# Universal relations in asymptotic formulas for orthogonal polynomials

D. R. Yafaev  Univ Rennes, CNRS, IRMAR-UMR 6625, F-35000 Rennes, France and
SPGU, Univ. Nab. 7/9, Saint Petersburg, 199034 Russia<EMAIL_ADDRESS>To the memory of Michail Zakharovitch Solomyak on the occasion of his 90th
anniversary

###### 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 coefficients of the operator $J$, amplitude and phase factors
in asymptotic formulas for $P_{n}(\lambda)$ are linked by certain universal
relations found in the paper.

Our approach relies on a study of operators diagonalizing Jacobi operators.
Diagonalizing operators are constructed in terms of orthogonal polynomials
$P_{n}(\lambda)$. They act from the space $L^{2}(\mathbb{R})$ of functions
into the space $\ell^{2}({\mathbb{Z}}_{+})$ of sequences. We consider such
operators in a rather general setting and find necessary and sufficient
conditions of their boundedness.

###### Key words and phrases:

Jacobi matrices, orthogonal polynomials, integral operators, oscillating
kernels.

###### 2000 Mathematics Subject Classification:

33C45, 47B38

Supported by RFBR grant No. 17-01-00668 A

##  1\. Introduction

###  1.1. Main result

Jacobi operators $J$ acting in the space $\ell^{2}({\mathbb{Z}}_{+})$ are
diagonalized (see, e.g., the book [1]) by unitary operators ${\bf
U}:L^{2}({\mathbb{R}};d\rho)\to\ell^{2}({\mathbb{Z}}_{+})$ ($d\rho$ is the
spectral measure of $J$) constructed in terms of the corresponding orthonormal
polynomials $P_{n}(\lambda)$:

$({\bf U}{\bf f})_{n}=\int_{\mathbb{R}}P_{n}(\lambda){\bf
f}(\lambda)d\rho(\lambda),\quad
n\in{\mathbb{Z}}_{+}=\\{0,1,\ldots\\},\quad{\bf f}\in
L^{2}({\mathbb{R}};d\rho).$ (1.1)

We are interested in an asymptotic behavior of the polynomials
$P_{n}(\lambda)$ as $n\to\infty$ for $\lambda$ in the absolutely continuous
spectrum of $J$. Typically, $P_{n}(\lambda)$ are oscillating functions such
that

$P_{n}(\lambda)=2\kappa(\lambda)n^{-r}\cos\big{(}\omega(\lambda)n^{s}+\Phi_{n}(\lambda)\big{)}+O(n^{-\delta})$
(1.2)

where $s\in(0,1]$, $r\geq 0$, $\delta>1/2$, $\omega^{\prime}(\lambda)>0$ and
$\Phi_{n}^{\prime}(\lambda)=o(n^{s})$ as $n\to\infty$. A more general
asymptotic formula of this type is given in Assumption 4.4.

Our main goal is to show that, irrespective of specific assumptions about
matrix elements of the operator $J$, or about the corresponding spectral
measure

$d\rho(\lambda)=\tau(\lambda)d\lambda,\quad\tau(\lambda)>0,$ (1.3)

the asymptotic coefficients in formula (1.2) are linked by the universal
relations

$\boxed{2r+s=1}$ (1.4)

and

$\boxed{2\pi\tau(\lambda)\kappa^{2}(\lambda)=s\omega^{\prime}(\lambda).}$
(1.5)

Given unitarity of the operators (1.1), we deduce these relations from
asymptotic formulas for time dependent evolution $e^{-i\Theta(J)t}f$ as
$t\to\infty$ for suitable functions $\Theta(\lambda)$. These results were
announced in [21]; their proofs are given in Sect. 4.

###  1.2. Boundedness of semidiscrete Fourier transforms

Unitarity (and hence boundedness) of operators (1.1) are consequences of the
orthogonal polynomials theory. In Sect. 3, we discuss sufficiently arbitrary
semidiscrete (that is, acting from a space of functions into a space of
sequences) Fourier operators $V$ imitating their structure and find conditions
of their boundedness by direct tools. A typical result about such operators is
stated as Theorem 1.1. Below $C$ and $c$ are various positive constants whose
precise values are irrelevant.

######  Theorem 1.1.

Let an operator $V:L^{2}({\mathbb{R}})\to\ell^{2}({\mathbb{Z}}_{+})$ be
defined by the formula

$(Vf)_{n}=v_{n}\int_{-\infty}^{\infty}e^{ix_{n}\lambda}w(\lambda)f(\lambda)d\lambda$
(1.6)

where $v_{n}\in{\mathbb{R}}$, $x_{n}\in{\mathbb{R}}$, $x_{n}<x_{n+1}$ and
$x_{n}\to\infty$. If

$\sup_{R>0}\sum_{n:x_{n}\in(R,R+1)}v_{n}^{2}<\infty$ (1.7)

and

$|w(\lambda)|~\leq C(1+|\lambda|)^{-1},$ (1.8)

then the operator $V:L^{2}({\mathbb{R}})\to\ell^{2}({\mathbb{Z}}_{+})$ is
bounded. Conversely, if $V$ is bounded for $w$ being the characteristic
function of some interval, then condition (1.7) is satisfied.

Note that (1.7) holds true, if

$x_{n}=n^{s},\quad s\in(0,1]\quad\mbox{and}\quad|v_{n}|~\leq
C(1+n)^{-r}\quad\mbox{with}\quad 2r\geq 1-s.$ (1.9)

Let us mention two particular cases of Theorem 1.1. Suppose that $w$ is a
bounded function with compact support. If $s=1$, $r=0$, then the boundedness
of the operator $V$ follows from the Parseval identity for the exponentials
$e^{in\lambda}$. The case $s=1/2$, $r=1/4$ can be deduced (see Sect. 2.4) from
classical results on Hermite (or Laguerre) polynomials. It is however clear
that

$1^{0}$ such statements should be true for sufficiently arbitrary sequences
$x_{n}$ and under less stringent assumptions on the function $w$

$2^{0}$ the theory of orthogonal polynomials is irrelevant for a study of
operators (1.6).

Our goal in Sect. 3 is to justify these conjectures. It is convenient to state
the problem in a more general setting. Let $\Phi$,

$(\Phi
f)(x)=(2\pi)^{-d/2}\int_{{\mathbb{R}}^{d}}e^{ix\xi}f(\xi)d\xi=:\hat{f}(x)$
(1.10)

be the (adjoint) Fourier transform, and let $W$ be the operator of
multiplication by a bounded function $w(\xi)$. We find in Theorem 3.2
generalizing Theorem 1.1 necessary and sufficient conditions of boundedness of
the operator

$A=\Phi W:L^{2}({\mathbb{R}}^{d})\to L^{2}({\mathbb{R}}^{d};dM)$ (1.11)

where $dM(x)$ is some measure on ${\mathbb{R}}^{d}$. This is of course a well
studied problem; nevertheless, Theorem 3.2 seems to be new. Note that the
boundedness of operator $A$ for $w(\xi)=(1+|\xi|)^{-\ell}$ is equivalent to
the embedding of the Sobolev space ${\sf H}^{\ell}({\mathbb{R}}^{d})$ into
$L^{2}({\mathbb{R}}^{d};dM)$.

Let us however mention an important relevant result. Obviously, the operator
$v\Phi w$ is bounded in the space $L^{2}({\mathbb{R}}^{d})$ if $v,w\in
L^{\infty}({\mathbb{R}}^{d})$. The case of functions $v$ and $w$ with local
singularities was studied in the paper [4] by M. Sh. Birman, G. E. Karadzhov
and M. Z. Solomyak. The conditions of boundedness of such operators $v\Phi w$
generalize classical Hardy inequalities.

##  2\. Jacobi operators and orthogonal polynomials

###  2.1. Basic facts

Let us briefly recall necessary results about Jacobi operators $J$. They act
in the space $\ell^{2}({\mathbb{Z}}_{+})$ by the formula

$(Ju)_{n}=a_{n-1}u_{n-1}+b_{n}u_{n}+a_{n}u_{n+1},\quad
n\in{\mathbb{Z}}_{+},\quad a_{-1}=0,$

where the sequences $a_{n}>0$ and $b_{n}=\bar{b}_{n}$ are given and
$u=\\{u_{n}\\}_{n=0}^{\infty}$. A minimal symmetric operator $J_{0}$ is
defined on a set of vectors with only a finite number of non-zero components
$u_{n}$. The spectra of all self-adjoint extensions $J$ of $J_{0}$ are simple
with $e_{0}=(1,0,0,\ldots)^{\top}$ being a generating vector. The spectral
measure of $J$ is defined by the relation
$d\rho_{J}(\lambda)=d(E_{J}(\lambda)e_{0},e_{0})$ where $E_{J}(\lambda)$ is
the spectral family of the operator $J$. If the Carleman condition

$\sum_{n=0}^{\infty}a_{n}^{-1}=\infty$ (2.1)

is satisfied, then the operator $J_{0}$ is essentially self-adjoint.

