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On spectral-spacing distributions for systems lacking asymptotics
Physics LettersA 158 (1991) 370—372 North-Holland
PHYSICS LETTERS A
On spectral-spacing distributions for systems lacking asymptotics H.P.W. Gottlieb Division ofScience and Technology, Griffith University, Nathan, Queensland, 4111, Australia Received 3 June 1991; accepted for publication 19 July 1991 Communicated by J.P. Vigier
The detennination of an optimal number of level spacings utilized to obtain a representative distribution is discussed for spectra of a one-dimensional system for which there is no asymptotic expansion when a governing parameter is irrational.
The study of the statistics of eigenvalue spacings has received a good deal of attention in many fields including nuclear [1] and atomic [2] physics, especially in relation to possible chaotic behaviour. It has been remarked [3] that the statistical properties of spectra of even the one-dimensional Schrodinger equation with a barrier may have interesting properties, not often discussed, depending on the rationally or irrationality of a governing parameter. Even in the classical case of the spectrum of a vibrating string with a discontinuity in density [4], there are marked differences depending on whether a parameter is rational [5] (when the pattern of eigenvalues is repeated) or irrational [6] (when the eigenvalues, or any subclasses, do not have an asymptotic expansion). In this paper, we address the question: how large a number of eigenvalues need be taken in order to obtain a, representative histogram of the spacings’ distnbution; in particular, is there an optimal number of data values within a chosen upper limit. Usually, taking more values gives a better representation of a statistical distribution, but this is not so in the cases considered here. We analyse the eigenfrequencies of vibrating strings with fixed ends and a single step discontinuity in otherwise constant density. The equation to be solved is written in the form sin(a.,J~Q) cos[(l —a)Q] + cos ( a~iAQ) sin [(1
—
a )Q] = 0,
(1)
where a (0< a < 1) is the fractional length of string 370
with density p~,the remainder having density P2, and R =p~/p2; Q is a dimensionless frequency parameter [4]. The spectral spacings (differences of successive roots Q of (1)) may be computed for various values of a and R. Let r— R a 1 a 2 —
—
If r is rational (= N/M, N and M coprime integers), then there are harmonic and/or odd-harmonic frequency sub-spectra [41. The overall pattern of roots repeats itself [5], and the solutions fall into distinct classes each of which is an asymptotic expansion (cf. ref. [61). Then the pattern of spectral spacings will repeat itself. It can be shown that there are M+ N spacings of Q of the basic interval of length ME! (1 a). They exhibit symmetry about the centre of the interval, with ~= [(M+N) /2] + (3) —
different values (where [ ] + indicates “the least integer greater than or equal to”). Within a cycle, the spacings therefore repeat themselves or are “folded back” about the “centre” in reverse order, where the centre corresponds to a mode number, or between two mode numbers depending on whether N+ M is odd or even respectively. Ifr is irrational, then the solutions do not have any asymptotic expansion [6], and the spacings do not repeat themselves: there is a spectral-spacing distribution to be determined by computation of a large number of roots.
0375-9601/91/$ 03.50 © 1991 Elsevier Science Publishers B.V. All rights reserved.
Volume 158, number 8
PHYSICS LE~FERSA
When r is irrational, the spacings should nearly repeat in cycles given by integers corresponding to nearby rational approximations; and within an approximate cycle, the spacings should approximately repeat themselves about the centre in reverse order. This can be seen even for low integers in table 1, solving (1) for R = 2, a = 1/5, i.e. r= 1 /2~/~ which is successively better approximated by N/M= 1/3; 4/11; 5/14; 6/17; Corresponding approximate “reversing” cycles of lengths 4; 15; 19; 23 (= M+ N) may be discerned, centred about mode “numbers” (= [M+ N+ 1] /2) 2 ~ 8; 10; 12 respectively, It follows from this analysis that the distribution of spacings given by the first 23 solutions for r= 1 / 2,,J~should be some sort of indication of the actual distribution, not very precise because of the small mode numbers involved here, but better than taking, say, 30 solutions because that would contain only part of the next approximately repeating cycle based on 23. Actually, because of the symmetry about the “centre” of the cycle, even just the first p= 12 spacings (with weighting 1/2 for the 12th value) would give about as r,~asonablean indication of the distribution to this accuracy, with greater efficiency. Only when the number of spacings corresponding to the next best rational approximation (N= 23, M=65, p=44) is computed could a closer approximation to the actual distribution be expected. It should be emphasized that one always solves the equation containing the exact r, it is the choice ofthe optimal number of solutions to be included in the ....
