Asymptotics of eigenvalues and eigenvectors of Toeplitz matrices

Assume that $\alpha\geqslant0$ and let $W^{\alpha}$ denote the weighted Wiener algebra of functions $a\colon\mathbb T\to\mathbb C$ with representations

$$\begin{equation*} a(t)=\sum_{j=-\infty}^\infty a_jt^j, \qquad t\in\mathbb T, \end{equation*}$$

where the Fourier coefficients satisfy

$$\begin{equation*} \|a\|_\alpha:=\sum_{j=-\infty}^\infty|a_j|(1+|j|)^\alpha<\infty. \end{equation*}$$

Along with $a\in W^\alpha$ we look at the function $g\colon[0,2\pi]\to\mathbb R$ given by $g(\sigma):= a(e^{i\sigma})$.

We define the class $\operatorname{SL}^\alpha$ as the set of symbols $a$ in $W^\alpha$ such that

• (i)  
  $a$ is real-valued;
• (ii)  
  the range of $g$ is an interval $[0,\mu]$ where $\mu>0$; $g(0)=g(2\pi)=0$, $g''(0)=g''(2\pi)>0$, and there exists a point $\varphi_0\in(0,2\pi)$ such that $g(\varphi_0)=\mu$, $g'(\sigma)>0$ for $\sigma\in(0,\varphi_0)$, $g'(\sigma)<0$ for $\sigma\in(\varphi_0,2\pi)$ and $g''(\varphi_0)<0$.

Note that (i) is equivalent to all the matrices $T_n(a)$, $n\in\mathbb Z_+$, being Hermitian (selfadjoint). If $a\in W^\alpha$, then $g\in C^{\lfloor\alpha\rfloor}[0,2\pi]$, where $\lfloor\alpha\rfloor$ is the integer part of $\alpha$. Thus, if $a\in\operatorname{SL}^\alpha$ with $\alpha\geqslant1$ then, in particular, $g$ belongs to $C^1[0,2\pi]$. Furthermore, in (ii) we assume that $g$ has second derivatives at the points $\sigma=0$, $\varphi_0$ and $2\pi$.

We will say that the symbol $a$ belongs to the class $\operatorname{MSL}^\alpha$ with $\alpha\geqslant1$ if $a\in\operatorname{SL}^\alpha$ and the following additional condition is satisfied:

• (iii)  
  there exist functions $q_1,q_2\in W^\alpha$ such that

  $$\begin{equation} a(t)=(t-1)q_1(t)\quad\text{and} \quad a(t)-a(e^{i\varphi_0})=(t-e^{i\varphi_0})q_2(t). \end{equation} \tag{ 2.1 }$$

By Lemma 3.1 in [41], if $a\in\operatorname{SL}^\alpha$, then the functions $q_1$ and $q_2$ defined by (2.1) must belong to $W^{\alpha-1}$. In the definition of the class $\operatorname{MSL}^\alpha$ we impose the stronger condition (iii), that is, we assume that the symbol has additional smoothness at the points where it takes the minimum and maximum values. Using the same lemma we can show that each $a\in\operatorname{MSL}^\alpha$ has the representations

$$\begin{equation} a(t)=(t-1)^2 q_3(t)\quad\text{and} \quad a(t)-a(e^{i\varphi_0})=(t-e^{i\varphi_0})^2 q_4(t), \end{equation} \tag{ 2.2 }$$

with $q_3,q_4\in W^{\alpha-1}$.

Let $a\in\operatorname{MSL}^{\alpha}$, $\alpha\geqslant1$. For each $\lambda\in[0,\mu]$ there exists precisely one point $\varphi_1(\lambda)$ on the interval $[0,\varphi_0]$ such that $g(\varphi_1(\lambda))=\lambda$ and precisely one point $\varphi_2(\lambda)\in[\varphi_0,2\pi]$ such that $g(\varphi_2(\lambda))=\lambda$. In other words $\varphi_1$ and $\varphi_2$ are the inverse functions of $g$ restricted to $[0,\varphi_0]$ and $[\varphi_0,2\pi]$, respectively. For each $\lambda\in[0,\mu]$, $g$ takes values not exceeding $\lambda$ on the intervals $[0,\varphi_1(\lambda)]$ and $[\varphi_2(\lambda),2\pi]$. Let $\varphi(\lambda)$ denote the arithmetic mean of the lengths of these two intervals:

$$\begin{equation*} \varphi(\lambda):=\frac{1}{2}(\varphi_1(\lambda)-\varphi_2(\lambda))+\pi =\frac{1}{2}\bigl|\{\sigma\in[0,2\pi]\colon\ g(\sigma)\leqslant\lambda\}\bigr|, \end{equation*}$$

where ${|\cdot|}$ denotes Lebesgue measure on $[0,2\pi]$. Note that $\varphi\colon[0,\mu]\to[0,\pi]$ is continuous and bijective and let $\psi\colon[0,\pi]\to[0,\mu]$ denote its inverse function. Up to a linear change of the argument, $\varphi$ and $\psi$ are the distribution function and the quantile function of the `random variable' $a$ on the probability space $[0,2\pi]$ with normalized Lebesgue measure. Set

$$\begin{equation*} \sigma_1(s)=\varphi_1(\psi(s))=\varphi_1(\lambda)\quad\text{and} \quad \sigma_2(s)=\varphi_2(\psi(s))=\varphi_2(\lambda). \end{equation*}$$

Note that the derivatives of $\varphi_1$ and $\varphi_2$ are unbounded in a neighbourhood of $0$ and $\mu$, whereas the functions $\sigma_1$ and $\sigma_2$ have continuous first derivatives on the whole of $[0,\pi]$. In addition, it is easy to see that

$$\begin{equation*} g(\sigma_1(s))=g(\sigma_2(s))=\psi(s)=\lambda. \end{equation*}$$

For each $s\in[0,\pi]$ the function $a-\psi(s)$ has two zeros, $t=e^{i\sigma_1(s)}$ and $t=e^{i\sigma_2(s)}$. We can define a positive function $b$ in terms of these:

$$\begin{equation} b(t,s):=\frac{(a(t)-\psi(s))e^{is}}{(t-e^{i\sigma_1(s)})(t^{-1}-e^{-i\sigma_2(s)})}, \qquad t\in\mathbb T, \quad s\in[0,\pi]. \end{equation} \tag{ 2.3 }$$

Let $\eta\colon[0,\pi]\to\mathbb R$ be defined by

$$\begin{equation} \eta(s):=\frac{1}{4\pi}\int_0^{2\pi}\frac{\log b(e^{i\sigma},s)}{\operatorname{tan}\frac{\sigma-\sigma_2(s)}{2}}\,d\sigma -\frac{1}{4\pi}\int_0^{2\pi}\frac{\log b(e^{i\sigma},s)}{\operatorname{tan}\frac{\sigma-\sigma_1(s)}{2}}\,d\sigma. \end{equation} \tag{ 2.4 }$$

The singular integrals above are taken in the sense of the Cauchy principal value.

