Entropy of Hermite polynomials with application to the harmonic oscillator

Transcription

1 Entropy of Hermite polynomials with application to the harmonic oscillator Walter Van Assche DedicatedtoJeanMeinguet Abstract We analyse the entropy of Hermite polynomials and orthogonal polynomials for the Freud weights wx =exp x m onrand show how these entropies are related to information entropy of the one-dimensional harmonic oscillator. The physical interest in such entropies comes from a stronger version of the Heisenberg uncertainty principle, due to Bialynicki-Birula and Mycielski, which is expressed as a lower bound for the sum of the information entropies of a quantum-mechanical system in the position space and in the momentum space. The information entropies of the harmonic oscillator The Schrödinger equation in D dimensions is given by + V ψ = Eψ, where the potential V and the wave function ψ are functions of x =x,...,x D and E is the energy. The wave function ψ is normalized in such a way that ρx = ψx is a probability density in position space. If ˆψp is the Fourier transform of the wave function ψ, then by the Plancherel formula γp = ˆψp is also a probability density, but now in the momentum space. The information entropy for the quantummechanical system with potential V is then given by S ρ = ρxlogρx dx R D The author is a Senior Research Associate of the Belgian National Fund for Scientific Research NFWO. This work is supported by INTAS project Mathematics Subject Classification : 33C45. Key words and phrases : Hermite polynomials, Freud weights, information entropy, harmonic oscillator.

2 86 W. Van Assche in the position space, and S γ = γplogγp dp R D in the momentum space. These two entropies have allowed Bialynicki-Birula and Mycielski [4] to find a new and stronger version of the Heisenberg uncertainty principle. For a quantum mechanical system in D dimensions this new uncertainty relation is. S ρ + S γ D + log π, which expresses in a quantitative way that it is impossible to get precise information in both position and momentum space. High values of S ρ are associated with low values of S γ,andviceversa. In this paper we restrict ourselves to the one-dimensional harmonic oscillator, in which case D = and the potential has the form Thepossibleenergylevelsare V x = λ x. E n = n +, n =0,,,..., and the wave functions are Hermite functions e λx / H n x λ, where H n n = 0,,,... are Hermite polynomials [0]. This gives λ/π ρ n x = n n! e λx Hnx λ, and for Hermite functions, the Fourier transform is again a Hermite function, hence γ n p = λπn n! e p /λ H np/ λ. The entropy in the position space then becomes. S ρn =log π/λ +log n n!+ πn n! and the entropy in the momentum space is.3 S γn =log πλ +log n n!+ πn n! y H n ye y dy H nylogh ny e y dy, y H πn n! nye y dy Hn πn n! ylogh n y e y dy.

3 Entropy of orthogonal polynomials 87 The problem thus consists of computing integrals of the form y H nye y dy and entropy integrals of the Hermite polynomials H nye y log H ny dy. We will consider a more general situation involving integrals for orthogonal polynomials p n x n =0,,... for the weight function wx =e x m on R. This weight function is known as a Freud weight and the case m = corresponds to Hermite polynomials. Freud studied these weights and the corresponding orthogonal polynomials in detail for m =, 4, 6 and formulated conjectures regarding the asymptotic behaviour of the largest zero of the orthogonal polynomials and the behaviour of the coefficients in the three-term recurrence relation for the orthonormal polynomials. If x n,n is the largest zero of p n, then Freud conjectured that /m.4 lim n n /m x n,n =, λ m where λ m = Γ m+ π Γ m = m x m π dx. x If the three-term recurrence relation for the orthonormal polynomials is xp n x =a n+ p n+ x+a n p n x, and the orthonormal polynomials are given by p n x =γ n x n +,withγ n > 0, then a n = γ n /γ n and Freud conjectured that.5 lim n /m a n = lim γ n n n /m n = /m. γ n λ m The conjecture regarding the largest zero behaviour was proved by Rakhmanov [8], the conjecture regarding the behaviour of the recurrence coefficients was proved by Alphonse Magnus [] for even integer values of m, and by Lubinsky, Mhaskar, and Saff [0] for arbitrary m>0. For more on these Freud conjectures we refer to Nevai [6] or to [, 4.]. For orthogonal polynomials with respect to Freud weights the quantities to be computed are.6 x m p n xe x m dx and.7 p nxe x m log p nx dx. Usually we will take the orthonormal polynomials in these integrals. For small n these integrals give information related to the ground state and a few excited states,