Orthonormal polynomials $P_{n}(z)$ can be defined as solutions of the
difference equation

$a_{n-1}u_{n-1}(z)+b_{n}u_{n}(z)+a_{n}u_{n+1}(z)=zu_{n}(z),\quad
n\in{\mathbb{Z}}_{+},\quad z\in{\mathbb{C}},$ (2.2)

satisfying the boundary condition $P_{0}(z)=1$. Clearly, $P_{n}(z)$ is a
polynomial of degree $n$: $P_{n}(z)=p_{n}z^{n}+\cdots$ with $p_{n}>0$. It can
be shown that these polynomials are orthogonal and normalized in the spaces
$L^{2}({\mathbb{R}};d\rho)$:

$\int_{-\infty}^{\infty}P_{n}(\lambda)P_{m}(\lambda)d\rho(\lambda)=\delta_{n,m};$

as usual, $\delta_{n,n}=1$ and $\delta_{n,m}=0$ for $n\neq m$. Here
$d\rho=d\rho_{J}$ is the spectral measure of an arbitrary self-adjoint
extension $J$ of the minimal operator $J_{0}$. Alternatively, given a
probability measure $d\rho(\lambda)$, the polynomials
$P_{0}(\lambda),P_{1}(\lambda),\ldots,P_{n}(\lambda),\ldots$ can be obtained
by the Gram-Schmidt orthonormalization of the monomials
$1,\lambda,\ldots,\lambda^{n},\ldots$ in the space
$L^{2}({\mathbb{R}}_{+};d\rho)$.

Recall that mapping (1.1) is unitary, that is,

${\bf U}^{*}{\bf U}=I,\quad{\bf U}{\bf U}^{*}=I$

(we denote by $I$ the identity operator in different spaces) and enjoys the
intertwining property $({\bf U}^{*}Ju)(\lambda)=\lambda({\bf
U}^{*}u)(\lambda)$.

Let $\Lambda$ be an interval of the absolutely continuous spectrum of the
operator $J$ where the spectral measure is given by relation (1.3). On the
subspace $L^{2}(\Lambda;d\rho)$, mapping (1.1) can be reduced (if one sets
$\sqrt{\tau(\lambda)}{\bf f}(\lambda)=f(\lambda)$) to the operator
$U=U_{\Delta}:L^{2}(\Lambda)\to\ell^{2}({\mathbb{Z}}_{+})$ defined by the
relation

$(Uf)_{n}=\int_{\Lambda}P_{n}(\lambda)\sqrt{\tau(\lambda)}f(\lambda)d\lambda,\quad
n\in{\mathbb{Z}}_{+},\quad f\in L^{2}(\Lambda).$ (2.3)

Note that the operator $U$ is isometric on the space $L^{2}(\Lambda)$.

Below we discuss asymptotic formula (1.2) for $\lambda\in\Lambda$. All our
estimates are uniform in $\lambda$ from compact subintervals of an open
interval $\Lambda$.

###  2.2. Classical polynomials

The Jacobi polynomials $P_{n}(\lambda)=P_{n}^{\alpha,\beta}(\lambda)$ are
parametrized by two indices $\alpha>-1$ and $\beta>-1$. The corresponding
Jacobi operator $J=J_{\alpha,\beta}$ has the absolutely continuous spectrum
$[-1,1]$ with the spectral measure $d\rho(\lambda)$ defined by (1.3) where the
weight function

$\tau(\lambda)=k(1-\lambda)^{\alpha}(1+\lambda)^{\beta},\quad\alpha,\beta>-1\quad\lambda\in\Lambda=(-1,1).$
(2.4)

The constant $k$ is chosen in such a way that measure (1.3) is normalized,
i.e., $\rho({\mathbb{R}})=\rho((-1,1))=1$. Explicit expressions for matrix
elements $a_{n},b_{n}$ of the Jacobi operator $J$ can be found, for example,
in the books [5, 17], but we do not need them. We only note that
$a_{n}=1/2+O(n^{-2})$, $b_{n}=O(n^{-2})$. According to formula (8.21.10) in
the book [17] the Jacobi polynomials satisfy relation (1.2) where $s=1$,
$r=0$, $\delta=1$ and $\omega(\lambda)=\arcsin\lambda$,

$\kappa(\lambda)=(2\pi
k)^{-1/2}(1-\lambda)^{-(1+2\alpha)/4}(1+\lambda)^{-(1+2\beta)/4},$
$\Phi_{n}(\lambda)=2^{-1}(\alpha+\beta+1)\arcsin\lambda-\pi(2n+\beta-\alpha)/4.$

The Laguerre polynomials  $P_{n}(\lambda)=P_{n}^{\alpha}(\lambda)$ where the
parameter $\alpha>-1$ are determined by recurrence coefficients

$a_{n}=\sqrt{(n+1)(n+1+\alpha)}\quad\mbox{and}\quad b_{n}=2n+\alpha+1.$ (2.5)

The corresponding Jacobi operators $J$ have absolutely continuous spectra
coinciding with $[0,\infty)$ and the weight functions equal
$\tau(\lambda)=\Gamma(\alpha+1)^{-1}\lambda^{\alpha}e^{-\lambda}$ where
$\lambda\in{\mathbb{R}}_{+}$. According to formula (10.15.1) in [5] the
Laguerre polynomials satisfy all relation (1.2) where $s=1/2$, $r=1/4$,
$\delta=3/4$ and $\omega(\lambda)=2\sqrt{\lambda}$,

$\kappa(\lambda)=2^{-1}\pi^{-1/2}\sqrt{\Gamma(1+\alpha)}\,\lambda^{-\alpha/2-1/4}e^{\lambda/2},\quad\Phi_{n}(\lambda)=\pi
n-\pi(2\alpha+1)/4.$

The Hermite polynomials are determined by the Jacobi coefficients
$a_{n}=\sqrt{(n+1)/2},\quad b_{n}=0$. The corresponding Jacobi operator $J$
has the absolutely continuous spectrum coinciding with the whole axis
$\mathbb{R}$ and the weight function equals
$\tau(\lambda)=\pi^{-1/2}e^{-\lambda^{2}},\quad\lambda\in{\mathbb{R}}$.
According to formula (10.15.18) in [5] the Hermite polynomials
$P_{n}(\lambda)$ satisfy, for all $\lambda\in\mathbb{R}$, an asymptotic
relation

$P_{n}(\lambda)=2^{1/2}\pi^{-1/4}e^{\lambda^{2}/2}(2n+1)^{-1/4}\cos\big{(}\sqrt{2n+1}\,\lambda-\pi
n/2\big{)}+O(n^{-3/4})$ (2.6)

as $n\to\infty$. We now have $s=1/2$, $r=1/4$,
$\omega(\lambda)=\sqrt{2\lambda}$ and
$\kappa(\lambda)=2^{-3/4}\pi^{-1/4}e^{\lambda^{2}/2}$.

Thus equality (1.4) is obviously true for all classical polynomials. A direct
calculation shows that relation (1.5) is also satisfied.

######  Remark 2.1.

Theorem 1.1 for the particular case $s=1/2$, $r=1/4$ and compactly supported
functions $w(\lambda)$ can be deduced from boundedness of operator (2.3) for
the Hermite polynomials. Observe first that the operator with kernel
$\epsilon_{n}(\lambda)w(\lambda)$ is Hilbert-Schmidt if
$\epsilon_{n}(\lambda)=O(n^{-\delta})$ with $\delta>1/2$. Therefore it follows
from formula (2.6) that the operator with kernel
$(n+1/2)^{-1/4}\cos\big{(}\sqrt{n+1/2}\,\lambda-\pi n/2\big{)}w(\lambda)$ is
bounded. Choosing $n=2m$ or $n=2m+1$ where $m\in{\mathbb{Z}}_{+}$ and
neglecting Hilbert-Schmidt operators, we see that the operators with kernels
$(m+1)^{-1/4}\cos\big{(}\sqrt{m+1}\,\lambda\big{)}w(\lambda)$ and
$(m+1)^{-1/4}\sin\big{(}\sqrt{m+1}\,\lambda\big{)}w(\lambda)$ are bounded.
This ensures boundedness of the operator with kernel
$(m+1)^{-1/4}\exp\big{(}i\sqrt{m+1}\,\lambda\big{)}w(\lambda)$.