Let $\lambda_{1,n},\dots,\lambda_{n,n}$ denote the eigenvalues of the matrix $T_n(a)$. We can now state the main results in [43].

Theorem 2.1.  Let $\alpha\geqslant1$ and $a\in\operatorname{MSL}^\alpha$. Then for each $n\geqslant1$:

(i) all eigenvalues of $T_n(a)$ are distinct, so that $\lambda_{1,n}<\dots<\lambda_{n,n}$;

(ii) the quantities $s_{j,n}:=\varphi(\lambda_{j,n})$, $j=1,\dots,n$, satisfy the equations

$$\begin{equation} (n+1)s_{j,n}+\eta(s_{j,n})=\pi j+\Delta^{(1)}_{j,n}, \end{equation} \tag{ 2.5 }$$

where $\Delta^{(1)}_{j,n}=o({1}/{n^{\alpha-1}})$ uniformly in $j$ as $n\to\infty$;

(iii) for sufficiently large $n$, equation (2.5) has a unique solution $s_{j,n}\in[0,\pi]$ for each $j=1,\dots,n$.

Set $d_{j,n}={\pi j}/(n+1)$.

Theorem 2.2.  If all the assumptions of Theorem 2.1 are fulfilled, then

$$\begin{equation*} s_{j,n}=d_{j,n}+\sum_{k=1}^{\lfloor\alpha\rfloor}\frac{p_k(d_{j,n})}{(n+1)^k}+\Delta^{(2)}_{j,n}, \end{equation*}$$

where $\Delta^{(2)}_{j,n}=o({1}/{n^{\alpha}})$ uniformly in $j$ as $n\to\infty$. The coefficients $p_k$ can be calculated explicitly; in particular,

$$\begin{equation*} p_1(s)=-\eta(s)\quad\text{and} \quad p_2(s)=\eta(s)\eta'(s). \end{equation*}$$

Theorem 2.3.  If all the assumptions of Theorem 2.1 are fulfilled, then

$$\begin{equation*} \lambda_{j,n}=\psi(d_{j,n})+\sum_{k=1}^{\lfloor\alpha\rfloor}\frac{r_k(d_{j,n})}{(n+1)^k} +\Delta^{(3)}_{j,n}, \end{equation*}$$

where $\Delta^{(3)}_{j,n}=o\big(\frac{1}{n^\alpha}(d_{j,n}(\pi-d_{j,n}))^{\alpha-1}\big)$ for $1\leqslant\alpha<2$ and $o\bigl(\frac{1}{n^\alpha}d_{j,n}(\pi-d_{j,n})\bigr)$ for $\alpha\geqslant 2$ uniformly in $j$ as $n\to\infty$. The coefficients $r_k$ can be calculated explicitly; in particular,

$$\begin{equation*} r_1(s)=-\psi'(s)\eta(s)\quad\text{and} \quad r_2(s)=\frac{1}{2}\psi''(s)\eta^2(s)+\psi'(s)\eta(s)\eta'(s). \end{equation*}$$

Remark 2.1.  Let $a\in\operatorname{MSL}^{\alpha}$ and $\alpha\in [1,2)$. Then the eigenvalues (for which ${\epsilon\leqslant d_{j,n}\leqslant\pi-\epsilon}$) have the following asymptotics:

$$\begin{equation*} \lambda_{j,n}=\psi(d_{j,n})-\frac{\psi'(d_{j,n})\eta(d_{j,n})}{n+1}+o\biggl(\frac{1}{n^{\alpha}}\biggr). \end{equation*}$$

This means that the distance between two successive eigenvalues has the two-sided estimate

$$\begin{equation} \frac{c}{n+1}<\lambda_{j+1,n}-\lambda_{j,n}<\frac{C}{n+1}, \end{equation} \tag{ 2.6 }$$

where the constants $c$ and $C$ are independent of $j$ and $n$. On the other hand the difference $\lambda_{j,n}-\psi(d_{j,n})$ also has order $O(1/n)$. In this connection an approximation

$$\begin{equation} \lambda_{j,n}\approx\psi(d_{j,n}) \end{equation} \tag{ 2.7 }$$

is often not sufficient for computations because the approximate formula (2.7) does not `separate' eigenvalues: a point $\psi(d_{j,n})$ need not lie closest to the $j$th eigenvalue, but perhaps to some other eigenvalue, for instance, $\lambda_{j+2,n}$.

The assertion of Theorem 2.3 holds for all eigenvalues of $T_{n}(a)$. However, bearing in mind the behaviour of $\psi(s)$ and $\eta(s)$ close to the endpoints of $[0,\pi]$ (see Lemmas 4.2 and 4.6 in [43]) we can refine the asymptotic formulae for the extreme eigenvalues (cf. Corollary 2.5 in [41]).

Theorem 2.4.  Let $a\in\operatorname{MSL}^{\alpha}$ for some $\alpha\geqslant2$.

(i) If $j/(n+1) \to 0$, then

$$\begin{equation} \lambda_{j,n}=g(0)+\frac{c_1 j^2}{(n+1)^2}+\frac{c_2 j^2}{(n+1)^3}+\Delta^{(4)}_{j,n}, \end{equation} \tag{ 2.8 }$$

where $c_1=\pi^2 g''(0) / 2$, $c_2=-\pi^2 g''(0)\eta'(0)$ and $\Delta^{(4)}_{j,n}=o(({j}/{n})^3)$ as $n \to \infty$.

(ii) If $j/(n+1) \to 1$, then

$$\begin{equation} \lambda_{j,n}=g(\pi)+\frac{c_3 (n+1-j)^2}{(n+1)^2}+\frac{c_4 (n+1-j)^2}{(n+1)^3}+ \Delta^{(5)}_{j,n}, \end{equation} \tag{ 2.9 }$$

where $c_3=\pi^2 g''(\phi_{0}) / 2$, $c_4=-\pi^2 g''(\phi_{0})\eta'(\pi)$ and $\Delta^{(5)}_{j,n}= o((({n+1-j})/{n})^3)$ as $n\to \infty$.

Theorem 2.1 and, as a consequence, also Theorems 2.2 and 2.3 are based on the following result, which provides an exact equation for the eigenvalues. We introduce some further notation. As functions of the form (2.3) are positive, the operator $T_{n}(b)$ is invertible for each $n\geqslant1$. Let $\Theta_k\colon\mathbb T\times[0,\pi]\to\mathbb C$ be defined by

$$\begin{equation*} \Theta_k(t,s):=[T_k^{-1}(b(\,\cdot\,,s))\chi_0](t). \end{equation*}$$

Here the Toeplitz matrix and the inverse matrix are realized as operators in the space

$$\begin{equation*} \ell^{(n)}_{2}:=\biggl\{p(t)\colon p(t)=\sum^{n-1}_{j=0}p_{k}t^{k},\ t\in\mathbb T\biggr\}, \end{equation*}$$

of polynomials of degree at most $n-1$ and the function $\chi_{0}$ is identically equal to one: $\chi_0(t)\equiv1$. In other words, $\Theta_k(t,s)$ is a polynomial of degree at most $n-1$ constructed from the entries in the zeroth column of $T_k^{-1}(b(\,\cdot\,,s))$. For brevity we set $z_k(s)=e^{i\sigma_k(s)}$, $k=1,2$, so that we take $z_k^m(s)$ to be treated as $e^{im\sigma_k(s)}$.