4 88 W. Van Assche the asymptotic behaviour as n gives information about the so called Rydberg states. The analytic computation of the integral in.6 is easy. By symmetry we have x m p nxe x m dx = x m p nxe xm dx, 0 and integration by parts gives By symmetry again this is x m p nxe x m dx = m = m x m p n xe x m dx = m 0 0 xp nxe xm dx [p nx+xp n xp nx]e xm dx. [p n x+xp nxp n x]e x m dx Observe that we can write xp n x =np nx+π n x, where π n is a polynomial of degree at most n. Then by orthonormality we finally get.8 x m p n xe x m dx = +n m. For orthonormal Hermite polynomials p n x = n n! π / H n x wechoosem = and find y Hn ye y dy =n + n n! π. Observe that for the ground state n = 0 this already gives since H 0 = S ρ0 =log π/λ +, S γ 0 =log πλ +, so that S ρ0 + S γ0 =+logπ, and thus the Bialynicki-Birula-Mycielski inequality. is sharp in this case. The evaluation of the entropy integral.7 is much more difficult and requires more knowledge of Hermite polynomials and orthogonal polynomials with Freud weights. A heuristic approach using logarithmic potentials If we denote the zeros of p n by x,n <x,n < <x n,n,thenp n x =γ n nj= x x j,n. The entropy integral.7 then becomes p nxe x m log p nx dx =logγ n + n p nxe x m j= log x x j,n dx. If we denote by Uz; µ = log z x dµx

5 Entropy of orthogonal polynomials 89 the logarithmic potential of a probability measure µ, andbyµ n the measure with density p nxe x m,thenµ n is a probability measure and Uz; µ n = p n xe x m log z x dx, so that the entropy integral is also given by p n xe x m n log p n x dx =logγ n Ux j,n ; µ n. j= This connection with the potential of µ n indicates that we can get an idea of the asymptotic behaviour of the entropy integral in.7 if we know the asymptotic behaviour of γ n, the weak convergence of the measures µ n and the asymptotic distribution of the zeros x j,n j n. This information is available in the literature. For the asymptotic behaviour of γ n we have the result that lim n n/m γn /n = eλ m /m, which follows from and is weaker than the Freud conjecture.5. For the behaviour of the measures µ n we have lim f λ m n n /m x p nxe x m dx = fx π dx = fx dµ e x, x where µ e is the equilibrium measure of the interval [, ] and f is an arbitrary continuous function of at most polynomial growth at ± [6]. Finally, for the distribution of the zeros we have lim n n n f j= λ m n /m x j,n = fxv m x dx, where f is a continuous function of at most polynomial growth at ± and v m is the Ullman weight v m x = m π x t m dt, x [, ]. t x This result was proved by Rakhmanov [8] and also follows from the Freud conjecture.5 see [, p. 3]. Observe that for m = the Ullman density is v x = π x, x [, ], which is also known as the semi-circle density and which is well known to describe the asymptotic distribution of eigenvalues of some random matrices Wigner [3]. For m the Ullman density tends to the density of the measure µ e v x = π, x [, ]. x