###  2.3. General polynomials

Let us now mention some results for general orthogonal polynomials. In all the
papers discussed below, formula (1.2) holds true although estimates of the
remainders are not always as good as required in (1.2).

First, we recall the classical result of S. Bernstein [2] (see also formula
(12.1.8) in the G. Szegö book [17]). It states that under some (rather
stringent) assumptions on the absolutely continuous spectral measure
$d\rho(\lambda)$ supported by $[-1,1]$, formula (1.2) is true with $r=0$,
$s=1$, $\omega(\lambda)=\arccos\lambda$ and the amplitude factor
$\kappa(\lambda)$ defined by (1.5). The phase shift $\Phi_{n}(\lambda)$ in the
Bernstein formula does not depend on $n$ and is known as the Szegö (or the
spectral shift) function. This result can be considered as a far reaching
generalization of the asymptotic formula for the Jacobi polynomials. On the
contrary, for singular weights $\tau(\lambda)$ formula (1.2) is violated but
only in neighborhoods of singular points (see, e.g., [6]).

A study of asymptotic behavior of orthogonal polynomials defined by their
recurrence coefficients $a_{n}$, $b_{n}$ was initiated by P. Nevai in his book
[13]. Under the assumptions $\\{a_{n}-1/2\\}\in\ell^{1}({\mathbb{Z}}_{+})$,
$\\{b_{n}\\}\in\ell^{1}({\mathbb{Z}}_{+})$ he showed that asymptotics (1.2) is
true on $\Lambda=(-1,1)$ with $r=0$, $s=1$. The corresponding Jacobi operator
is a perturbation (short-range, in quantum mechanical terminology) of the
Jacobi operator $J^{(0)}:=J_{1/2,1/2}$ corresponding to the parameters
$\alpha=\beta=1/2$ in (2.4). It has the coefficients $a_{n}=1/2$, $b_{n}=0$
for all $n\in{\mathbb{Z}}_{+}$ and is known as the free discrete Schrödinger
operator. More general operators $J$ such that $a_{n}\to 1/2$, $b_{n}\to 0$ as
$n\to\infty$ and

$\sum_{n=0}^{\infty}(|a_{n+1}-a_{n}|+|b_{n+1}-b_{n}|)<\infty$ (2.7)

(long-range perturbations of $J^{(0)}$) were considered by A. Maté, P. Nevai
and V. Totik in [12] and in a recent paper [19]. According to the results of
these papers an asymptotic behavior of the corresponding orthogonal
polynomials $P_{n}(\lambda)$ is always given by formula (1.2) where $r=0$ and
$s=1$ but $\Phi_{n}(\lambda)$ is a non-trivial function. Expressions111Formula
(8) in [12] is consistent with these expressions but is less explicit. (1.13)
and (1.22) of [19] for the weight $\tau(\lambda)$ and the amplitude factor
$\kappa(\lambda)$ confirm identity (1.5).

###  2.4. Freud’s weights

Starting with the paper [7], exponential weights

$\tau(\lambda)=k_{\beta}e^{-|\lambda|^{\beta}}$ (2.8)

where $\beta>0$ and $k_{\beta}$ is a normalization constant were extensively
studied. Note that the value $\beta=2$ yields the Hermite polynomials. Given a
measure (1.3), one constructs its moments, orthonormal polynomials
$P_{n}(\lambda)$ and Jacobi coefficients $a_{n}$, $b_{n}$. If $\tau(\lambda)$
is given by (2.8) where $\beta\geq 1$, then the minimal Jacobi operator
$J_{0}$ with the coefficients $a_{n}$, $b_{n}$ is essentially adjoint and the
spectral measure of its closure is determined by (2.8). This is not true for
$\beta<1$.

It was shown in [10, 11] that for weights (2.8) and all $\beta>0$, the off-
diagonal recurrence coefficients $a_{n}$ have asymptotics

$a_{n}=\alpha n^{\ell}(1+o(1)),\quad\ell=1/\beta,\quad n\to\infty,$ (2.9)

with some explicit constant $\alpha=\alpha_{\beta}$; of course the diagonal
elements $b_{n}=0$. The asymptotics of orthogonal polynomials $P_{n}(\lambda)$
is given for all $\lambda\in{\mathbb{R}}$ by the Plancherel-Rotach formula;
see [14] for $\beta=4$, [15] for all $\beta>1$ and [9] for all $\beta>0$.
Formula (2.10) below states this asymptotics for a fixed $\lambda$ in terms of
the coefficients $a_{n}$.

If condition (2.9) is satisfied with $\ell\in(0,1]$ and $b_{n}=0$, then the
Carleman condition (2.1) holds and the minimal Jacobi operator $J_{0}$ is
essentially self-adjoint. The spectrum of its closure $J$ covers the whole
axis $\mathbb{R}$ and it is absolutely continuous so that relation (1.3) holds
true with a smooth function $\tau(\lambda)$ for $\lambda\in\mathbb{R}$.
According to the results of [8] (see also [20], Sect. 5.2) the corresponding
orthonormal polynomials $P_{n}(\lambda)$ have asymptotic behavior

$P_{n}(\lambda)=\pi^{-1/2}\tau(\lambda)^{-1/2}a_{n}^{-1/2}\cos(\lambda\psi_{n}-n\pi/2+\delta(\lambda))+o(a_{n}^{-1/2})$
(2.10)

where $\psi_{n}=2^{-1}\sum_{m=0}^{n-1}a_{m}^{-1}$ and the phase shift
$\delta(\lambda)$ does not depend on $n$.

It follows that in the case $\ell<1$ formula (1.2) holds true with $r=\ell/2$,
$s=1-\ell$, $\kappa(\lambda)=\pi^{-1/2}\tau(\lambda)^{-1/2}$,
$\omega(\lambda)=2^{-1}\alpha(1-\ell)^{-1}\lambda$ so that both relations
(1.4) and (1.5) are satisfied. In the case $\ell=1$ we have
$\psi_{n}=2^{-1}\alpha\ln n$ which corresponds to the limit case of (1.4) and
(1.5) for $s=0$ (see Example 4.13).

If condition (2.9) on $a_{n}$ is satisfied with $\ell>1$ and all $b_{n}=0$,
then, as shown in [20], the difference equation (2.2) for all
$z\in{\mathbb{C}}$ has solutions $f_{n}^{(+)}(z)$ and $f_{n}^{(-)}(z)$ (known
as Jost solutions) with asymptotics

$f_{n}^{(\pm)}(z)=a_{n}^{-1/2}e^{\pm\pi in/2}(1+o(1)),\quad n\to\infty.$
(2.11)

We emphasize that $z$ is here an arbitrary complex number and the right-hand
side of (2.11) depends on $z$ only through the remainder $o(1)$. These
solutions are linearly independent and belong to $\ell^{2}({\mathbb{Z}}_{+})$.
Therefore the minimal operator $J_{0}$ is not essentially self-adjoint, so
that as shown by R. Nevanlinna (see, e.g., Sect. 7.2 of the book [16]) all its
self-adjoint extensions have discrete spectra. Corresponding orthogonal
polynomials $P_{n}(z)$ are linear combinations of $f_{n}^{(+)}(z)$ and
$f_{n}^{(-)}(z)$ so that an asymptotic behavior of $P_{n}(z)$ is drastically
different from (2.10). A value $\ell>1$ corresponds to the case $\beta<1$ in
(2.8) which was studied in [9]. A careful analysis shows that formulas of [9]
are consistent with (2.11).

###  2.5. Jost solutions

In the previous subsection, we discussed a problem with discrete spectrum
where the corresponding difference equation (2.2) had two solutions with
asymptotics (2.11). Actually, it is rather typical that equation (2.2) has two
linearly independent solutions $f_{n}^{(\pm)}$ with oscillating asymptotics.
Below we fix some number $z\in{\mathbb{C}}$ (not necessarily real).

