Random Matrix Theory to Topological Insulators. Philippe Jacquod. U of Arizona
Symmetries in Quantum Transport : From Random Matrix Theory to Topological Insulators Philippe Jacquod U of Arizona UA Phys colloquium - feb 1, 2013 Continuous symmetries and conservation laws Noether
More informationInteraction Matrix Element Fluctuations
Interaction Matrix Element Fluctuations in Quantum Dots Lev Kaplan Tulane University and Yoram Alhassid Yale University Interaction Matrix Element Fluctuations in Quantum Dots mpipks Dresden March 5-8,
More informationIntegrability and disorder in mesoscopic systems: Application to orbital magnetism
Integrability and disorder in mesoscopic systems: Application to orbital magnetism Klaus Richter Institut für Physik, Memminger Strasse. 6, 86135 Augsburg, Germany and Max-Planck-Institut für Physik komplexer
More informationarxiv:nlin/ v1 [nlin.cd] 8 Jan 2001
The Riemannium P. Leboeuf, A. G. Monastra, and O. Bohigas Laboratoire de Physique Théorique et Modèles Statistiques, Bât. 100, Université de Paris-Sud, 91405 Orsay Cedex, France Abstract arxiv:nlin/0101014v1
More informationSemiclassical formula for the number variance of the Riemann zeros
Nonlinearity 1 (1 988) 399-407. Printed in the UK Semiclassical formula for the number variance of the Riemann zeros M V Berry H H Wills Physics Laboratory, Tyndall Avenue, Bristol BS8 ltl, UK Received
More information2. As we shall see, we choose to write in terms of σ x because ( X ) 2 = σ 2 x.
Section 5.1 Simple One-Dimensional Problems: The Free Particle Page 9 The Free Particle Gaussian Wave Packets The Gaussian wave packet initial state is one of the few states for which both the { x } and
More informationElectron transport through quantum dots
Electron transport through quantum dots S. Rotter, B. Weingartner, F. Libisch, and J. Burgdörfer Inst. for Theoretical Physics/E136 December 3, 2002 Closed billiards have long served as prototype systems
More informationRandom Wave Model in theory and experiment
Random Wave Model in theory and experiment Ulrich Kuhl Fachbereich Physik, Philipps-Universität Marburg, D-35032 Marburg, Deutschland Maribor, 23-27 February 2009 ulrich.kuhl@physik.uni-marburg.de Literature
More informationTheory of Mesoscopic Systems
Theory of Mesoscopic Systems Boris Altshuler Princeton University, Columbia University & NEC Laboratories America Lecture 2 08 June 2006 Brownian Motion - Diffusion Einstein-Sutherland Relation for electric
More informationThe Transition to Chaos
Linda E. Reichl The Transition to Chaos Conservative Classical Systems and Quantum Manifestations Second Edition With 180 Illustrations v I.,,-,,t,...,* ', Springer Dedication Acknowledgements v vii 1
More informationCluster Distribution in Mean-Field Percolation: Scaling and. Universality arxiv:cond-mat/ v1 [cond-mat.stat-mech] 6 Jun 1997.
Cluster Distribution in Mean-Field Percolation: Scaling and Universality arxiv:cond-mat/9706064v1 [cond-mat.stat-mech] 6 Jun 1997 Joseph Rudnick and Paisan Nakmahachalasint Department of Physics, UCLA,
More informationLecture 3 Dynamics 29
Lecture 3 Dynamics 29 30 LECTURE 3. DYNAMICS 3.1 Introduction Having described the states and the observables of a quantum system, we shall now introduce the rules that determine their time evolution.
More information1 Intro to RMT (Gene)
M705 Spring 2013 Summary for Week 2 1 Intro to RMT (Gene) (Also see the Anderson - Guionnet - Zeitouni book, pp.6-11(?) ) We start with two independent families of R.V.s, {Z i,j } 1 i
More informationAn Inverse Mass Expansion for Entanglement Entropy. Free Massive Scalar Field Theory
in Free Massive Scalar Field Theory NCSR Demokritos National Technical University of Athens based on arxiv:1711.02618 [hep-th] in collaboration with Dimitris Katsinis March 28 2018 Entanglement and Entanglement
More informationPartial Dynamical Symmetry in Deformed Nuclei. Abstract
Partial Dynamical Symmetry in Deformed Nuclei Amiram Leviatan Racah Institute of Physics, The Hebrew University, Jerusalem 91904, Israel arxiv:nucl-th/9606049v1 23 Jun 1996 Abstract We discuss the notion
More informationarxiv:cond-mat/ v1 1 Apr 1999
Dimer-Type Correlations and Band Crossings in Fibonacci Lattices Ignacio G. Cuesta 1 and Indubala I. Satija 2 Department of Physics, George Mason University, Fairfax, VA 22030 (April 21, 2001) arxiv:cond-mat/9904022
More informationPHY 435 / 635 Decoherence and Open Quantum Systems Instructor: Sebastian Wüster, IISERBhopal,2018
Week 10 PHY 435 / 635 Decoherence and Open Quantum Systems Instructor: Sebastian Wüster, IISERBhopal,2018 These notes are provided for the students of the class above only. There is no warranty for correctness,
More informationBrief review of Quantum Mechanics (QM)
Brief review of Quantum Mechanics (QM) Note: This is a collection of several formulae and facts that we will use throughout the course. It is by no means a complete discussion of QM, nor will I attempt
More informationacta physica slovaca vol. 54 No. 2, April 2004 SIMPLE EXPERIMENTAL METHODS TO CONTROL CHAOS IN A DOUBLE PLASMA MACHINE 1
acta physica slovaca vol. 54 No. 2, 89 96 April 2004 SIMPLE EXPERIMENTAL METHODS TO CONTROL CHAOS IN A DOUBLE PLASMA MACHINE 1 D. G. Dimitriu 2, C. Găman, M. Mihai-Plugaru, G. Amarandei, C. Ioniţă, E.