Table 1 The first 23 eigenvalue spacings iS.Q of eq. (1) with r=l/2,/~ (a=l/5,R=2)formodenumbern (roundedoffto2decimal places here for ease of perusal, and “wrapped around” n= 12).
n
iS.Q
2 3 4 5 6 7 8 9 10 11 12
3.06 2.72 2.77 3.09 3.01 2.68 2.84 3.11 2.96 2.66 2.90 3.12
distribution that depends on the integers in the rational approximation to r. The general approach to the efficient accurate determination of the distribution of spacings of zeros of eq. (1) for irrational parameter r (eq. (2)) is therefore as follows: (i) Determine successively better rational approximations N/M for r. (ii) Determine the largest corresponding “halfcycle” number p (eq. (3)) not greater than the maximum number of solutions one wishes to compute based no doubt on accuracy and speed of the computer. (Preferably there should be a large gap to the next p value.) (iii) Plot a histogram, normalizing so that, in its bin, the last spacing values (mode numberp) is given weighting 1/2 if M+N is odd. [(ii’) If the equation to be considered, unlike (1), does not have the “folding” symmetry about the midpoint of a cycle, then one would simply look at the largest number v=M+N, without any non-unit weighting in (iii).] Unlike conventional sampling, there is nothing to be gained by taking a sample larger than a p-value, such as a convenient round power of 10, but less than the next p-value; taking a sample size between p-values would actually worsen the distribution picture. We illustrate the above procedure for the three cases listed in table 2. (The second case is the golden —
Table 2 Chi-square values for p level spacings distributed into 40 bins, compared with distribution for large number L spacings. Case I: a=l/5, R=2, r=l/2.~/~, L=4552. Case II: a=~J~—2,R=4, r=(~J~_l)/2 (golden mean) L=4l8i. Case III: a=(9— 3.,J~)/14, R= 1/9, r= 3_2.J~(Hald), L= 5333.
n 3.07 2.73 2.76 3.09 3.02 2.68 2.83 3.11 2.97 2.66 2.89
23 22 21 20 19 18 17 16 15 14 13
16 September 1991
casel
N/M 204/577 —
1189/3363 case II
377/610 610/987 —
1597/2584 case HI
204/1189 —
781/4552
x2
p 391 1000 2276
(nextp) (1495) (1495) (8714)
494 799 1000 2091
(799) (1292) (1292) (3383)
0.482 0.412 0.841 0.166
697 1000 2667
(2667) (2667) (3363)
0.724 3.547 0.413
0.972 i.928 0.099
371
Volume 158, number 8
PHYSICS LETTERS A
mean. The third case corresponds to the eigenvalue equation of Hald [61, which used erroneous density values: they should be squared there.) In each case the chi-squared value is computed for a distribution over 40 bins, with the base distribution taken as a large value ofL=N+Mlevels in the vicinity of.5000, depending on the actual value of r. Files of length ~ were read, with a weighting of only 1/2 for the last value if (N+M) corresponding to that p is odd, as explained above. In all cases, bin contents were at least 6, which is greater than the 5 usually given in statistical texts as the minimum desirable number, Table 2 gives the chi-squared values (39 degrees of freedom) for various approximations N/M to r, as well as for the round value of 1000 spacings. In each case, for better rational approximations the chisquared value decreases, indicating a better fit to the expected distribution. The last column corresponds toi~=4UL = [L/2] ÷.For 1000 spacings, the is larger than for the smaller p listed; e.g. for case I, taking the significantly smaller number of 391 spacings in fact gives a better fit. For case I, p= 2276 is highly optimal, since the next p value (8714) is so much larger. For case II (the golden mean: the “most irrational number”), the p sequence is Fibonacci number-like, and there are no spectacular jumps. It is now evident that fig. 3 in ref. [6] need not havetaken as many as the 1000 spacings used to build up the 20-bin histogram depicted there. Indeed, for r=3—2~handL=5333,with 2Obins, takingp=458 (corresponding to rational approximation N/M= 134/781) gives chi-squared value 0.283 compared with 0.850 for 1000 spacings. Thus this low p value would have yielded a more accurate distribution, with less computational effort. (Ref. [6] actually dealt with normalized solution minus mode number, rather than spacings.)