Theorem 2.5.  Assume that $\alpha\geqslant1$, and let $a\in\operatorname{MSL}^\alpha$ and $n\geqslant1$. Then $\lambda=\psi(s)$ is an eigenvalue of the operator $T_n(a)$ if and only if

$$\begin{equation} z_2^{n+1}(s)\Theta_{n+2}(z_1(s),s)\overline{\Theta_{n+2}(z_2(s),s)} =z_1^{n+1}(s)\Theta_{n+2}(z_2(s),s)\overline{\Theta_{n+2}(z_1(s),s)}. \end{equation} \tag{ 2.10 }$$

Note that (2.10) involves the action of the inverse operator $T^{-1}_{n+2}(b(\,\cdot\,,s))$, so it is rather difficult to analyze directly. However, it can be analyzed asymptotically as $n\to\infty$, when we replace $T^{-1}_{n+2}(b(\,\cdot\,,s))$ by the function ${\Theta(t,s)\!:=\}$, where $T^{-1}(b)$ is the inverse of the (infinite) Toeplitz operator. As $\Theta(t,s)$ is easy to calculate in terms of the Wiener-Hopf factorization of $b$, we obtain equation (2.5) in this way.

We conclude this subsection by observing that the formulae in Theorem 2.3 have proved to be numerically highly efficient and yield nice results for $n=40,\dots,100$ (see [37], [41] and [43]).

The arguments resulting in the exact equation (2.10) enable us to deduce exact formulae for the components of the eigenvectors too. The results presented in this subsection were published in [42]. Keeping the notation used in §2.1, for brevity we write

$$\begin{equation*} \theta(t):=\Theta(t,s_{j,n})=[T_{n+2}^{-1}(b(\,\cdot\,,s_{j,n}))\chi_0](t), \end{equation*}$$

where we recall that $s_{j,n}$ is related to the $j$th eigenvalue of the matrix $T_n(a)$ by $\lambda_{j,n}=\psi(s_{j,n})$.

Theorem 2.6.  Assume that $\alpha\geqslant1$ and let $a\in\operatorname{MSL}^\alpha$, $n\geqslant 1$ and $1\leqslant j\leqslant n$. Then the vector

$$\begin{equation} X^{(j,n)}=M^{(j,n)}+L^{(j,n)}+R^{(j,n)} \end{equation} \tag{ 2.11 }$$

whose $p$th component, $p=0,1,\dots,n-1$, is given by

$$\begin{equation*} \begin{aligned} \, M^{(j,n)}_p &:= z_1^{(n-1)/{2}-p}|\theta(z_1)|+(-1)^{n-j}z_2^{(n-1)/{2}-p}|\theta(z_2)|, \\ L^{(j,n)}_p &:=-\frac{z_1^{(n+1)/{2}}\,\overline{\theta(z_1)}}{2\pi i|\theta(z_1)|} \int_\mathbb T\biggl(\frac{\theta(t)-\theta(z_1)}{t-z_1}-\frac{\theta(t)-\theta(z_2)}{t-z_2}\biggr)\frac{dt}{t^{p+1}}, \\ R^{(j,n)}_p&:=\overline{L^{(j,n)}_{n-p-1}}, \end{aligned} \end{equation*}$$

is an eigenvector corresponding to the eigenvalue $\lambda_{j,n}$. In addition, $M^{(j,n)}$ is conjugation symmetric: $M^{(j,n)}_p=\overline{M^{(j,n)}_{n-p-1}}$.

That is, the eigenvector $X^{(j,n)}$ can be represented as a sum of three vectors: the leading term $M^{(j,n)}$, the term $L^{(j,n)}$, concentrated in a neighbourhood of the left-hand endpoint (for small $p$ its components $L^{(j,n)}_p$ exceed significantly the other terms in absolute value) and the term $R^{(j,n)}$, concentrated in a neighbourhood of the right-hand endpoint. In other words, $M^{(j,n)}$ yields a rough approximation of the eigenvector, which is satisfactory for central components, while $L^{(j,n)}$ and $R^{(j,n)}$ play the roles of left- and right-hand corrections, respectively.

In the symmetric case, that is, when the symbol has the property $a(t)=a(t^{-1})$ (which is equivalent to $g(\sigma)=g(2\pi-\sigma)$), the statement of Theorem 2.6 can be simplified slightly.

Corollary 2.1.  Assume that $\alpha\geqslant1$ and let $a\in\operatorname{MSL}^\alpha$ be a function with real Fourier coefficients (so that $a(t)=a(t^{-1})$). Then for each $s\in[0,2\pi]$

$$\begin{equation*} \begin{gathered} \, \varphi_0=\pi, \qquad \varphi(\lambda)=[g|_{[0,\pi]}]^{-1}(\lambda), \qquad\psi(s)=g|_{[0,\pi]}(s), \\ \sigma_1(s)=s, \qquad \sigma_2(s)=2\pi-s, \qquad b(e^{i\sigma},s)=\frac{g(\sigma)-g(s)}{2(\cos s-\cos\sigma)}, \end{gathered} \end{equation*}$$

and the formulae in Theorem 2.6 can be written as follows:

$$\begin{equation} \begin{aligned} \, M^{(j,n)}_p &=2i^{j+1}|\theta(z_1)|\sin\biggl(\biggl(p-\frac{n-1}{2}\biggr)s_j^{(n)}-\frac{j\pi}{2}\biggr), \\ L^{(j,n)}_p &=\frac{z_1^{(n+1)/{2}}\,\overline{\theta(z_1)}}{2\pi i|\theta(z_1)|} \int_\mathbb T\biggl(\frac{\theta(t)-\theta(z_1)}{t-z_1}-\frac{\theta(t)-\overline{\theta(z_1)}}{t-\overline z_1}\biggr)\frac{dt}{t^{p+1}}, \\ R^{(j,n)}_p &:=\overline{L^{(j,n)}_{n-p-1}}. \end{aligned} \end{equation} \tag{ 2.12 }$$