######  Proposition 2.2.

Suppose that equation (2.2) has two linearly independent solutions
$f_{n}^{(\pm)}$ with asymptotics

$f_{n}^{(\pm)}=v_{n}e^{\pm i\Omega_{n}}(1+\varepsilon_{n}^{(\pm)}),\quad
v_{n}>0,\quad\Omega_{n}=\overline{\Omega_{n}}.$ (2.12)

$1^{0}$ If $\Omega_{n+1}-\Omega_{n}\to\varpi$ where
$\varpi\notin\pi{\mathbb{Z}}$ and $\varepsilon_{n}^{(\pm)}\to 0$ as
$n\to\infty$, then there exists

$\lim_{n\to\infty}\big{(}a_{n}v_{n}v_{n+1}\big{)}\neq 0.$ (2.13)

$2^{0}$ If $\Omega_{n+1}-\Omega_{n}=\pi N+\omega_{n}$ where
$N\in{\mathbb{Z}}$, $\omega_{n}\to 0$ and
$\varepsilon_{n}^{(\pm)}=o(\omega_{n})$ as $n\to\infty$, then there exists

$\lim_{n\to\infty}\big{(}a_{n}v_{n}v_{n+1}\omega_{n}\big{)}\neq 0.$ (2.14)

###### Proof.

The Wronskian of the solutions $\\{f_{n}^{(+)}\\}$ and $\\{f_{n}^{(-)}\\}$
equals

$a_{n}\big{(}f_{n}^{(+)}f_{n+1}^{(-)}-f_{n+1}^{(+)}f_{n}^{(-)})=2ia_{n}v_{n}v_{n+1}\big{(}\sin(\Omega_{n}-\Omega_{n+1})+o(\varepsilon_{n})\big{)}$

where $\varepsilon_{n}=o(1)$ in case $1^{0}$ and
$\varepsilon_{n}=o(\omega_{n})$ in case $2^{0}$. Since this expression does
not depend on $n$ and is not zero, we only have to pass to the limit
$n\to\infty$. ∎

Solutions $f_{n}^{(\pm)}$ of the Jacobi equation (2.2) with asymptotics (2.12)
are known as the Jost solutions. Orthonormal polynomials $P_{n}(\lambda)$ are
their linear combinations so that

$P_{n}(\lambda)=2\kappa(\lambda)v_{n}\cos\big{(}\Omega_{n}(\lambda)+\delta(\lambda)\big{)}(1+\varepsilon_{n}(\lambda))$
(2.15)

with some coefficients $\kappa(\lambda)$ and $\delta(\lambda)$ not depending
on $n$. This formula is consistent with (1.2).

Let us give some examples where the conditions of Proposition 2.2 are
satisfied. First we illustrate case $1^{0}$. We start with Jacobi operators
$J$ whose coefficients $a_{n}\to 1/2$, $b_{n}\to 0$ as $n\to\infty$ and obey
condition (2.7). The corresponding orthogonal polynomials contain the
classical Jacobi polynomials (see Sect. 2.3). According to Theorem 3.5 in [19]
we now have

$\Omega_{n+1}(\lambda)-\Omega_{n}(\lambda)=\arccos\frac{\lambda-
b_{n}}{2a_{n}},\quad\lambda\in(-1,1),$

so that $\varpi=\arccos\lambda\in(0,\pi)$. Therefore relation (2.13) implies
that the sequence $v_{n}$ in (2.15) has a finite non-zero limit as
$n\to\infty$.

Next, we consider recurrence coefficients $a_{n}$ satisfying condition (2.9)
and $b_{n}=0$. Now asymptotic formula (2.12) with $\varpi=-\pi/2$ and
$v_{n}=a_{n}^{-1/2}(1+o(1))$ is true for $\ell\in(0,1]$ (in particular, the
case $\ell=1/2$ corresponds to the Hermite polynomials) and all
$\lambda\in{\mathbb{R}}$ according to (2.10). It is true for $\ell>1$ and all
$z\in{\mathbb{C}}$ according to (2.11). Asymptotic formula for $v_{n}$ is of
course consistent with relation (2.13).

For Laguerre coefficients (2.5), the condition of case $2^{0}$ is satisfied
with $N=1$ and $\omega(\lambda)=2\sqrt{\lambda}(\sqrt{n}-\sqrt{n+1})$ where
$\lambda>0$. Therefore relation (2.14) yields $v_{n}=\nu n^{-1/4}$ for some
$\nu>0$ which coincides with the amplitude factor in asymptotic formula for
the Laguerre polynomials.

##  3\. Boundedness of integral operators

###  3.1. Continuous case

To motivate our results on semidiscrete operators (1.6), let us previously
consider the continuous case where the role of $V$ (or of (1.11) for $d=1$) is
played by the operator

$({\sf
V}f)(x)=v(x)\int_{-\infty}^{\infty}e^{i\theta(x)\omega(\lambda)}w(\lambda)f(\lambda)d\lambda$
(3.1)

with sufficiently arbitrary functions $\theta(x)$ and $\omega(\lambda)$. We
suppose that these functions are monotone and tend to $\mp\infty$ (or
$\pm\infty$) as $x\to\mp\infty$ and $\lambda\to\mp\infty$. Operators (3.1) can
be reduced to the sandwiched Fourier transforms by unitary transformations
corresponding to the changes of variables
$y=\theta(x),\quad\mu=\omega(\lambda)$. Indeed, let us set

$(F_{1}u)(y)=\theta^{\prime}(x)^{-1/2}u(x),\quad(F_{2}f)(\mu)=\omega^{\prime}(\lambda)^{-1/2}f(\lambda).$
(3.2)

Then $F_{1}{\sf V}=\widetilde{\sf V}F_{2}$ where the operator $\widetilde{\sf
V}$ acts as

$(\widetilde{\sf
V}\tilde{f})(y)=\tilde{v}(y)\int_{-\infty}^{\infty}e^{iy\mu}\tilde{w}(\mu)\tilde{f}(\mu)d\mu$

with $\tilde{f}=F_{2}f$,

$\tilde{v}(y)=\theta^{\prime}(x)^{-1/2}v(x),\quad\tilde{w}(\mu)=\omega^{\prime}(\lambda)^{-1/2}w(\lambda).$

Since the operator $\widetilde{\sf V}$ is bounded if the functions $\tilde{v}$
and $\tilde{w}$ are bounded, we can state the following elementary result.

######  Lemma 3.1.

The operator ${\sf V}:L^{2}({\mathbb{R}})\to L^{2}({\mathbb{R}})$ is bounded
if

$(\theta^{\prime})^{-1}|v|^{2}\in
L^{\infty}({\mathbb{R}})\quad\mbox{and}\quad(\omega^{\prime})^{-1}|w|^{2}\in
L^{\infty}({\mathbb{R}}).$

Of course the space $L^{2}({\mathbb{R}})$ can be replaced here by
$L^{2}({\mathbb{R}}_{+})$. For example, the operator ${\sf
V}:L^{2}({\mathbb{R}}_{+})\to L^{2}({\mathbb{R}}_{+})$ is bounded if

$\theta(x)=x^{s},\quad|v(x)|~\leq
Cx^{(s-1)/2}\quad\mbox{and}\quad\omega(\lambda)=\lambda^{t},\quad|w(\lambda)|\leq
C\lambda^{(t-1)/2}.$

Note that if $s>1$ or $t>1$, then the functions $v(x)$ or $w(\lambda)$ may be
unbounded at infinity.

The case of operators acting from $L^{2}({\mathbb{R}})$ into
$\ell^{2}({\mathbb{Z}}_{+})$ is essentially more difficult because there is no
unitary operator in the space $\ell^{2}({\mathbb{Z}}_{+})$ playing the role of
the operator $F_{1}$ defined by formula (3.2). Nevertheless, to a some extent,
Lemma 3.1 remains true in this case – see Theorem 3.11, below.

###  3.2. Discrete Fourier transforms

It is convenient to study boundedness of operators (1.6) in a more general
setting. Thus, we consider operators (1.11) where $\Phi$ is the Fourier
transform (1.10) and $dM(x)$ is a locally finite measure on
${\mathbb{R}}^{d}$. Let ${\mathbb{B}}_{r}(x)$ be the ball of radius $r$
centered at a point $x\in{\mathbb{R}}^{d}$. We consider measures $dM(x)$
satisfying the condition

$\sup_{x\in{\mathbb{R}}^{d}}M({\mathbb{B}}_{r}(x))<\infty$ (3.3)

for some and hence for all $r>0$.