2
x
372
16 September 1991
Another application of our idea, now in a quantum-mechariical context, is to the above-mentioned work ofIvanovski et al. [31 which gives a histogram of level spacings of the first 800 states of the spectrum of the Schrodinger equation for a piecewise linear one-dimensional oscillator with a ô-function barrier, in the nonquasiclassicalregion, for a parameter q= 1 )/2 (golden mean) (fig. 2 in ref. [3]). For rational q=N/M, the spacings are repeated almost periodically with a period N+ M states [3]. This situation concerning a governing parameter is similar to what we have been discussing in this paper (see step (ii’) above). The rational approximation N/M= 233/377 gives N+M= 610, with the next N+ M= 987. Thus our work indicates that taking only 610 spacings should give a better representation of the statistical distribution than the “round” larger but unrelated number of 800 values of ref. [3]. Our procedure should have implications for determining statistical distributions for other systems governed by a parameter for which there is no asymptotic expansion for the spectral levels for irrational parameter value but for which the solution is found to have a repetitive pattern when the parameter is rational, with period determined in terms of the integers involved. (\/‘~—
References [110. Bohigas and H.A. Weidenmuller, Annu. Rev. Nucl. Part. Sci. 38 (1988) K.T. 421. Taylor and G. Wunner, Comm. At. Mol. [2] T.S. Monteiro, Phys.25 (1991) 253. [3] G. Ivanovski, N. Novkovski and V. Uromov, Phys. Lett. A 125 (1987) 1. [4] [5] [6]
H.P.W. Gottlieb, J. Sound Vib. 108 (1986) 63, 385 (Addendum). E.R. Lapwood, Geophys. 1. R. Astron. Soc. 40 (1975) 453. 0.H. Hald, Commun. Pure AppI. Math. 37 (1984)539.
PHYSICS LETTERS A
On spectral-spacing distributions for systems lacking asymptotics H.P.W. Gottlieb Division ofScience and Technology, Griffith University, Nathan, Queensland, 4111, Australia Received 3 June 1991; accepted for publication 19 July 1991 Communicated by J.P. Vigier
The detennination of an optimal number of level spacings utilized to obtain a representative distribution is discussed for spectra of a one-dimensional system for which there is no asymptotic expansion when a governing parameter is irrational.
The study of the statistics of eigenvalue spacings has received a good deal of attention in many fields including nuclear [1] and atomic [2] physics, especially in relation to possible chaotic behaviour. It has been remarked [3] that the statistical properties of spectra of even the one-dimensional Schrodinger equation with a barrier may have interesting properties, not often discussed, depending on the rationally or irrationality of a governing parameter. Even in the classical case of the spectrum of a vibrating string with a discontinuity in density [4], there are marked differences depending on whether a parameter is rational [5] (when the pattern of eigenvalues is repeated) or irrational [6] (when the eigenvalues, or any subclasses, do not have an asymptotic expansion). In this paper, we address the question: how large a number of eigenvalues need be taken in order to obtain a, representative histogram of the spacings’ distnbution; in particular, is there an optimal number of data values within a chosen upper limit. Usually, taking more values gives a better representation of a statistical distribution, but this is not so in the cases considered here. We analyse the eigenfrequencies of vibrating strings with fixed ends and a single step discontinuity in otherwise constant density. The equation to be solved is written in the form sin(a.,J~Q) cos[(l —a)Q] + cos ( a~iAQ) sin [(1
—
a )Q] = 0,
(1)
where a (0< a < 1) is the fractional length of string 370
with density p~,the remainder having density P2, and R =p~/p2; Q is a dimensionless frequency parameter [4]. The spectral spacings (differences of successive roots Q of (1)) may be computed for various values of a and R. Let r— R a 1 a 2 —
—
If r is rational (= N/M, N and M coprime integers), then there are harmonic and/or odd-harmonic frequency sub-spectra [41. The overall pattern of roots repeats itself [5], and the solutions fall into distinct classes each of which is an asymptotic expansion (cf. ref. [61). Then the pattern of spectral spacings will repeat itself. It can be shown that there are M+ N spacings of Q of the basic interval of length ME! (1 a). They exhibit symmetry about the centre of the interval, with ~= [(M+N) /2] + (3) —
different values (where [ ] + indicates “the least integer greater than or equal to”). Within a cycle, the spacings therefore repeat themselves or are “folded back” about the “centre” in reverse order, where the centre corresponds to a mode number, or between two mode numbers depending on whether N+ M is odd or even respectively. Ifr is irrational, then the solutions do not have any asymptotic expansion [6], and the spacings do not repeat themselves: there is a spectral-spacing distribution to be determined by computation of a large number of roots.