Theorem 2.6 reveals the structure of eigenvectors, but can only be implemented if we calculate the exact eigenvalues and can invert $(n+2)\times(n+2)$-matrices. To simplify the formulae in Theorem 2.6 we look at the `basic' approximation (see Theorem 2.1) of the equation for the eigenvalues,

$$\begin{equation} (n+1)s+\eta(s)=\pi j. \end{equation} \tag{ 2.13 }$$

Note that if it has roots $\widehat s_{j,n}$, then

$$\begin{equation*} |s_{j,n}-\widehat s_{j,n}|=o\biggl(\frac1{n^{\alpha-1}}\biggr), \qquad n\to\infty, \end{equation*}$$

uniformly in $j$. Equation (2.13) is easy to solve numerically. In addition, asymptotic approximations to its solution are given by the formulae in Theorem 2.3. For ${s\in[0,\pi]}$ we have the Wiener-Hopf factorization

$$\begin{equation*} b(t,s)=b_{+}(t,s)b_{-}(t,s), \end{equation*}$$

where

$$\begin{equation} b_\pm(t,s)=\exp\biggl(\frac{1}{2}\log b(t,s)\pm\frac{1}{2\pi i}\int_\mathbb T\frac{\log b(\tau,s)}{\tau-t}\,d\tau\biggr). \end{equation} \tag{ 2.14 }$$

Theorem 2.7.  Let $a \in\operatorname{MSL}^\alpha$, $\alpha\geqslant3$. Let $\widehat z_k:=e^{i\sigma_k(s_{j,n})}$, $k=1,2$, and for $p=0,1,\dots,n-1$ set

$$\begin{equation*} \begin{aligned} \, \widehat M^{(j,n)}_p &:= \frac{\widehat z_1^{\,(n-1)/{2}-p}}{|b_+(\widehat z_1)|}+(-1)^{n-j}\frac{\widehat z_2^{\,(n-1)/{2}-p}}{|b_+(\widehat z_2)|}, \\ \widehat L^{(j,n)}_p &:= -\frac{\widehat z_1^{\,(n+1)/{2}}b_+(\widehat z_1)}{2\pi i|b_+(\widehat z_1)|} \int_\mathbb T\biggl(\frac{b_+^{-1}(t)-b_+^{-1}(\widehat z_1)}{t-\widehat z_1}-\frac{b_+^{-1}(t)-b_+^{-1}(\widehat z_2)}{t-\widehat z_2}\biggr)\frac{dt}{t^{p+1}}, \\ \widehat R^{(j,n)}_p &:=\overline{\widehat L^{(j,n)}_{n-p-1}}. \end{aligned} \end{equation*}$$

Let $\Omega^{(j,n)}$ be the vector defined as the remainder term in the formula

$$\begin{equation} X^{(j,n)}=\widehat M^{(j,n)}+\widehat L^{(j,n)}+\widehat R^{(j,n)}+\Omega^{(j,n)}. \end{equation} \tag{ 2.15 }$$

Then $\Omega^{(j,n)}_p=o(1/n^{\alpha-3})$ as $n\to\infty$ and this relation holds uniformly in $j$ and $p$.

The integral involved in Theorem 2.7 can often be simplified. For example, if the symbol is rational, then the integral can be calculated using residue theory. In fact, in this case the function $b(\,\cdot\,,s)$ given by (2.3) can be expressed by

$$\begin{equation*} b(t,s)=\frac{P(t,s)}{Q(t,s)}, \end{equation*}$$

where $P$ and $Q$ are polynomials in $t$. It is easy to see that the symbol $b(\,\cdot\,,s)$ has a Wiener-Hopf factorization $b(t,s)=b_-(t,s)b_+(t,s)$, where

$$\begin{equation*} b_-(t,s)=\frac{\prod_{j=1}^\rho\bigl(1-\frac{1}{\overline v_j(s)t}\bigr)}{\prod_{j=1}^q\bigl(1-\frac{1}{\overline u_j(s)t}\bigr)}, \qquad b_+(t,s)=\beta_0(s)\frac{\prod_{j=1}^\rho\bigl(1-\frac{t}{v_j(s)}\bigr)} {\prod_{j=1}^q\bigl(1-\frac{t}{u_j(s)}\bigr)}; \end{equation*}$$

here $\rho$ and $q$ are positive integers, $v_j(s)$ and $\overline v_j^{\,-1}(s)$ are formed from the $2\rho$ zeros of $P(\,\cdot\,,s)$, $u_j(s)$ and $\overline u^{\,-1}_j(s)$ are formed from the $2q$ zeros of $Q(\,\cdot\,,s)$, and $\beta_0(s)$ is a constant. Again, for brevity we write $b_\pm(t)$ instead of $b_\pm(t,s_{j,n})$.

Theorem 2.8.  Let $a$ be a rational symbol in $\operatorname{MSL}^\alpha$. Assume further that $b(\,\cdot\,,s)$ has only simple poles. Then an eigenvector can be expressed in the following form:

$$\begin{equation*} X^{(j,n)}=\widehat M^{(j,n)}+\widehat L^{(j,n)}+\widehat R^{(j,n)}+\Omega^{(j,n)}, \end{equation*}$$

where, for $p=0,1,\dots,n-1$,

$$\begin{equation*} \begin{aligned} \, \widehat M^{(j,n)}_p &:=\frac{\widehat z_1^{\,(n-1)/{2}-p}}{|b_+(\widehat z_1)|}+(-1)^{n-j}\frac{\widehat z_2^{\,(n-1)/{2}-p}}{|b_+(\widehat z_2)|}, \\ \widehat L^{(j,n)}_p &:=\frac{\widehat z_1^{\,(n+1)/{2}}(\widehat z_1\,{-}\,\widehat z_2)b_+(\widehat z_1)}{|b_+(\widehat z_1)|} \!\sum_{m=1}^q\!\frac{v_m^{-p-1}(s_{j,n})}{\frac{\partial b_+}{\partial t}(v_m(s_{j,n}))(v_m(s_{j,n})\,{-}\,\widehat z_1)(v_m(s_{j,n})\,{-}\,\widehat z_2)}, \\ \widehat R^{(j,n)}_p &:=\overline{\widehat L^{(j,n)}_{n-p-1}}, \end{aligned} \end{equation*}$$

and for each $\gamma>0$ the limit relation $\Omega^{(j,n)}_p=o(1/n^\gamma)$ holds uniformly in $j$ and $p$ as $n\to\infty$.

Note that if some of the poles of $b(\,\cdot\,,s)$ are multiple, then we can also find an expression for $\widehat L^{(j,n)}_p$ using residue theory.

In [42] we presented numerical experiments validating the formulae in Theorems 2.7 and 2.8 and revealing the meaning of the different terms in these formulae.

Consider a symbol of the form (3.1) and assume that

• (1)  
  the curve $\mathscr{R}(a)$ does not intersect itself; in particular, $a(1) \ne a(-1)$;
• (2)  
  $a'(t) \ne 0$ for $t \in \mathbb T \setminus \{-1,1\}$;
• (3)  
  $a''(\pm 1) \ne 0$.