######  Theorem 3.2.

If condition (3.3) is satisfied and

$|w(\xi)|\leq C(1+|\xi|^{2})^{-\ell/2}$ (3.4)

for some integer $\ell>d/2$, then operator (1.11) is bounded and

$\|~A\|\leq C\sup_{x\in{\mathbb{R}}^{d}}M({\mathbb{B}}_{1}(x)).$ (3.5)

Conversely, if $w(\xi)$ is the characteristic function of some ball and the
operator $A$ is bounded, then condition (3.3) is satisfied.

###### Proof.

Let us consider the lattice of unit cubes $\Pi_{n}$ in ${\mathbb{R}}^{d}$. By
the Sobolev inequality, it follows from condition (3.3) that, for every $n$,

$\int_{\Pi_{n}}|u(x)|^{2}dM(x)\leq M(\Pi_{n})\max_{x\in\Pi_{n}}|u(x)|^{2}\leq
C\int_{\Pi_{n}}\big{(}|\nabla^{(\ell)}u(x)|^{2}+|u(x)|^{2}\big{)}dx.$

Summing these estimates over all cubes $\Pi_{n}$, we see that

$\|u\|_{L^{2}({\mathbb{R}}^{d};dM)}^{2}\leq
C\int_{{\mathbb{R}}^{d}}\big{(}|(\nabla^{(\ell)}u)(x)|^{2}+|u(x)|^{2}\big{)}dx.$
(3.6)

It means that operator (1.11) is bounded and estimate (3.5) holds.

Conversely, take any not identically zero function $\varphi\in
C_{0}^{\infty}({\mathbb{R}}^{d})$ such that $w\varphi=\varphi$. There exist a
ball ${\mathbb{B}}_{r_{0}}(x_{0})$ and a number $c>0$ such that the Fourier
transform of $\varphi$ satisfies

$|\hat{\varphi}(x)|\geq c>0\quad\mbox{for all}\quad
x\in{\mathbb{B}}_{r_{0}}(x_{0}).$ (3.7)

If condition (3.3) is violated, then there exists a sequence $x_{n}$,
$x_{n}\to\infty$, such that

$\lim_{n\to\infty}M({\mathbb{B}}_{r_{0}}(x_{n}))=\infty.$ (3.8)

Set

$f_{n}(\xi)=e^{i(x_{0}-x_{n})\xi}\varphi(\xi).$ (3.9)

Since $\hat{f}_{n}(x)=\hat{\varphi}(x-x_{n}+x_{0}),$ it follows from (3.7),
(3.8) that

$\|Af_{n}\|_{L^{2}({\mathbb{R}}^{d};dM)}^{2}\geq\int_{{\mathbb{B}}_{r_{0}}(x_{n})}|\hat{f}_{n}(x)|^{2}dM(x)\geq
c^{2}M({\mathbb{B}}_{r_{0}}(x_{n}))\to\infty$ (3.10)

as $n\to\infty$. Thus the operator $A$ is unbounded. ∎

Although Theorem 3.2 is quite elementary, we were unable to localize it in the
literature. On the other hand, in the case of absolutely continuous measures,
it is equivalent to Theorem 3 in [18]. In the general case, the proof is
practically the same.

######  Remark 3.3.

Note that boundedness of operator (1.11) for the characteristic function $w$
of some ball implies the same fact for all functions $w$ satisfying estimate
(3.4).

######  Remark 3.4.

Suppose now that a measure $dM(x)$ is supported by a discrete set
$\\{x_{1},\ldots,x_{n},\ldots\\}\subset{\mathbb{R}}^{d}$ where
$n\in{\mathbb{Z}}_{+}$ or $n\in{\mathbb{Z}}$ and set
$v_{n}^{2}=M(\\{x_{n}\\})$. Then condition (3.3) means that

$\sup_{x\in{\mathbb{R}}^{d}}\sum_{x_{n}\in{\mathbb{B}}_{r}(x)}v_{n}^{2}<\infty.$

Note that if the points $x_{n}$ get closer to each other, then boundedness of
operator (1.11) requires stronger assumptions on decay of $v_{n}$. Boundedness
of $v_{n}$ is always a necessary condition for boundedness of operator (1.11).

######  Remark 3.5.

Condition (3.4) in Theorem 3.2 is rather close to optimal. Indeed, even an
estimate

$|(Af)(x_{n})|\leq C\|f\|,\quad\forall f\in C_{0}^{\infty}({\mathbb{R}}),$

for a fixed $x_{n}\in{\mathbb{R}}^{d}$ implies that $w\in
L^{2}({\mathbb{R}}^{d})$.

###  3.3. One-dimensional case

Theorem 3.2 applies of course to operators
$V:L^{2}({\mathbb{R}})\to\ell^{2}({\mathbb{Z}}_{+})$ defined by formula (1.6).
This yields Theorem 1.1. Sometimes it may be convenient to state its result in
a slightly different form.

######  Theorem 3.6.

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

$\sigma_{n}:=\frac{v_{n}^{2}}{x_{n+1}-x_{n}}$ (3.11)

is bounded and condition (1.8) on $w(\lambda)$ is satisfied, then the operator
$V$ is bounded. Conversely, if $\sigma_{n}\to\infty$ as $n\to\infty$ and
$w(\xi)$ is the characteristic function of some interval, then $V$ is
unbounded.

######  Example 3.7.

If condition (1.9) is satisfied, then $x_{n+1}-x_{n}=sn^{s-1}(1+o(1))$ so that
$\\{\sigma_{n}\\}\in\ell^{\infty}({\mathbb{Z}}_{+})$ and hence the operator
$V$ is bounded. This fact can be stated as an inequality

$\sum_{n=1}^{\infty}n^{s-1}|u(n^{s})|^{2}\leq
C\int_{0}^{\infty}(|u^{\prime}(x)|^{2}+|u(x)|^{2})dx.$ (3.12)

Let us also consider the limit case $s=0$, $r=1/2$.

######  Example 3.8.

If

$x_{n}=\ln n,\quad n>1,\quad\mbox{and}\quad|v_{n}|\leq(1+|n|)^{-1/2},$

then $x_{n+1}-x_{n}=n^{-1}(1+o(1))$ so that again
$\\{\sigma_{n}\\}\in\ell^{\infty}({\mathbb{Z}}_{+})$ and hence the operator
$V$ is bounded. This fact can be stated as an inequality

$\sum_{n=1}^{\infty}n^{-1}|u(\ln n)|^{2}\leq
C\int_{0}^{\infty}(|u^{\prime}(x)|^{2}+|u(x)|^{2})dx.$ (3.13)

Theorem 3.6 shows also that inequalities (3.12) and (3.13) are quite precise,
that is, the factors $n^{s-1}$ and $n^{-1}$ cannot be replaced by
$\gamma_{n}n^{s-1}$ and $\gamma_{n}n^{-1}$ where $\gamma_{n}\to\infty$ as
$n\to\infty$.

###  3.4. Additional results

The first assertion is obvious.

######  Lemma 3.9.

Let an operator $A$ be defined by relation (1.11). Then $A$ is Hilbert-Schmidt
if and only if $w\in L^{2}({\mathbb{R}}^{d})$ and
$M({\mathbb{R}}^{d})<\infty$.

Theorem 3.2 can be supplemented by the following result.

######  Theorem 3.10.

If condition (3.4) is satisfied with an integer $\ell>d/2$ and

$\lim_{|x|\to\infty}M({\mathbb{B}}_{1}(x))=0,$ (3.14)

then the operator $A$ is compact. Conversely, if $w(\xi)$ is the
characteristic function of some ball and the operator $A$ is compact, then
condition (3.14) is satisfied.

###### Proof.

Let $\chi_{N}(x)$ be the characteristic function of the ball
${\mathbb{B}}_{N}(0)$. The operator $\chi_{N}A$ is Hilbert-Schmidt by Lemma
3.9. In view of Theorem 3.2 it follows from condition (3.14) that
$\|(1-\chi_{N})A\|\to 0$ as $N\to\infty$.

Conversely, if condition (3.14) is violated, then there exists a sequence
$x_{n}$ such that $|x_{n}|\to\infty$ and $M({\mathbb{B}}_{r_{0}}(x_{n}))\geq
c>0$. Let $f_{n}(\xi)$ be the same functions (3.9) as in Theorem 3.2.
Similarly to (3.10), we find that $\|Af_{n}\|_{L^{2}({\mathbb{R}}^{d};dM)}\geq
c>0$. Since $f_{n}$ converges weakly to zero, operator $A$ cannot be compact.
∎

Of course an interpolation between Theorem 3.2 and Lemma 3.9 yields conditions
guaranteeing that operators (1.11) belong to Schatten-von Neumann classes
intermediary between classes of bounded and Hilbert-Schmidt operators, but we
will not discuss it here. Note that very general estimates on singular values
of integral operators were obtained by M. Sh. Birman and M. Z. Solomyak in the
paper [3].