0375-9601/91/$ 03.50 © 1991 Elsevier Science Publishers B.V. All rights reserved.
Volume 158, number 8
PHYSICS LE~FERSA
When r is irrational, the spacings should nearly repeat in cycles given by integers corresponding to nearby rational approximations; and within an approximate cycle, the spacings should approximately repeat themselves about the centre in reverse order. This can be seen even for low integers in table 1, solving (1) for R = 2, a = 1/5, i.e. r= 1 /2~/~ which is successively better approximated by N/M= 1/3; 4/11; 5/14; 6/17; Corresponding approximate “reversing” cycles of lengths 4; 15; 19; 23 (= M+ N) may be discerned, centred about mode “numbers” (= [M+ N+ 1] /2) 2 ~ 8; 10; 12 respectively, It follows from this analysis that the distribution of spacings given by the first 23 solutions for r= 1 / 2,,J~should be some sort of indication of the actual distribution, not very precise because of the small mode numbers involved here, but better than taking, say, 30 solutions because that would contain only part of the next approximately repeating cycle based on 23. Actually, because of the symmetry about the “centre” of the cycle, even just the first p= 12 spacings (with weighting 1/2 for the 12th value) would give about as r,~asonablean indication of the distribution to this accuracy, with greater efficiency. Only when the number of spacings corresponding to the next best rational approximation (N= 23, M=65, p=44) is computed could a closer approximation to the actual distribution be expected. It should be emphasized that one always solves the equation containing the exact r, it is the choice ofthe optimal number of solutions to be included in the ....
Table 1 The first 23 eigenvalue spacings iS.Q of eq. (1) with r=l/2,/~ (a=l/5,R=2)formodenumbern (roundedoffto2decimal places here for ease of perusal, and “wrapped around” n= 12).
n
iS.Q
2 3 4 5 6 7 8 9 10 11 12
3.06 2.72 2.77 3.09 3.01 2.68 2.84 3.11 2.96 2.66 2.90 3.12
distribution that depends on the integers in the rational approximation to r. The general approach to the efficient accurate determination of the distribution of spacings of zeros of eq. (1) for irrational parameter r (eq. (2)) is therefore as follows: (i) Determine successively better rational approximations N/M for r. (ii) Determine the largest corresponding “halfcycle” number p (eq. (3)) not greater than the maximum number of solutions one wishes to compute based no doubt on accuracy and speed of the computer. (Preferably there should be a large gap to the next p value.) (iii) Plot a histogram, normalizing so that, in its bin, the last spacing values (mode numberp) is given weighting 1/2 if M+N is odd. [(ii’) If the equation to be considered, unlike (1), does not have the “folding” symmetry about the midpoint of a cycle, then one would simply look at the largest number v=M+N, without any non-unit weighting in (iii).] Unlike conventional sampling, there is nothing to be gained by taking a sample larger than a p-value, such as a convenient round power of 10, but less than the next p-value; taking a sample size between p-values would actually worsen the distribution picture. We illustrate the above procedure for the three cases listed in table 2. (The second case is the golden —
Table 2 Chi-square values for p level spacings distributed into 40 bins, compared with distribution for large number L spacings. Case I: a=l/5, R=2, r=l/2.~/~, L=4552. Case II: a=~J~—2,R=4, r=(~J~_l)/2 (golden mean) L=4l8i. Case III: a=(9— 3.,J~)/14, R= 1/9, r= 3_2.J~(Hald), L= 5333.