As usual, set $g(\varphi) := a(e^{i\varphi})$. Then it follows from (3.2) that

$$\begin{equation*} \mathscr{R}(a)=\{z \in \mathbb C \colon\ z= g(\varphi), \ \varphi \in [0,\pi] \}, \qquad g'(0)=g'(\pi)=0, \quad a'(\pm 1)=0. \end{equation*}$$

For sufficiently small $\delta > 0$ set $\mathscr{R}_\delta(a):= g(\Omega_\delta)$, where

$$\begin{equation*} \Omega_\delta := \{\varphi \in \mathbb C \colon 0 < \operatorname{Re}\varphi < \pi,\ |\operatorname{Im} \varphi| < \delta \}. \end{equation*}$$

First we look at the equation $a(z) - \lambda=0$ for $\lambda \in \mathscr{R}(a) \setminus \{a(1), a(-1) \}$. It has $2r$ roots in the complex plane, which we arrange in the order of ascending absolute values. Taking the symmetry (3.2) into account we can represent this set of roots as follows:

$$\begin{equation} \biggl\{u_{1},u_{2},\dots,u_{r-1},u_{r},\frac 1{u_{r}},\frac1{u_{r-1}},\dots,\frac1{u_2},\frac1{u_1} \biggr\}. \end{equation} \tag{ 3.3 }$$

Thus, the root $u_k$, $1 \leqslant k \leqslant r-1$, which lies outside the unit circle $\mathbb T$, corresponds to the root $1/u_k$ inside $\mathbb T$. By conditions 1 and 2 precisely two roots lie on $\mathbb T$:

$$\begin{equation} u_r=e^{i \varphi}\quad\text{and} \quad \frac 1{u_r}=e^{-i \varphi}, \qquad \varphi \in (0,\pi). \end{equation} \tag{ 3.4 }$$

The other $2r-2$ roots are separated from $\mathbb T$ uniformly in $\lambda \in \mathscr{R}(a) \setminus \{a(1),a(-1)\}$. Now let $\lambda \in \mathscr{R}_\delta(a)$. It is easy to show that by 2 and 3, for sufficiently small $\delta$ the equation $a(z) - \lambda=0$ has a unique root in a neighbourhood of the upper half-circle, which has the form $z_r=e^{i \varphi}$, where $\varphi := \varphi(\lambda)$ is an analytic function in $\Omega_\delta$. In accordance with this definition $g(\varphi(\lambda))=\lambda$ for $\lambda \in \mathscr{R}_\delta(a)$. We will regard the $u_k := u_k(\lambda)=u_k(g(\varphi))$ as functions of $\varphi$ and denote them by $u_k := u_k(\varphi)$, where $\varphi \in \Omega_\delta$. Now there exists a positive $\delta_0$ such that

$$\begin{equation} \inf_{k=1,2,\dots,r-1} \inf_{\lambda \in \mathscr{R}_\delta(a) } \mathrm{dist}(u_k, \mathbb T) \geqslant e^{\delta_0}-1 \quad (> \delta_0). \end{equation} \tag{ 3.5 }$$

Consider the functions

$$\begin{equation} \begin{gathered} \, h(z)=\prod_{k=1}^{r-1} \biggl( 1 - \frac{z}{u_k(\varphi)}\biggr), \qquad z \in \mathbb{C}, \quad \varphi \in \Omega_\delta, \end{gathered} \end{equation} \tag{ 3.6 }$$

and

$$\begin{equation} \begin{gathered} \, \theta(\varphi)=- i\log \frac{h(e^{i \varphi})}{h(e^{-i \varphi})}, \qquad \varphi \in \Omega_\delta. \end{gathered} \end{equation} \tag{ 3.7 }$$

It follows from the definition (3.6) of $h$ that $h(z) \ne 0$ for $z \in \Gamma_\delta := \exp\{i \Omega_\delta \}$. Hence we can take a continuous branch of the function $\theta(\varphi)$ in the domain $\varphi \in \overline{\Omega}_\delta$ such that $\theta(0)=\theta(\pi)=0$. We introduce the quantities

$$\begin{equation} d_{j,n} := \frac{\pi j}{n+1}\quad\text{and} \quad e_{j,n} := d_{j,n} - \frac{\theta(d_{j,n})}{n+1} \end{equation} \tag{ 3.8 }$$

and the domains

$$\begin{equation} \Omega_{j,n} := \biggl\{ \varphi \in \mathbb C\colon |\varphi - e_{j,n}| \leqslant \frac{c_{j,n}}{(n+1)^2}\biggr\}, \end{equation} \tag{ 3.9 }$$

where $j=1, 2,\dots,n$, $c_{j,n}=3 M_{j,n}' M_{j,n}$,

$$\begin{equation} M_{j,n}=\sup_{\varphi \in \Omega_{j,n}} |\theta(\varphi)| \quad\text{and} \quad M_{j,n}'=\sup_{\varphi \in \Omega_{j,n}} |\theta'(\varphi)|. \end{equation} \tag{ 3.10 }$$

Note that $c_{j,n}$ is bounded uniformly in $j$ and $n$.

Now we state the main result of this subsection.

Theorem 3.1.  Let $a$ be a function of the form (3.1) such that conditions 1–3 are satisfied. Then the following results hold for sufficiently large $n$.

(i) All the eigenvalues of $T_n(a)$ are simple and $\lambda_{j,n} \in g(\Omega_{j,n})$ for $j=1,2,\dots,n$.

(ii) The points $\varphi_{j,n}$ ($\in \Omega_\delta$) such that $\lambda_{j,n}= g(\varphi_{j,n})$ satisfy

$$\begin{equation} (n+1) \varphi_{j,n}+\theta(\varphi_{j,n})=\pi j+\Delta_n(\varphi_{j,n}), \qquad j=1,2,\dots,n, \end{equation} \tag{ 3.11 }$$

where $|\Delta_n(\varphi)|=O(e^{-\Delta \cdot n})$ and $|\Delta'_n(\varphi)|=O(n e^{-\Delta \cdot n})$ for some positive constant $\Delta$, uniformly in $\varphi \in \Omega_{j,n}$ and $j$ as $n \to \infty$. Equation (3.11) has a unique solution in the domain $\Omega_{j,n}$.

(iii) Each equation of the form

$$\begin{equation} (n+1) \varphi_{j,n}^\ast+\theta(\varphi_{j,n}^*)=\pi j, \qquad j=1,2,\dots,n, \end{equation} \tag{ 3.12 }$$

has a unique solution in $\Omega_{j,n}$, and $\lambda_{j,n}=g(\varphi_{j,n}^*)+O(e^{-\Delta \cdot n}/n)$ for some positive constant $\Delta$ uniformly in $j$ as $n \to \infty$.