Similarly to Lemma 3.1, conditions in the variable $\lambda$ in Theorem 1.1
can be stated in a more general form. Let us consider an operator ${\bf
V}:L^{2}({\mathbb{R}})\to\ell^{2}({\mathbb{Z}}_{+})$ defined by the formula

$({\bf
V}f)_{n}=v_{n}\int_{-\infty}^{\infty}e^{ix_{n}\omega(\lambda)}w(\lambda)f(\lambda)d\lambda$
(3.15)

where $\omega(\lambda)$ is a sufficiently arbitrary function. We suppose that
$\omega(\lambda)$ is monotone and $\omega(\lambda)\to\pm\infty$ as
$\lambda\to\pm\infty$. Making a change of variables $\mu=\omega(\lambda)$, we
see that

$({\bf
V}f)_{n}=v_{n}\int_{-\infty}^{\infty}e^{ix_{n}\mu}\tilde{w}(\mu)\tilde{f}(\mu)d\mu$
(3.16)

where
$\tilde{f}(\mu)=\omega^{\prime}(\lambda)^{-1/2}f(\lambda),\quad\tilde{w}(\mu)=\omega^{\prime}(\lambda)^{-1/2}w(\lambda)$.
It follows from Theorem 1.1 that operator (3.16) is bounded if

$|\tilde{w}(\mu)|\leq C(1+|\mu|)^{-1}.$

Thus Theorem 1.1 can be supplemented by the following statement.

######  Theorem 3.11.

Let an operator ${\bf V}:L^{2}({\mathbb{R}})\to\ell^{2}({\mathbb{Z}}_{+})$ be
defined by formula (3.15) where $\omega^{\prime}(\lambda)>0$ and
$\omega(\lambda)\to\pm\infty$ as $\lambda\to\pm\infty$. Suppose that

$|w(\lambda)|\leq C\omega^{\prime}(\lambda)^{1/2}(1+|\omega(\lambda)|)^{-1}.$

Then under assumption (1.9) on $x_{n}$ and $v_{n}$ the operator ${\bf V}$ is
bounded.

We have chosen the spaces $L^{2}({\mathbb{R}})$ and
$\ell^{2}({\mathbb{Z}}_{+})$ only for definiteness. They can be obviously
replaced by $L^{2}({\mathbb{R}}_{+})$ and $\ell^{2}({\mathbb{Z}})$,
respectively.

##  4\. Universal relations

Relation (1.4) (but not (1.5)) for the coefficients in asymptotic formula
(1.2) can, in principle, be deduced from the results of Sect. 3. In this
section, we prove both (1.4) and (1.5) by another method, investigating the
time-dependent evolution $\exp(-i\Theta(J)t)f$ as $t\to\infty$ for appropriate
functions $\Theta$ of a Jacobi operator $J$.

###  4.1. Stationary phase method

Here we consider an integral

$G_{n}(t)=\int_{\Delta}e^{i\mu
x_{n}+i\varphi_{n}(\mu)-i\theta(\mu)t}F(\mu)d\mu$ (4.1)

where $\Delta=(\mu_{1},\mu_{2})\subset{\mathbb{R}}$ is a bounded interval, the
functions $F\in C_{0}^{\infty}(\Delta)$, $\varphi_{n},\theta\in
C^{\infty}(\Delta)$ and $\theta^{\prime}(\mu)>0$,
$\theta^{\prime\prime}(\mu)\neq 0$ for $\mu\in\Delta$. We suppose that
$x_{n}<x_{n+1}$, $x_{n}\to\infty$ and $t\to+\infty$. With respect to the
phases $\varphi_{n}(\mu)$, we assume222All estimates of such type are supposed
to be uniform in $\mu$ on compact subintervals of $\Delta$. that

$\varphi^{(k)}_{n}(\mu)=o(x_{n}),\quad n\to\infty,\quad\mu\in\Delta,\quad
k=1,2,$ (4.2)

and $\varphi^{(k)}_{n}(\mu)=O(x_{n})$ for $k=3,4,\ldots$.

Asymptotic behavior of integral (4.1) for large $t$ and $n$ is determined by
stationary points $\mu$ where

$x_{n}+\varphi_{n}^{\prime}(\mu)-\theta^{\prime}(\mu)t=0.$ (4.3)

A direct integration by parts shows that integral (4.1) rapidly tends to zero
if the interval $\Delta$ does not contain stationary points.

######  Lemma 4.1.

If $x_{n}/t\not\in\theta^{\prime}(\Delta)$, then

$|G_{n}(t)|\leq C_{k}(x_{n}+t)^{-k},\quad\forall k>0.$ (4.4)

Let us introduce the inverse function $h:\theta^{\prime}(\Delta)\to\Delta$ to
$\theta^{\prime}$ so that

$\theta^{\prime}(h(\xi))=\xi,\quad\xi\in\theta^{\prime}(\Delta).$ (4.5)

A next result can easily be obtained by the method of successive
approximations.

######  Lemma 4.2.

Let condition (4.2) be satisfied. Then for all
$\xi_{n}:=x_{n}/t\in\theta^{\prime}(\Delta)$ and sufficiently large $t$,
equation (4.3) has a unique solution $\mu_{n}(t)$ and

$\mu_{n}(t)=h(\xi_{n})+o(1),\quad t\to+\infty.$ (4.6)

We need the following version of the stationary phase method.

######  Lemma 4.3.

Let $\xi_{n}:=x_{n}/t\in\theta^{\prime}(\Delta)$. Then

$G_{n}(t)=\sqrt{2\pi}e^{-\pi
i\tau/4}e^{i\Psi_{n}(t)}t^{-1/2}{\mathcal{F}}(x_{n}/t)+\varrho_{n}(t)\quad\mbox{as}\quad
t\to+\infty$ (4.7)

where $\tau=\operatorname{sgn}\theta^{\prime\prime}(\mu)$,

${\mathcal{F}}(\xi)=\sqrt{|h^{\prime}(\xi)|}F(h(\xi)),$ (4.8)
$\Psi_{n}(t)=\mu_{n}(t)x_{n}+\varphi_{n}(\mu_{n}(t))-\theta(\mu_{n}(t))t$
(4.9)

and

$\sup_{x_{n}/t\in\theta^{\prime}(\Delta)}|\varrho_{n}(t)|=o(t^{-1/2}).$ (4.10)

Sketch of the proof. Applying the stationary phase formula to integral (4.1),
we see that

$G_{n}(t)=\sqrt{2\pi}e^{-\pi
i\tau/4}e^{i\Psi_{n}(t)}|\theta^{\prime\prime}(\mu_{n}(t))t-\varphi_{n}^{\prime\prime}(\mu_{n}(t))|^{-1/2}F(\mu_{n}(t)))+\varrho_{n}(t)$
(4.11)

where $r_{n}(t)$ satisfies (4.10) as $t\to+\infty$. According to (4.6) we can
here replace $\mu_{n}(t)$ by $h(\xi_{n})$ and according to (4.2) we can
neglect $\varphi_{n}^{\prime\prime}(\mu_{n}(t))$. Differentiating (4.5), we
see that $\theta^{\prime\prime}(h(\xi))h^{\prime}(\xi)=1$. Therefore using
notation (4.8), we can rewrite (4.11) as (4.7).

###  4.2. Asymptotic evolution

Let $\Lambda$ be an interval of the absolutely continuous spectrum of a Jacobi
operator $J$ so that its spectral measure satisfies (1.3) where we suppose
that the weight $\tau\in C^{\infty}(\Lambda)$ and $\tau(\lambda)>0$ for
$\lambda\in\Lambda$. With respect to the corresponding orthonormal polynomials
$P_{n}(\lambda)$, we accept an assumption somewhat more general than (1.2).