6 90 W. Van Assche Observe that Uz; µ n = log z/c n x/c n p nxe x m dx log c n = Uz/c n ;ˆµ n log c n, where ˆµ n is the rescaled measure with ˆµ n A =µ n c n A. Then the entropy integral becomes p n nxlogp nx e x m dx = log γ /n n n U n j= λ m n /m x j,n ;ˆµ n log λ m As n the right hand side is of the form log + log eλ m /m Ux; µ e v m x dx log λ m /m, n /m. and since the potential of the equilibrium measure µ e has the constant value log on [, ], this quantity reduces to /m. Hence, one expects to have lim p n n n xlogp n x e x m dx = m. This is, however, a heuristic reasoning: the limit lim n n n U j= λ m n /m x j,n ;ˆµ n = Ux; µ e v m x dx does not follow immediately from the results given earlier. In fact the weak convergence of the measure ˆµ n and the asymptotic distribution of the zeros only gives lim inf n n n U j= λ m n /m x j,n ;ˆµ n Ux; µ e v m x dx see [7, Thm.. on p. 68], hence more work is needed to get the required result. Furthermore, this only shows that the entropy integral.7 behaves like n/m + on, and we would like know the term on in more detail. 3 Entropy of Hermite polynomials Let us consider the entropy integral for orthonormal Hermite polynomials p n ylogp n y e y dy, where p n x = n n! π / H n x. It is more convenient to consider E n = p n ye y log[p n ye y ] dy,

7 Entropy of orthogonal polynomials 9 since we already know exactly the integral p nye y log e y dy which is given in.8. It is known that the weighted polynomial e x / H n x attains its extremum on the finite interval [ n +, n + ] see [0, Sonin s theorem in 7.3 combined with 7.6]. Moreover the successive relative maxima of e x / H n x form an increasing sequence for x 0 [0, Thm on p. 77], hence the maximum of e x Hn x will be attained near the turning point x n = n +. We can use the asymptotic formula near this turning point [0, p. 0] e x / H n x = 3/3 n/+/4 n! [At+On /3 ], π 3/4 n / where x = x n / 3 /3 n /6 t and A is the Airy function, which is defined as the only bounded solution up to a constant factor of the differential equation y + xy/3 = 0, to find that max <x< e x p n x = max e x p x n x =On /6. n x x n Therefore, we split up the integration as xn E n = p n ye y log[p n ye y ] dy + p n ye y log[p n ye y ] dy x n x >x n :=E,n + E,n. Here E,n will be the dominating term and E,n will be small compared to E,n.For E,n we can use the Plancherel-Rotach asymptotics [0, Thm. 8..9] [ e x / n n! n + H n x = sin [sin θ θ]+ 3π ] + O/n, 4 4 πn/sinθ which holds for x = n +cosθ and ɛ θ π ɛ,withɛ>0. For the orthonormal polynomials this is [ e x p n x = n + π sin [sin θ θ]+ 3π n sin θ 4 4 ] + O/n, for x = n +cosθ and ɛ θ π ɛ. Withthiswehave E,n = π sin n + θ θ sin + 3π π 0 4 log π sin n + [sin θ θ]/+ 3π 4 dθ [ + O/n]. n sin θ The integral consists of three terms E = π π 0 sin n + θ θ sin + 3π 4 dθ log π, n

8 9 W. Van Assche E = π π 0 sin n + θ θ sin + 3π 4 log sin n + θ θ sin + 3π 4 dθ, and E 3 = π sin n + θ θ sin + 3π log sin θdθ. π 0 4 For the integral in E we have π sin n + θ θ sin + 3π dθ = π sin φdφ+ O/n, π 0 4 π 0 so that E = log π + O/n. n For E and E 3 we can use a lemma from [, Lemma.] to evaluate these integrals asymptotically: Lemma. Let g be a continuous real function which is periodic with period π, f L [0,π] and γ a measurable function, almost everywhere finite on [0,π]. Then π lim gnθ + γθfθ dθ = π gθ dθ π fθ dθ. n π 0 π 0 π 0 Observe that [sinθ θ]/ is an increasing function, mapping [0,π] to[0,π]. By a change of variables, we can then use this lemma to find lim E = π sin φ log sin φdφ= log, n π 0 and lim E 3 = π sin φdφ π log sin θdθ=log. n π 0 π 0 Together this gives E,n = log π log n ++o. The remaining term E,n can be shown to be of smaller order, so that finally p nye y log[p nye y ] dy = log π log n ++o. Together with.8 this gives p n ye y log p n y dy = n + 3 log π log n + o.