(iv) The function

$$\begin{equation} H_{j,n}(\varphi) := d_{j,n} - \frac{\theta(\varphi)}{n+1} \end{equation} \tag{ 3.13 }$$

is a contraction mapping of $\Omega_{j,n}$ into itself; if

$$\begin{equation} \varphi_{j,n}^{(1)}=e_{j,n}\quad\textit{and} \quad \varphi_{j,n}^{(k)}=H_{j,n}(\varphi_{j,n}^{(k-1)}), \quad k \geqslant 2, \end{equation} \tag{ 3.14 }$$

then

$$\begin{equation} |\varphi_{j,n}^{*} - \varphi_{j,n}^{(k)}|=6 \biggl(\frac{M_{j,n}'}{n+1} \biggr)^k \biggl(\frac{M_{j,n}}{n+1}\biggr), \end{equation} \tag{ 3.15 }$$

$$\begin{equation} |\lambda_{j,n}^{*} - g(\varphi_{j,n}^{(k)})|=6 \biggl(\frac{M_{j,n}'}{n+1}\biggr)^k \biggl(\frac{M_{j,n} K_{j,n}'}{n+1}\biggr), \end{equation} \tag{ 3.16 }$$

where $M_{j,n}$ and $M_{j,n}'$ are defined by (3.10) and $K_{j,n}'=\sup_{\varphi \in \Omega_j} |g(\varphi)|$.

Setting $k=2$ in (3.15) and (3.16) we obtain the following asymptotic expansions for $\varphi_{j,n}$ and $\lambda_{j,n}$.

Theorem 3.2.  Let $a(t)$ be a symbol of the form (3.1) such that conditions 1–3 are satisfied. Then

(i)

$$\begin{equation} \varphi_{j,n}=d_{j,n} - \frac{\theta(d_{j,n})}{n+1}+ \frac{\theta(d_{j,n})\theta'(d_{j,n})}{(n+1)^2}+\Delta^{(6)}_{j,n}, \end{equation} \tag{ 3.17 }$$

where $\Delta^{(6)}_{j,n}=O(1 / n^3)$ as $n \to \infty$ uniformly in $j=1,2,\dots,n$;

(ii)

$$\begin{equation} \lambda_{j,n}=g(d_{j,n})+\frac{c_1(d_{j,n})}{n+1}+\frac{c_2(d_{j,n})}{(n+1)^2}+ \Delta^{(7)}_{j,n}, \end{equation} \tag{ 3.18 }$$

where $\Delta^{(7)}_{j,n}=O(d_{j,n} (\pi - d_{j,n}) / n^3)$ uniformly in $j=1,2,\dots,n$ as $n \to \infty$,

$$\begin{equation*} c_1(d)=- g'(d) \theta(d)\quad\text{and} \quad c_2(d)=\frac{1}{2} g''(d) \theta^2(d)+g'(d) \theta(d) \theta'(d). \end{equation*}$$

We have given the asymptotic formulae (3.17) and (3.18) as examples. Using estimates (3.15) and (3.16) for $k>2$ we can find arbitrarily many terms of the expansions for $\lambda_{j,n}$ and $\varphi_{j,n}$.

As a consequence of Theorem 3.2, we obtain a result on the extreme eigenvalues.

Theorem 3.3.  Let $a$ be a symbol of the form (3.1) and assume that conditions 1–3 are satisfied. Then the following results hold:

(i) if $j^2/(n+1) \to 0$, then

$$\begin{equation} \lambda_{j,n}=g(0)+\frac{c_3 j^2}{(n+1)^2}+\frac{c_4 j^2}{(n+1)^3}+\Delta^{(8)}_{j,n}, \end{equation} \tag{ 3.19 }$$

where $c_3=\pi^2 g''(0) / 2$, $c_4=-\pi^2 g''(0) \theta'(0)$ and $\Delta^{(8)}_{j,n}=O(j^4/n^4)$ as $n \to \infty$;

(ii) if $(n+1-j)^2/(n+1) \to 0$, then

$$\begin{equation} \lambda_{j,n}=g(\pi)+\frac{c_5 (n+1-j)^2}{(n+1)^2}+\frac{c_6 (n+1-j)^2}{(n+1)^3}+ \Delta^{(9)}_{j,n}, \end{equation} \tag{ 3.20 }$$

where $c_5=\pi^2 g''(\pi) / 2$, $c_6=-\pi^2 g''(\pi) \theta'(\pi)$ and $\Delta^{(9)}_{j,n}= O((n+1-j)^4/n^4)$ as $n \to \infty$.

Remark 3.1.  The leading term of the asymptotic expansion (3.18) corresponds to a point $g(d)$ on the curve $\mathscr{R}(a)$. This approximation to the eigenvalue $\lambda_{j,n} \approx g(d)$ has accuracy $O(1/n)$ and cannot be considered satisfactory: the distance between successive eigenvalues also has order $O(1/n)$. Thus, this approximation `does not separate' eigenvalues (see Remark 2.1). On the other hand the approximation $\lambda_{j,n} \approx g(e_{j,n})$ has accuracy $O(1/n^2)$ (see (3.16) for $k=1$) and can be treated as an individual approximation to the eigenvalue $\lambda_{j,n}$. In particular, the expression $g(e_{j,n})$ can be used to analyze the position of $\lambda_{j,n}$ relative to the curve $\mathscr{R}(a)$. In fact,

$$\begin{equation*} g(e_{j,n})=g\biggl(d - \frac{\theta(d)}{n+1}\biggr) =g\biggl(d - \frac{\operatorname{Re} \theta(d)}{n+1} - i\,\frac{\operatorname{Im} \theta(d)}{n+1}\biggr), \end{equation*}$$

where

$$\begin{equation*} \operatorname{Re}\theta(d)=\arg \frac{h(e^{i d})}{h(e^{-i d})}, \qquad \operatorname{Im}\theta(d)=-\log \frac{|h(e^{i d})|}{|h(e^{-i d})|}. \end{equation*}$$

Setting

$$\begin{equation*} \widetilde{e}_{j,n} := d - \frac{\operatorname{Re}\theta(d)}{n+1}=\frac{\pi j - \operatorname{Re} \theta(d)}{n+1} \end{equation*}$$

we obtain

$$\begin{equation*} g(e_{j,n})=g(\widetilde{e}_{j,n}) - i\,\frac{g'(\widetilde{e}_{j,n}) \operatorname{Im} \theta(d)}{n+1}+ O\biggl(\frac1{n^2}\biggr). \end{equation*}$$

That is, to within $O(1/n^2)$ the eigenvalue $\lambda_{j,n}$ lies on the normal to $\mathscr{R}(a)$ at the point $g(\widetilde{e}_{j,n})$. The distance from $\lambda_{j,n}$ along the normal has order $O(1/n)$, provided that $\operatorname{Im} \theta(d) \ne 0$. Note that depending on the sign of $\operatorname{Im} \theta(d)$ the point $\lambda_{j,n}$ lies to the right or the left of $\mathscr{R}(a)$ in the plane.

The numerical gain from the formulae in Theorems 3.1–3.3 was fairly comprehensively analyzed in [44].