More informationAnderson Localization from Classical Trajectories
Anderson Localization from Classical Trajectories Piet Brouwer Laboratory of Atomic and Solid State Physics Cornell University Support: NSF, Packard Foundation With: Alexander Altland (Cologne) Quantum
More informationdoi: /PhysRevC
doi:.3/physrevc.67.5436 PHYSICAL REVIEW C 67, 5436 3 Nuclear collective tunneling between Hartree states T. Kohmura Department of Economics, Josai University, Saado 35-95, Japan M. Maruyama Department
More informationOn the synchronization of a class of electronic circuits that exhibit chaos
Chaos, Solitons and Fractals 13 2002) 1515±1521 www.elsevier.com/locate/chaos On the synchronization of a class of electronic circuits that exhibit chaos Er-Wei Bai a, *, Karl E. Lonngren a, J.C. Sprott
More informationIllustrating Dynamical Symmetries in Classical Mechanics: The Laplace-Runge-Lenz Vector Revisited
Illustrating Dynamical Symmetries in Classical Mechanics: The Laplace-Runge-Lenz Vector Revisited Ross C. O Connell and Kannan Jagannathan Physics Department, Amherst College Amherst, MA 01002-5000 Abstract
More informationTorque on a wedge and an annular piston. II. Electromagnetic Case
Torque on a wedge and an annular piston. II. Electromagnetic Case Kim Milton Homer L. Dodge Department of Physics and Astronomy, University of Oklahoma, Norman, OK 73019, USA Collaborators: E. K. Abalo,
More informationOrigin of the first Hund rule in He-like atoms and 2-electron quantum dots
in He-like atoms and 2-electron quantum dots T Sako 1, A Ichimura 2, J Paldus 3 and GHF Diercksen 4 1 Nihon University, College of Science and Technology, Funabashi, JAPAN 2 Institute of Space and Astronautical
More informationFrom time series to superstatistics
From time series to superstatistics Christian Beck School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London E 4NS, United Kingdom Ezechiel G. D. Cohen The Rockefeller University,
More informationGeometrical theory of diffraction and spectral statistics
mpi-pks/9907008 Geometrical theory of diffraction and spectral statistics arxiv:chao-dyn/990006v 6 Oct 999 Martin Sieber Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Str. 38, 087 Dresden,
More informationsingle uniform density, but has a step change in density at x = 0, with the string essentially y(x, t) =A sin(!t k 1 x), (5.1)
Chapter 5 Waves II 5.1 Reflection & Transmission of waves et us now consider what happens to a wave travelling along a string which no longer has a single uniform density, but has a step change in density
More informationPossible Color Octet Quark-Anti-Quark Condensate in the. Instanton Model. Abstract
SUNY-NTG-01-03 Possible Color Octet Quark-Anti-Quark Condensate in the Instanton Model Thomas Schäfer Department of Physics, SUNY Stony Brook, Stony Brook, NY 11794 and Riken-BNL Research Center, Brookhaven
More informationThe non-interacting Bose gas
Chapter The non-interacting Bose gas Learning goals What is a Bose-Einstein condensate and why does it form? What determines the critical temperature and the condensate fraction? What changes for trapped
More informationarxiv:quant-ph/ v2 28 Mar 2004
Counterintuitive transitions in the multistate Landau-Zener problem with linear level crossings. N.A. Sinitsyn 1 1 Department of Physics, Texas A&M University, College Station, Texas 77843-4242 arxiv:quant-ph/0403113v2
More informationTitleQuantum Chaos in Generic Systems.