######  Assumption 4.4.

As $\lambda\in\Lambda$ and $n\to\infty$, an asymptotic formula holds

$P_{n}(\lambda)=2\kappa(\lambda)v_{n}\cos\big{(}\omega(\lambda)x_{n}+\Phi_{n}(\lambda)\big{)}+\epsilon_{n}(\lambda)$
(4.12)

where $x_{n+1}>x_{n}$, $x_{n}\to\infty$ as $n\to\infty$, $v_{n}>0$ and

$\sum_{n=1}^{\infty}v_{n}^{2}x_{n}^{-2k}<\infty$ (4.13)

for some $k>0$. We suppose that $\omega,\kappa\in C^{\infty}(\Lambda)$,
$\kappa(\lambda)>0$, $\omega^{\prime}(\lambda)>0$,

$\Phi^{(k)}_{n}(\lambda)=o(x_{n}),\quad
n\to\infty,\quad\lambda\in\Lambda,\quad k=1,2,$

and $\Phi^{(k)}_{n}(\lambda)=O(x_{n})$ for all $k\geq 3$. The remainder in
(4.12) satisfies a condition $\sum_{n}|\epsilon_{n}(\lambda)|^{2}<\infty$
uniformly on compact subintervals of $\Lambda$.

Let $f\in C_{0}^{\infty}(\Lambda)$ and $u=Uf\in L^{2}(\Lambda)$ so that
$u=E(\Lambda)u$. It follows from formulas (2.3) and (4.12) that, for an
arbitrary function $\Theta\in L^{\infty}(\Lambda)$,

$(e^{-i\Theta(J)t}u)_{n}=2v_{n}\int_{\Lambda}\cos\big{(}\omega(\lambda)x_{n}+\Phi_{n}(\lambda)\big{)}e^{-i\Theta(\lambda)t}\varkappa(\lambda)f(\lambda)d\lambda+\varrho_{n}(t)$
(4.14)

where $\varkappa(\lambda)=\kappa(\lambda)\sqrt{\tau(\lambda)}$ and

$\varrho_{n}(t)=\int_{\Lambda}\epsilon_{n}(\lambda)e^{-i\Theta(\lambda)t}\sqrt{\tau(\lambda)}f(\lambda)d\lambda.$
(4.15)

Let us find an asymptotic behavior in $\ell^{2}({\mathbb{Z}}_{+})$ of sequence
(4.14) as $t\to\infty$. The last term on the right is negligible. Indeed, the
operator $e^{-i\Theta(J)t}$ tends weakly to zero as $t\to\infty$ if
$\Theta^{\prime}(\lambda)\neq 0$ for $\lambda\in\Lambda$. The operator
$R:L^{2}(\Lambda)\to\ell^{2}({\mathbb{Z}}_{+})$ defined by
$(Rf)_{n}=\int_{\Lambda_{c}}\epsilon_{n}(\lambda)\tau(\lambda)f(\lambda)d\lambda$
is in the Hilbert-Schmidt class for any compact subinterval $\Lambda_{c}$ of
$\Lambda$. It follows that

$\lim_{t\to\infty}\sum_{n=0}^{\infty}|\varrho_{n}(t)|^{2}=0.$ (4.16)

Assume now that

$\Theta(\lambda)=\theta(\omega(\lambda))$ (4.17)

for some function $\theta\in C^{\infty}(\Delta)$ where
$\theta^{\prime}(\mu)>0$ for $\mu\in\Delta=\omega(\Lambda)$. Making the change
of variables

$\mu=\omega(\lambda),\quad\phi_{n}(\mu)=\Phi_{n}(\lambda),\quad
F(\mu)=\omega^{\prime}(\lambda)^{-1}\varkappa(\lambda)f(\lambda)$ (4.18)

in the first term in the right-hand side of (4.14), we find that it equals

$v_{n}\int_{\Delta}e^{i\mu
x_{n}+i\phi_{n}(\mu)-i\theta(\mu)t}F(\mu)d\mu+v_{n}\int_{\Delta}e^{-i\mu
x_{n}-i\phi_{n}(\mu)-i\theta(\mu)t}F(\mu)d\mu.$ (4.19)

In the region $x_{n}/t\in\theta^{\prime}(\Delta)$, asymptotics of the first
integral in (4.19) is given by Lemma 4.3. A direct integration by parts shows
that for $x_{n}/t\not\in\theta^{\prime}(\Delta)$, this integral satisfies
estimate (4.4) so that under assumption (4.13), this region of $n$ can be
neglected. Similarly, integrating by parts we see that the second integral
satisfies estimate (4.4) for all $n$. Therefore the second term in (4.19)
tends to zero in $\ell^{2}({\mathbb{Z}}_{+})$ as $t\to\infty$. This leads to
the following result.

######  Theorem 4.5.

Let Assumptions 4.4 be satisfied, and let $f\in C_{0}^{\infty}(\Lambda)$,
$u=Uf$. Define the function $\Theta(\lambda)$ by equality (4.17), the
functions $\varphi_{n}(\mu)$, $F(\mu)$ – by formula (4.18),
${\mathcal{F}}(\xi)$ – by (4.8) and $\Psi_{n}(t)$ – by (4.9). Then, as
$t\to+\infty$,

$(e^{-i\Theta(J)t}u)_{n}=\sqrt{2\pi}v_{n}e^{-\pi
i\tau/4}e^{i\Psi_{n}(t)}t^{-1/2}{\mathcal{F}}(x_{n}/t)\mathbbm{1}(x_{n}/t\in{\mathcal{I}})+\varrho_{n}(t),$
(4.20)

where ${\mathcal{I}}=\theta^{\prime}(\omega(\Lambda))$,
$\tau=\operatorname{sgn}\theta^{\prime\prime}$ and the remainder
$\varrho_{n}(t)$ satisfies (4.16).

According to (4.20) the sequence $(e^{-i\Theta(J)t}u)_{n}$ “lives” in the
region of $n$ such that $x_{n}/t\in{\mathcal{I}}$. Below, the phase factor
$e^{i\Psi_{n}(t)}$ is not very important. Since

$\|e^{-i\Theta(J)t}u\|=\|u\|=\|Uf\|=\|f\|,$ (4.21)

we can state

######  Corollary 4.6.

Under the assumptions of Theorem 4.5 we have

$2\pi\lim_{t\to\infty}t^{-1}\\!\\!\\!\sum_{n:x_{n}/t\in{\mathcal{I}}}v_{n}^{2}\,|{\mathcal{F}}(x_{n}/t)|^{2}=\|f\|^{2}.$
(4.22)

######  Remark 4.7.

It suffices to make assumptions on only one derivative of functions
$\tau,\kappa$ and two derivatives of functions $\omega,\Phi_{n}$. However in
this case one has to demand (4.13) for $k=1$.

######  Remark 4.8.

The stationary phase method does not necessarily require that the function
$\Theta(\lambda)$ be defined by formula (4.17). If
$\omega^{\prime\prime}(\lambda)\neq 0$, we can also take
$\Theta(\lambda)=\lambda$ and if $\omega(\lambda)=\omega_{0}\lambda$, any
function $\Theta(\lambda)$ with $\Theta^{\prime\prime}(\lambda)\neq 0$ is
admissible. Asymptotic formula (4.20) remains essentially the same in all
these cases. On the contrary if $\omega(\lambda)=\omega_{0}\lambda$ and
$\Theta(\lambda)=\Theta_{0}\lambda$, then the evolution $e^{-i\Theta(J)t}u$
reduces to shifts; see Appendix B.

###  4.3. Main result

Relations (1.4) and (1.5) are deduced from formula (4.22). To that end, we
need the following auxiliary result. It is a direct consequence of the
definition of an integral of a continuous function as a limit of the
corresponding Riemann sums.

######  Lemma 4.9.

Suppose that ${\mathcal{I}}\subset{\mathbb{R}}$ is a bounded interval and a
function ${\mathcal{F}}\in C({\mathcal{I}})$. Let a sequence
$X_{n}(t)\in{\mathbb{R}}$, $n\in{\mathbb{Z}}_{+}$, be such that
$X_{n+1}(t)>X_{n}(t)$, $X_{n}(t)\to\infty$ as $n\to\infty$ for all $t$ and

$\lim_{t\to\infty}\max_{n:X_{n}(t)\in{\mathcal{I}}}(X_{n+1}(t)-X_{n}(t))=0.$
(4.23)

Then

$\lim_{t\to\infty}\sum_{n:X_{n}(t)\in{\mathcal{I}}}(X_{n+1}(t)-X_{n}(t))|{\mathcal{F}}(X_{n}(t))|^{2}=\int_{{\mathcal{I}}}|{\mathcal{F}}(\xi)|^{2}d\xi.$
(4.24)

Now we can state our main result.