Theorem 3.4.  Let $a(t)$ be a symbol of the form (3.1) and assume that conditions 1–3 are satisfied. Then eigenvectors $X^{(j,n)}=(X^{(j,n)}_p)_{p=1}^n$ associated with the eigenvalues $\lambda_{j,n}$ can be chosen so that the following asymptotic expressions (putting $\lambda:=\lambda_{j,n}$ for brevity) hold:

$$\begin{equation} X^{(j,n)}_p=a_{j,p}(\lambda)+b_{j,p}(\lambda)+c_{j,p}(\lambda)+O(e^{-\delta_{1}n}), \qquad p= 1, 2, \dots,n, \end{equation} \tag{ 3.21 }$$

where the remainder estimate is uniform with respect to $\lambda$, and $\delta_{1}$ is independent of $n$, $j$ and $p$. Furthermore,

$$\begin{equation} a_{j,p}(\lambda) = \frac{(-1)^{j}e^{-ip\varphi(\lambda)}}{h_{\lambda}(e^{i\varphi(\lambda)})} - \frac{(-1)^{j}e^{ip\varphi(\lambda)}}{h_{\lambda}(e^{-i\varphi(\lambda)})}, \end{equation} \tag{ 3.22 }$$

$$\begin{equation} b_{j,p}(\lambda) = (-1)^{j} \sum_{\nu=1}^{r-1} \biggl[\frac{2i\sin(\varphi(\lambda))}{u_{\nu}^{p}(\lambda)} \cdot \frac{1}{(u_{\nu}(\lambda) - e^{i\varphi(\lambda)})(u_{\nu}(\lambda) - e^{-i\varphi(\lambda)}) h_{\lambda}'(u_{\nu}(\lambda))}\biggr] \end{equation} \tag{ 3.23 }$$

$$\begin{equation} \text{and} \qquad c_{j,p}(\lambda) = (-1)^{j-1} b_{j,n+1-p}(\lambda). \end{equation} \tag{ 3.24 }$$

Remark 3.2.  Theorem 3.4 was established as Theorem 2.3 in [45], albeit under the additional assumption that the function $a(t)-\lambda$ has only simple zeros for each $\lambda$ in $\mathscr{R}_{\delta}(a)$. Now, after [42], we can drop this technical condition.

Remark 3.3.  Let us analyze the formulae in Theorem 3.4. As the root $u_{\nu}(\lambda)$ lies outside the unit circle, the term $b_{j,p}(\lambda)$ is small for components with large index $p$, and the term $c_{j,p}(\lambda)$ is small when $n-p$ is large. In addition, it is easy to see that, as $n$ grows, the contribution of these terms can be significant for a few of the first and last components, respectively. Note however, that the number of these components does not depend significantly on $n$. Components in the middle part of the vector $X^{(j,n)}$ are mostly well approximated by $a_{j,p}(\lambda)$ from (3.22). A numerical illustration of this observation can be found in [45].

Remark 3.4.  In (3.22)–(3.24) we can replace the exact eigenvalues by their approximations (asymptotic expressions), for instance, the ones given by (3.18).

Remark 3.5.  Formulae (3.22)–(3.24) become much simpler for eigenvectors corresponding to the extreme eigenvalues (see Theorem 2.4 in [45]).

Dai, Geary and Kadanoff [46] and Kadanoff [47] considered symbols of the form

$$\begin{equation} a(t)=\biggl(2-t-\frac{1}{t}\biggr)^\gamma(-t)^{\beta}, \qquad t\in\mathbb T, \end{equation} \tag{ 4.1 }$$

where $0<\gamma<-\beta<1$. They conjectured that for large $n$ the corresponding eigenvalues $\lambda=\lambda_{j,n}$ satisfy

$$\begin{equation} \lambda_{j,n}\approx a\biggl(n^{(2\gamma+1)/n}\exp\biggl\{-i\biggl(\frac{2\pi j}{n}\biggr)-\frac{i}{n}d\biggl(\frac{2\pi j}{n}\biggr)\biggr\}+o\biggl(\frac{1}{n}\biggr)\biggr), \end{equation} \tag{ 4.2 }$$

where $d$ is some function (unknown to them). This conjecture was supported by numerical experiment and a certain heuristic argument.

Now we state the main results of [48] and compare these with (4.2).

As usual, let $H^\infty$ be the Hardy space of bounded analytic functions in the unit disc $\mathbb D$. For a fixed function $a\in C(\mathbb T)$ let $\operatorname{wind}_\lambda(a)$ denote the number of windings of the image of $a$, that is, the curve $\mathscr R(a):=\{z\in \mathbb C\colon\ z=a(t),\ t\in\mathbb T\}$, about the point $\lambda\in\mathbb C\setminus\mathscr R(a)$, and let $\mathscr D(a)$ be the set of points $\lambda\in\mathbb C$ for which $\operatorname{wind}_\lambda(a)\neq0$. In this subsection we treat matrices $T_n(a)$ with symbols of the form $a(t)=t^{-1}h(t)$, where

• (1.1)  
  $h\in H^\infty$ and $h_0:= h(0)\neq 0$;
• (2.1)  
  $h(t)=(1-t)^\alpha f(t)$, where $\alpha\in [0,\infty)\setminus\mathbb Z$ and $f\in C^\infty(\mathbb T)$;
• (3.1)  
  $h$ extends analytically to an open neighbourhood $W$ of the set $\mathbb T\setminus\{1\}$ which does not contain $1$;
• (4.1)  
  $\mathscr R(a)$ is a Jordan curve in $\mathbb C$ and $\operatorname{wind}_\lambda(a)=-1$ for each $\lambda\in\mathscr D(a)$.

Note that if we set $\gamma={\alpha}/{2}$ and $\beta=\gamma-1$ in (4.1), then the symbol satisfies conditions (1.1)–4.1, with $f(t)\equiv(-1)^{2\gamma-1}$.

Let $D_n(a)$ denote the determinant of $T_n(a)$. Then the eigenvalues of the matrix $T_n(a)$ are the roots of the equation $D_n(a-\lambda)=0$. By our assumptions $T_n(a)$ is a Hessenberg matrix, that is, it can be obtained from a lower triangular matrix by adding one nontrivial diagonal. In combination with the Baxter-Schmidt formula (see [5], §2.3, for instance) this enables us to represent $D_n(a-\lambda)$ as a Fourier integral. Its value depends significantly on the pole at $\lambda$ and the singularity of the term $(1-t)^\alpha$ at $1$. Let $W_0$ be a small neighbourhood of zero in $\mathbb C$. We can show that for each $\lambda\in\mathscr D(a)\cap(a(W)\setminus W_0)$ there exists a unique point $t_\lambda\notin\overline{\mathbb D}$ such that $a(t_\lambda)=\lambda$. An asymptotic analysis of the Fourier integral just mentioned yields an asymptotic representation for $D_n(a-\lambda)$.

