[Paper Review] Integrals of products of Hermite functions

Weimin Wang|ArXiv.org|Jan 26, 2009

Polynomial and algebraic computation2 references4 citations

TL;DR

This paper computes integrals of products of four normalized Hermite functions using generating functions and orthogonal polynomial identities, deriving sharp asymptotic bounds for the resulting coupling coefficients. The key result establishes that these integrals decay like $ \frac{1}{\sqrt{j+k}} e^{-\frac{(j-k)^2}{3(j+k)}} $ for large $ j+k $, with optimal polynomial corrections, extending prior stability results for harmonic oscillator perturbations.

ABSTRACT

We compute the integrals of products of Hermite functions using the generating functions. The precise asymptotics of 4 Hermite functions are presented below. This estimate is relevant for the corresponding cubic nonlinear equation.

Motivation & Objective

• To compute the integrals $ W_{jpqk} = \int_{-\infty}^\infty h_j(x)h_p(x)h_q(x)h_k(x)\,dx $ for arbitrary non-negative integers $ j,p,q,k $.
• To generalize the asymptotic estimate for $ W_{j00k} $ previously used in stability analysis of the harmonic oscillator under time-dependent perturbations.
• To establish uniform bounds on $ |W_{jpqk}| $ that capture the optimal Gaussian decay and polynomial prefactor for all $ j,k $, with dependence on $ p,q $.
• To show that the Schur norm of the coupling operator is of the same order as the operator norm, enabling extension of stability results to potentials with exponentially decaying Hermite coefficients.

Proposed method

• Use of generating functions for Hermite polynomials to express products $ H_j(x)H_k(x) $ and $ H_p(x)H_q(x) $ as linear combinations of $ H_\ell(\sqrt{2}x) $.
• Expansion of the product of four Hermite functions into a sum over $ \ell $, with coefficients derived from the overlap of the two polynomial expansions.
• Application of the orthogonality relation $ \int H_\ell^2(\sqrt{2}x) e^{-2x^2} dx = 2^{\ell - 1/2} \ell! \sqrt{\pi} $ to evaluate the integral.
• Use of Stirling's approximation to derive asymptotic estimates for the factorial ratios in the coefficients.
• Estimation of binomial-like sums via bounds on $ \sum_r (-1)^r C_k^r C_j^{\ell - r} $, exploiting generating function identities and binomial coefficient decay.
• Derivation of uniform bounds by controlling the sum over $ \ell $, using exponential and factorial decay to dominate the $ X = \frac{j-k}{\sqrt{j+k}} $ dependence.

Experimental results

Research questions

• RQ1 What is the precise asymptotic behavior of the integral $ W_{jpqk} = \int_{-\infty}^\infty h_j h_p h_q h_k \,dx $ for large $ j+k $, with fixed $ p,q $?
• RQ2 How does the decay rate of $ |W_{jpqk}| $ depend on the difference $ j-k $ and the sum $ j+k $?
• RQ3 Can the bound for $ W_{j00k} $ be generalized to arbitrary $ p,q $, and does the same Gaussian decay structure persist?
• RQ4 What is the optimal polynomial prefactor in front of the Gaussian decay, and is it sharp?
• RQ5 Does the Schur norm of the coupling operator scale like the operator norm, and what does this imply for stability of the harmonic oscillator?

Key findings

• The integral $ W_{jpqk} $ vanishes when $ j+p+q+k $ is odd, due to parity selection rules.
• For $ \frac{j-k}{\sqrt{j+k}} \geq \sqrt{p+q} $, the bound $ |W_{jpqk}| \lesssim \frac{C_{p,q}}{\sqrt{j+k}} e^{-\frac{(j-k)^2}{3(j+k)}} $ holds, with $ C_{p,q} \leq a^{p+q} $ for some $ a>1 $.
• The leading-order decay is $ \sim \frac{1}{\sqrt{j+k}} e^{-\frac{(j-k)^2}{2(j+k)}} $, matching the $ p=q=0 $ case, and the polynomial prefactor is optimal.
• The Schur norm of the coupling operator is of the same order as the operator norm, implying that the decay is sufficient for stability in nonlinear Schrödinger equations.
• The result extends the stability analysis in [W] to potentials with exponentially decaying Hermite coefficients, as the decay rate ensures diminishing spatial influence for high modes.
• The method via generating functions and orthogonal polynomial expansions yields a systematic framework for computing arbitrary products of Hermite functions.