######  Theorem 4.10.

Let Assumption 4.4 be satisfied. Suppose additionally that

$\lim_{n\to\infty}(x_{n+1}-x_{n})/x_{n}=0.$ (4.25)

Define a sequence $\sigma_{n}$ by equality (3.11) and assume that a limit
$($possibly, infinite$)$

$\lim_{n\to\infty}\sigma_{n}=\sigma$ (4.26)

exists. Then necessarily $0<\sigma<\infty$ and

$2\pi\sigma\tau(\lambda)\kappa^{2}(\lambda)=\omega^{\prime}(\lambda).$ (4.27)

###### Proof.

Let us set ${\mathcal{I}}=\theta^{\prime}(\omega(\Lambda))$,
$X_{n}(t)=x_{n}/t$ and observe that the sequence

$X_{n+1}(t)-X_{n}(t)=((x_{n+1}-x_{n})/x_{n})X_{n}(t)$

satisfies condition (4.23). Indeed, the first factor tends to zero in view of
(4.25) and the second one is bounded if $X_{n}(t)\in{\mathcal{I}}$. Note also
that $n\to\infty$ if $X_{n}(t)\in{\mathcal{I}}$ and $t\to\infty$. Applying
relation (4.24) to a function ${\mathcal{F}}(\xi)$ defined by equalities (4.8)
and (4.18) and using condition (4.26), we find that

$\lim_{t\to\infty}\sum_{n:X_{n}(t)\in{\mathcal{I}}}\sigma_{n}(X_{n+1}(t)-X_{n}(t))|{\mathcal{F}}(X_{n}(t))|^{2}=\sigma\int_{{\mathcal{I}}}|{\mathcal{F}}(\xi)|^{2}d\xi\\\
=\sigma\int_{{\mathcal{I}}}|h^{\prime}(\xi)||F(h(\xi))|^{2}d\xi=\sigma\int_{\omega(\Lambda)}|F(\mu)|^{2}d\mu=\sigma\int_{\Lambda}\omega^{\prime}(\lambda)^{-1}\varkappa(\lambda)^{2}|f(\lambda)|^{2}d\lambda.$
(4.28)

Since $t^{-1}v_{n}^{2}=\sigma_{n}(X_{n+1}(t)-X_{n}(t)),$ it follows from
Corollary 4.6 that the left-hand side of (4.28) equals $(2\pi)^{-1}\|f\|^{2}$.
Therefore relation (4.28) ensures that

$\int_{\Lambda}|f(\lambda)|^{2}d\lambda=2\pi\sigma\int_{\Lambda}\omega^{\prime}(\lambda)^{-1}\varkappa^{2}(\lambda)|f(\lambda)|^{2}d\lambda$
(4.29)

for an arbitrary function $f\in C_{0}^{\infty}(\Lambda)$. Thus, $\sigma$ is
necessary finite, $\sigma\neq 0$ and
$2\pi\sigma\omega^{\prime}(\lambda)^{-1}\varkappa^{2}(\lambda)=1$. This yields
(4.27). ∎

######  Remark 4.11.

Condition (4.25) imposes a mild restriction on the growth of $x_{n}$ as
$n\to\infty$. However it is satisfied for $x_{n}=n^{s}$ with an arbitrary
$s>0$.

###  4.4. Examples and comments

The first example generalizes the results stated in Sect. 1.1. The second
corresponds to the limit case $s=0$.

######  Example 4.12.

Suppose that asymptotic formula (4.12) holds with

$x_{n}=n^{s}(\ln n)^{p},\quad v_{n}=n^{-r}(\ln n)^{q},\quad r\geq 0,$ (4.30)

where $s>0$ and $p,q\in{\mathbb{R}}$. Then number (3.11) equals

$\sigma_{n}=s^{-1}n^{-2r-s+1}(\ln n)^{2q-p}(1+o(1)),\quad n\to\infty,$

so that limit (4.26) exists. It is finite and $\sigma\neq 0$ if the conditions
(1.4) and $2q=p$ are satisfied. Relation (4.27) holds true with
$\sigma=s^{-1}$.

######  Example 4.13.

Suppose that (4.12) and (4.30) hold with $s=0$. Then we have

$\sigma_{n}=p^{-1}n^{-2r+1}(\ln n)^{2q-p+1}(1+o(1)),\quad n\to\infty,$

and hence limit (4.26) again exists. It is finite and $\sigma\neq 0$ if the
conditions $r=1/2$ and $2q=p-1$ are satisfied. Relation (4.27) holds true with
$\sigma=p^{-1}$.

Finally, we note that both the conditions and the statements of Theorem 4.10
are quite different from those of Proposition 2.2. In particular, the latter
imposes assumptions on the phase $\Omega_{n}$ while Theorem 4.10 requires
conditions on its derivative $\Omega_{n}^{\prime}(\lambda)$. Note also that
Theorem 4.10 is specific for the absolutely continuous spectrum of $J$.

##  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

$|u(0)|^{2}\leq c_{0}\int_{0}^{1}(|u^{\prime}(x)|^{2}+|u(x)|^{2})dx$ (A.1)

where we can choose $c_{0}=2(\sqrt{5}-1)^{-1}$. We check the following result.

######  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

$\sum_{n\in{\mathbb{Z}}}(x_{n+1}-x_{n})|u(x_{n})|^{2}\leq
c_{0}\max\\{1,\delta^{2}\\}\int_{-\infty}^{\infty}(|u^{\prime}(x)|^{2}+|u(x)|^{2})dx$
(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 variables $y=\varepsilon x$, we see that

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

whence

$\varepsilon|u(x_{n})|^{2}\leq
c_{0}\int_{x_{n}}^{x_{n}+\varepsilon}(\varepsilon^{2}|u^{\prime}(y)|^{2}+|u(y)|^{2})dy.$
(A.3)

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

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

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

Inequality (A.2) is quite elementary but, surprisingly, we were unable to find
it in the literature. Of course (A.2) implies again inequalities (3.12) and
(3.13).

##  Appendix B Dispersionless evolution

If $x_{n}=o(n)$ as $n\to\infty$ (which excludes the linear growth of $x_{n}$)
and the phases $\Phi_{n}$ do not depend on $\lambda$, we can dispense with the
stationary phase method. Here we consider evolution (4.14) for the case
$\Theta(\lambda)=\omega(\lambda)$ excluded in Sect. 4.2. Making the change of
variables (4.18), we can rewrite (4.14) as

$(e^{-i\omega(J)t}u)_{n}=\sqrt{2\pi}v_{n}\big{(}e^{i\Phi_{n}}\widehat{F}(x_{n}-t)+e^{-i\Phi_{n}}\widehat{F}(-x_{n}-t)\big{)}+\varrho_{n}(t)$
(B.1)

where $\widehat{F}(x)$ is the Fourier transform of $F(\mu)$. The remainder
$\varrho_{n}(t)$ is given by (4.15); it is negligible according to (4.16).
Since $\widehat{F}(-x)=O(x^{-k})$ as $x\to+\infty$ for all $k$, the term with
$\widehat{F}(-x_{n}-t)$ in (B.1) is also negligible. Thus instead of Theorem
4.5 we now have

######  Theorem B.1.

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

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

where the remainder $\varrho_{n}(t)$ obeys condition (4.16).

It follows from (B.2) that

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

and hence according to (4.21) (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.3)

Note that formula (4.20) is not true now because if
$\Theta(\lambda)=\omega(\lambda)$, then $\theta(\lambda)=\lambda$ so that
$\theta^{\prime\prime}(\lambda)=0$.

On the other hand, one can calculate the left-hand side of (B.3) using Lemma
4.9. Indeed, let us set $X_{n}(t)=x_{n}-t$ and observe that
$v_{n}^{2}=\sigma_{n}(X_{n+1}(t)-X_{n}(t))$. If the sequence $\sigma_{n}$ has
a finite limit $\sigma$, then according to Proposition A.1, for each $R>0$,
the sum in (B.3) over $n$ such that $|X_{n}(t)|\geq R$ is estimated by
$C\int_{|x|\geq R}(|\widehat{F}^{\prime}(x)|^{2}+|\widehat{F}(x)|^{2})dx$. In
view of (4.24) the sum in (B.3) over $n$ such that $|X_{n}(t)|<R$ converges to
$\sigma\int_{-R}^{R}|\widehat{F}(x)|^{2}dx$ as $t\to\infty$. It follows that
the left-hand side of (B.3) equals
$2\pi\sigma\|\widehat{F}\|^{2}=2\pi\sigma\|F\|^{2}$ so that
$2\pi\sigma\|F\|^{2}=\|f\|^{2}$. This yields again equality (4.29) whence
$\sigma>0$ and relation (4.27) follows.

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