9 Entropy of orthogonal polynomials 93 4 Entropy of orthogonal polynomials with Freud weights An asymptotic formula of Plancherel-Rotach type for orthonormal polynomials with Freud weight e x m was conjectured by Nevai in [6], and this conjecture was proved by Rakhmanov [9]. He showed that p n xe x m = π cos Φ n x [ + o], x n x for x ɛx n,wherex n =n/λ m /m and n + xn Φ n x = xm dt x t x m n t π 4, x [0,x n]. With the scaling x = x n y this gives n + Φ n x n y= ym dt y t π m t 4 n + := φy π 4, so that p n x nye n y m /λ m = cos [n + φy π/4] [ + o], y ɛ. πx n y Observe that for m =wehave φcos θ = sin θ θ, so that this is compatible with the Plancherel-Rotach formula for Hermite polynomials, taking into account that now we use x n = n rather than x n = n +. A similar analysis as in the previous section is now possible, but we will not give the details, which can be found in []. It suffices to say that with this analysis one again has lim p n n nxe x m log p nx dx = m, as we obtained by heuristic reasoning in Section. A more accurate formula can be obtained in a similiar way as for the Hermite polynomials, using some of the results obtained by Rakhmanov [8], [9] and Lubinsky and Saff [9], [], [5]. This has been done in [], where it was proved that p nxe x m log p nx dx = n + m log n m + m log πγm/ + log π + o. Γm +/ Observe that this is compatible with the case m = for Hermite polynomials. Of independent interest for the asymptotic behaviour of orthogonal polynomials with Freud weights, would be to find an asymptotic formula in the neighborhood of

10 94 W. Van Assche the turning point x n =n/λ m /m, similar to the Airy type asymptotics for Hermite polynomials. Such an asymptotic formula is not known yet, but the existence of such a formula is made plausible by an asymptotic formula for the largest zeros of certain Freud polynomials. Máté, Nevai and Totik [3] [4] proved that for positive even integers m the largest zeros of the orthogonal polynomials p n x for the weight function e x m satisfy /m n /m i k x n k+,n = + on /3, λ m 6n m /3 where i <i < are the zeros of the Airy function. 5 Conclusion The described analysis more details are given in [] gives for the orthonormal Hermite polynomials p n x = n n! π / H n x asn p nxlogp nx e x dx = n +log n 3 +logπ + o. In terms of the information entropies. and.3 this gives S ρn + S γn =logn + log + log π + o. This shows that the sum S ρn + S γn grows like log n as n. The result described in this paper is a joint effort of J. S. Dehesa and R. J. Yáñez from Granada Spain, A. I. Aptekarev and V. S. Buyarov from Moscow Russia, and the the present author. Together we also examined the harmonic oscillator in D > dimensions and the hydrogen atom in D dimensions. For these quantum mechanical systems the information entropy in position space and in momentum space can be expressed in terms of the entropy of Laguerre polynomials, and the angular part also requires the entropy of Gegenbauer polynomials. For these classical orthogonal polynomials we were able to obtain similar results as those described in the present paper. See [] [3], [5], [], and [4] for these extensions. References [] A. I. Aptekarev, J. S. Dehesa, R. J. Yáñez Spatial entropy of central potential and strong asymptotics of orthogonal polynomials, J. Math. Phys , [] A.I.Aptekarev,V.Buyarov,J.S.DehesaAsymptotic behaviour of L p -norms and entropy for general orthogonal polynomials, RAN Mat. Sb , 3 30 Russian Russian Acad. Sci. Sb. Math , [3] A.I.Aptekarev,V.S.Buyarov,W.VanAssche,J.S.DehesaAsymptotics of entropy integrals for orthogonal polynomials, Doklady Akad. Nauk , Russian Doklady Math , 47 49

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