TitleQuantum Chaos in Generic Systems Author(s) Robnik, Marko Citation 物性研究 (2004), 82(5): 662-665 Issue Date 2004-08-20 URL http://hdl.handle.net/2433/97885 Right Type Departmental Bulletin Paper Textversion
More informationBeautiful Graphene, Photonic Crystals, Schrödinger and Dirac Billiards and Their Spectral Properties
Beautiful Graphene, Photonic Crystals, Schrödinger and Dirac Billiards and Their Spectral Properties Cocoyoc 2012 Something about graphene and microwave billiards Dirac spectrum in a photonic crystal Experimental
More informationWhy Complexity is Different
Why Complexity is Different Yaneer Bar-Yam (Dated: March 21, 2017) One of the hardest things to explain is why complex systems are actually different from simple systems. The problem is rooted in a set
More informationNanoscale shift of the intensity distribution of dipole radiation
Shu et al. Vol. 26, No. 2/ February 2009/ J. Opt. Soc. Am. A 395 Nanoscale shift of the intensity distribution of dipole radiation Jie Shu, Xin Li, and Henk F. Arnoldus* Department of Physics and Astronomy,
More informationQuantum annealing for problems with ground-state degeneracy
Proceedings of the International Workshop on Statistical-Mechanical Informatics September 14 17, 2008, Sendai, Japan Quantum annealing for problems with ground-state degeneracy Yoshiki Matsuda 1, Hidetoshi
More informationAn Accurate Fourier-Spectral Solver for Variable Coefficient Elliptic Equations
An Accurate Fourier-Spectral Solver for Variable Coefficient Elliptic Equations Moshe Israeli Computer Science Department, Technion-Israel Institute of Technology, Technion city, Haifa 32000, ISRAEL Alexander
More informationLandau quantization, Localization, and Insulator-quantum. Hall Transition at Low Magnetic Fields
Landau quantization, Localization, and Insulator-quantum Hall Transition at Low Magnetic Fields Tsai-Yu Huang a, C.-T. Liang a, Gil-Ho Kim b, C.F. Huang c, C.P. Huang a and D.A. Ritchie d a Department
More informationarxiv:chao-dyn/ v1 12 Feb 1996
Spiral Waves in Chaotic Systems Andrei Goryachev and Raymond Kapral Chemical Physics Theory Group, Department of Chemistry, University of Toronto, Toronto, ON M5S 1A1, Canada arxiv:chao-dyn/96014v1 12
More informationComplex WKB analysis of energy-level degeneracies of non-hermitian Hamiltonians
INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS A: MATHEMATICAL AND GENERAL J. Phys. A: Math. Gen. 4 (001 L1 L6 www.iop.org/journals/ja PII: S005-4470(01077-7 LETTER TO THE EDITOR Complex WKB analysis
More informationConductance fluctuations at the integer quantum Hall plateau transition
PHYSICAL REVIEW B VOLUME 55, NUMBER 3 15 JANUARY 1997-I Conductance fluctuations at the integer quantum Hall plateau transition Sora Cho Department of Physics, University of California, Santa Barbara,
More informationChaotic Mixing in a Diffeomorphism of the Torus
Chaotic Mixing in a Diffeomorphism of the Torus Jean-Luc Thiffeault Department of Applied Physics and Applied Mathematics Columbia University with Steve Childress Courant Institute of Mathematical Sciences
More informationarxiv: v1 [cond-mat.stat-mech] 21 Nov 2007
Quantum anharmonic oscillator and its statistical properties arxiv:0711.3432v1 [cond-mat.stat-mech] 21 Nov 2007 Maciej M. Duras Institute of Physics, Cracow University of Technology, ulica Podchor ażych
More informationThe Twisting Trick for Double Well Hamiltonians
Commun. Math. Phys. 85, 471-479 (1982) Communications in Mathematical Physics Springer-Verlag 1982 The Twisting Trick for Double Well Hamiltonians E. B. Davies Department of Mathematics, King's College,
More informationEnergy spectrum for a short-range 1/r singular potential with a nonorbital barrier using the asymptotic iteration method
Energy spectrum for a short-range 1/r singular potential with a nonorbital barrier using the asymptotic iteration method A. J. Sous 1 and A. D. Alhaidari 1 Al-Quds Open University, Tulkarm, Palestine Saudi
More informationThermalization in Quantum Systems
Thermalization in Quantum Systems Jonas Larson Stockholm University and Universität zu Köln Dresden 18/4-2014 Motivation Long time evolution of closed quantum systems not fully understood. Cold atom system
More informationAn efficient numerical method for hydraulic transient computations M. Ciccotelli," S. Sello," P. Molmaro& " CISE Innovative Technology, Segrate, Italy
An efficient numerical method for hydraulic transient computations M. Ciccotelli," S. Sello," P. Molmaro& " CISE Innovative Technology, Segrate, Italy Abstract The aim of this paper is to present a new
More informationTobias Holck Colding: Publications
Tobias Holck Colding: Publications [1] T.H. Colding and W.P. Minicozzi II, The singular set of mean curvature flow with generic singularities, submitted 2014. [2] T.H. Colding and W.P. Minicozzi II, Lojasiewicz
More information