n 3.07 2.73 2.76 3.09 3.02 2.68 2.83 3.11 2.97 2.66 2.89
23 22 21 20 19 18 17 16 15 14 13
16 September 1991
casel
N/M 204/577 —
1189/3363 case II
377/610 610/987 —
1597/2584 case HI
204/1189 —
781/4552
x2
p 391 1000 2276
(nextp) (1495) (1495) (8714)
494 799 1000 2091
(799) (1292) (1292) (3383)
0.482 0.412 0.841 0.166
697 1000 2667
(2667) (2667) (3363)
0.724 3.547 0.413
0.972 i.928 0.099
371
Volume 158, number 8
PHYSICS LETTERS A
mean. The third case corresponds to the eigenvalue equation of Hald [61, which used erroneous density values: they should be squared there.) In each case the chi-squared value is computed for a distribution over 40 bins, with the base distribution taken as a large value ofL=N+Mlevels in the vicinity of.5000, depending on the actual value of r. Files of length ~ were read, with a weighting of only 1/2 for the last value if (N+M) corresponding to that p is odd, as explained above. In all cases, bin contents were at least 6, which is greater than the 5 usually given in statistical texts as the minimum desirable number, Table 2 gives the chi-squared values (39 degrees of freedom) for various approximations N/M to r, as well as for the round value of 1000 spacings. In each case, for better rational approximations the chisquared value decreases, indicating a better fit to the expected distribution. The last column corresponds toi~=4UL = [L/2] ÷.For 1000 spacings, the is larger than for the smaller p listed; e.g. for case I, taking the significantly smaller number of 391 spacings in fact gives a better fit. For case I, p= 2276 is highly optimal, since the next p value (8714) is so much larger. For case II (the golden mean: the “most irrational number”), the p sequence is Fibonacci number-like, and there are no spectacular jumps. It is now evident that fig. 3 in ref. [6] need not havetaken as many as the 1000 spacings used to build up the 20-bin histogram depicted there. Indeed, for r=3—2~handL=5333,with 2Obins, takingp=458 (corresponding to rational approximation N/M= 134/781) gives chi-squared value 0.283 compared with 0.850 for 1000 spacings. Thus this low p value would have yielded a more accurate distribution, with less computational effort. (Ref. [6] actually dealt with normalized solution minus mode number, rather than spacings.)
2
x
372
16 September 1991
Another application of our idea, now in a quantum-mechariical context, is to the above-mentioned work ofIvanovski et al. [31 which gives a histogram of level spacings of the first 800 states of the spectrum of the Schrodinger equation for a piecewise linear one-dimensional oscillator with a ô-function barrier, in the nonquasiclassicalregion, for a parameter q= 1 )/2 (golden mean) (fig. 2 in ref. [3]). For rational q=N/M, the spacings are repeated almost periodically with a period N+ M states [3]. This situation concerning a governing parameter is similar to what we have been discussing in this paper (see step (ii’) above). The rational approximation N/M= 233/377 gives N+M= 610, with the next N+ M= 987. Thus our work indicates that taking only 610 spacings should give a better representation of the statistical distribution than the “round” larger but unrelated number of 800 values of ref. [3]. Our procedure should have implications for determining statistical distributions for other systems governed by a parameter for which there is no asymptotic expansion for the spectral levels for irrational parameter value but for which the solution is found to have a repetitive pattern when the parameter is rational, with period determined in terms of the integers involved. (\/‘~—
References [110. Bohigas and H.A. Weidenmuller, Annu. Rev. Nucl. Part. Sci. 38 (1988) K.T. 421. Taylor and G. Wunner, Comm. At. Mol. [2] T.S. Monteiro, Phys.25 (1991) 253. [3] G. Ivanovski, N. Novkovski and V. Uromov, Phys. Lett. A 125 (1987) 1. [4] [5] [6]
H.P.W. Gottlieb, J. Sound Vib. 108 (1986) 63, 385 (Addendum). E.R. Lapwood, Geophys. 1. R. Astron. Soc. 40 (1975) 453. 0.H. Hald, Commun. Pure AppI. Math. 37 (1984)539.