Universal Relations in Asymptotic Formulas for Orthogonal Polynomials

Abstract

Orthogonal polynomials $P_{n}(\lambda)$ are oscillating functions of $n$ as $n\to\infty$ for $\lambda$ in the absolutely continuous spectrum of the corresponding Jacobi operator $J$. We show that, irrespective of any specific assumptions on the coefficients of the operator $J$, the amplitude and phase factors in asymptotic formulas for $P_{n}(\lambda)$ are linked by certain universal relations found in the paper. Our proofs rely on the study of a time-dependent evolution generated by suitable functions of the operator $J$.

Appendix A. An Elementary Inequality

As a by-product of our considerations, we obtain a simple inequality for functions in the Sobolev space ${\sf H}^1({\mathbb{R}})$. First, we note that

$$| u(0)|^2 \leq c_{0} \int_{0}^{1}(| u'(x)|^2 + | u(x)|^2)\,dx, $$

(A.1)

where we can choose $c_{0} =2(\sqrt{5}-1)^{-1}$. The following statement holds.

Proposition A.1.

Let a sequence $x_{n}\in {\mathbb{R}}$ , $n\in {\mathbb{Z}}$ , be such that $x_{n} \to \pm \infty$ as $n\to \pm \infty$ and $x_{n}< x_{n+1}< x_{n}+ \delta$ for some $\delta>0$ and all $n\in {\mathbb{Z}}$ . Then

$$\sum_{n\in {\mathbb{Z}}} (x_{n+1}-x_{n}) |u(x_{n})|^2 \leq c_{0}\max\{1, \delta^2 \}\int_{-\infty}^\infty (| u'(x)|^2 + | u(x)|^2)\,dx $$

(A.2)

for all $u\in{\sf H}^1({\mathbb{R}})$ .

Proof.

Applying estimate (A.1) to the function $u_{\varepsilon} (x)= u(\varepsilon x)$ and making the change of variable $y=\varepsilon x$, we see that

$$| u(0)|^2 \leq c_{0} \int_{0}^{1}(\varepsilon^2 | u '(\varepsilon x)|^2 + | u (\varepsilon x)|^2) \,dx =c_{0} \int_{0}^{\varepsilon}(\varepsilon | u'(y)|^2 + \varepsilon^{-1} | u (y)|^2)\,dy,$$

whence

$$\varepsilon | u(x_{n})|^2 \leq c_{0} \int_{x_{n}}^{x_{n} +\varepsilon}(\varepsilon^2 | u'(y)|^2 + | u (y)|^2)\,d y. $$

(A.3)

Let us set $\varepsilon =x_{n+1} -x_{n}$. Then $\varepsilon\leq\delta$, and (A.3) yields

$$(x_{n+1} -x_{n}) | u(x_{n})|^2 \leq c_{0} \max\{1, \delta^2 \} \int_{x_{n}}^{x_{n+1} }(| u'(y)|^2 + | u (y)|^2)\,d y.$$

It remains to take the sum of these estimates over all $n\in {\mathbb{Z}}$. $\Box$

Inequality (A.2) is quite elementary, but, surprisingly, we were unable to find it in the literature. Of course, (A.2) implies inequalities (3.11) and (3.12).

Appendix B. Dispersionless Evolution

If

$$\lim_{n\to\infty} (x_{n+1} -x_{n})=0 $$

(B.1)

(which excludes the linear growth of $x_{n}$) and the phases $\Phi_{n}$ do not depend on $\lambda$, then we can dispense with the stationary phase method. Here we consider the evolution (4.16) for the case $\Theta(\lambda)=\omega(\lambda)$, which was excluded in Section 4.2. Making the change of variables (4.20), we can rewrite (4.16) as

$$(e^{-i \omega (J) t} u)_{n}= \sqrt{2\pi} v_{n} (e^{i\Phi_{n}} \widehat {F} (x_{n}-t) + e^{-i\Phi_{n}} \widehat {F} (-x_{n}-t))+ r_{n} (t), $$

(B.2)

where $\widehat {F} (x)$ is the Fourier transform of $F(\mu)$. The remainder term $r_{n} (t)$ is given by (4.17); it is negligible according to (4.18). Since $\widehat {F} (-x)=O(x^{-k})$ as $x\to+ \infty$ for all $k$, the term with $\widehat {F} (- x_{n}-t)$ in (B.2) is also negligible. Thus, instead of Theorem 4.5, we have the following theorem.

Theorem B.1.

Let Assumption 4.4 be satisfied, and let $f\in C_{0}^\infty (\Lambda)$. Then

$$(e^{-i \omega (J) t} u)_{n}= \sqrt{2\pi}\,v_{n} e^{i\Phi_{n}} \widehat {F} (x_{n}-t) + r_{n} (t), $$

(B.3)

where the remainder term $r_{n} (t)$ obeys condition (4.18).

It follows from (B.3) that

$$\| e^{-i\omega (J) t} u\|^2= 2\pi \sum_{n\in {\mathbb{Z}}_{+}}v_{n}^2 | \widehat {F} (x_{n}-t)|^2+ o(1), \qquad t\to +\infty,$$

and hence according to (4.23) (where $\Theta=\omega$),

$$2\pi \lim_{t\to + \infty } \sum_{n\in {\mathbb{Z}}_{+}}v_{n}^2 | \widehat {F} (x_{n}-t)|^2 = \| f\|^2. $$

(B.4)

Note that formula (4.22) is not true in the case under consideration, because if $\Theta(\lambda)=\omega(\lambda)$, then $\theta(\lambda)=\lambda$, so that $\theta'' (\lambda)=0$ for all $\lambda\in{\mathbb{R}}$.

On the other hand, one can calculate the left-hand side of (B.4) by using Lemma 4.10. Indeed, let us set $X_{n} (t)=x_{n } -t$ and observe that $v_{n}^2= \sigma_{n} (X_{n+1} (t) -X_{n} (t))$ by definition (3.10). Suppose that the sequence $\sigma_{n}$ has a finite limit $\sigma$. Then, according to Proposition A.1, for each $R>0$, the sum in (B.4) over $n$ such that $|X_n (t)|\geq R$ is estimated by $C \int_{| x |\geq R} (| \widehat {F}'(x)|^2 + | \widehat {F}(x)|^2)d x$. In view of (B.1) condition (4.26) is satisfied. Therefore, according to (4.27), the sum in (B.4) over $n$ such that $|X_n (t)| < R$ converges to $\sigma \int_{-R}^R | \widehat {F}(x) |^2 d x$ as $t\to\infty$. It follows that the left-hand side of (B.4) equals $2\pi \sigma \| \widehat {F}\|^2 = 2\pi \sigma \| F\|^2$, so that $2\pi \sigma \| F\|^2 = \| f\|^2$. This yields (4.32), whence the relations (4.30) and $\sigma>0$ follow.