In all the statements in this subsection we take

$$\begin{equation*} \alpha_0 := \min\{\alpha,1\}\quad\text{and} \quad c_\alpha := \frac{\pi}{f(1)\Gamma(\alpha+1)\sin(\alpha\pi)}. \end{equation*}$$

Theorem 4.1.  Let $a(t)=t^{-1}h(t)$ be a symbol satisfying conditions (1.1)–4.1. Then for each small open neighbourhood $W_0$ of zero in $\mathbb C$ and points $\lambda\in\mathscr D(a)\cap(a(W)\setminus W_0)$

$$\begin{equation} D_n(a-\lambda)=(-h_0)^{n+1}\biggl(\frac{1}{t_\lambda^{n+2}a'(t_\lambda)} -\frac{1}{c_\alpha \lambda^2 n^{\alpha+1}}+R_1(n,\lambda)\biggr), \end{equation} \tag{ 4.3 }$$

where $R_1(n,\lambda)=O(1/n^{\alpha+\alpha_0+1})$ as $n\to\infty$ uniformly in $\lambda\in a(W)\setminus W_0$.

We now turn to the main results in this subsection. Set $\omega_n:=\exp(-2\pi i/n)$. For the values of $n$ under consideration there exist positive integers $n_1$ and $n_2$ such that $\omega_n^{n_1},\omega_n^{n-n_2}\in a^{-1}(W_0)$, but $\omega_n^{n_1+1},\omega_n^{n-n_2-1}\notin a^{-1}(W_0)$. Recall that $a(t_\lambda)=\lambda$.

Theorem 4.2.  Let $a(t)=t^{-1}h(t)$ be a symbol with properties (1.1)–4.1. Then for each small open neighbourhood $W_0$ of zero and for each $j$ between $n_1$ and $n-n_2$,

$$\begin{equation} t_{\lambda_{j,n}}=n^{(\alpha+1)/n}\omega_n^j \biggl(1+\frac{1}{n}\log\biggl(\frac{c_\alpha a^2(\omega_n^j)}{a'(\omega_n^j)\omega_n^{2j}}\biggr) +R_2(n,j)\biggr), \end{equation} \tag{ 4.4 }$$

where $R_2(n,j)=O(1/n^{\alpha_0+1})+O(\log n/n^2)$ as $n\to\infty$ uniformly in $j$.

Thus formula (4.4) proves the conjecture (4.2) in the special case when $\beta=\gamma-1$. In addition, we have found the function

$$\begin{equation*} d(t):= t\log\biggl(\frac{c_\alpha a^{2}(t)}{a'(t)t^{2}}\biggr) \end{equation*}$$

and have refined the order of the error term.

Taking the value of $a$ at the point (4.4) we obtain the following asymptotic expression for $\lambda_{j,n}$.

Theorem 4.3.  Let $a(t)=t^{-1}h(t)$ be a symbol with properties (1.1)–4.1. Then for each small open neighbourhood $W_0$ of zero in $\mathbb C$ and $j$ between $n_1$ and $n-n_2$,

$$\begin{equation} \lambda_{j,n}=a(\omega_n^j) +(\alpha+1)\omega_n^ja'(\omega_n^j)\frac{\log n}{n} +\frac{\omega_n^ja'(\omega_n^j)}{n}\log\biggl(\frac{c_\alpha a^2(\omega_n^j)}{a'(\omega_n^j)\omega_n^{2j}}\biggr) +R_3(n,j), \end{equation} \tag{ 4.5 }$$

where $R_3(n,j)=O(1/n^{\alpha_0+1})+O(\log^2n/n^2)$ as $n\to\infty$ uniformly for $j$ between $n_1$ and $n-n_2$.

Remark 4.1.  Here we have written out only a few of the first terms of the asymptotic expansion. Note however that using our method an arbitrary number of terms in this expansion can be obtained.

A numerical illustration of Theorem 4.3 can be found in [48].

To present the formulae for eigenvectors we define a function $b^{(j,n)}$ on $\mathbb T$ by

$$\begin{equation*} b^{(j,n)}(t):=\frac{1}{h(t)-\lambda_{j,n} t} \end{equation*}$$

and let $b^{(j,n)}_p$ denote its $p$th Fourier coefficient.

Theorem 4.4.  If $b_{n-1}^{(j,n)} \neq 0$, then

$$\begin{equation} X^{(j,n)} :=(b^{(j,n)}_p)_{p=0}^{n-1} \end{equation} \tag{ 4.6 }$$

is an eigenvector of $T_n(a)$ corresponding to the eigenvalue $\lambda_{j,n}$.

The next result describes the asymptotic behaviour of $b^{(j,n)}_p$ for large $p$ and $n$.

Theorem 4.5.  The relation

$$\begin{equation} b_p^{(j,n)}=\frac{D_2(\omega_n^j)\omega_n^{-jp}}{(D_1(\omega_n^j)n^{\alpha+1})^{p/n}}(1+R_1(j,n,p)) +\frac{D_3(\omega_n^j)}{p^{\alpha+1}}(1+R_2(j,n,p)) \end{equation} \tag{ 4.7 }$$

holds, where

$$\begin{equation*} \begin{gathered} \, D_1(u):=\frac{c_\alpha a^2(u)}{u^2a'(u)}, \qquad D_2(u):=\frac{-1}{u^2a'(u)}, \qquad D_3(u):=\frac{1}{c_\alpha a^2(u)}, \\ R_1(j,n,p)=O\biggl(\frac{p}{n^{\alpha_0+1}}\biggr)+O\biggl(\frac{\log n}{n}\biggr)\quad\text{uniformly in }j \text{ as }p,n\to\infty, \\ R_2(j,n,p)=O\biggl(\frac{\log n}{n}\biggr)+O\biggl(\frac1p^{\alpha_0}\biggr)\quad\text{uniformly in $n$ and $j$}. \end{gathered} \end{equation*}$$

We stress that we only consider points $\omega_n^j$ in $W\setminus a^{-1}(W_0)$. In this domain the functions $a$ and $a'$ are bounded and separated away from zero. Hence the $D_\ell$, $\ell=1,2,3$, are also bounded and separated away from zero there. Note also that since $p<n$, $R_1$ tends to zero as $n\to\infty$.

Remark 4.2.  We compare the above results on the asymptotics of eigenvectors and the results in [46] and [47] concerning symbols of the form (4.1).

We have already mentioned that for $\beta=\gamma-1$ the function (4.1) becomes `our' symbol of the form $a(t)=t^{-1}h(t)$, with $h(t)=(-1)^{2\gamma-1}(1-t)^{2\gamma}$. In [46] the authors conjectured that an eigenvector has the form

$$\begin{equation} X^{(j,n)} \approx \biggl(\frac{1+O(1/n)}{t_{\lambda_{j,n}}^p}\biggr)_{p=0}^{n-1},\quad\text{where} \quad t_{\lambda_{j,n}}\approx n^{(\alpha+1)/n}\omega_n^j \end{equation} \tag{ 4.8 }$$

as $n\to\infty$. Theorem 4.2 refines this conjecture and proves it while giving a rigorous estimate for the error term. Note that

$$\begin{equation*} D_1^{p/n}(\omega_n^j)=\exp\biggl(\frac{p}{n}\log D_1(\omega_n^j)\biggr) =\exp\biggl(\frac{p}{n}\,O(1)\biggr)=1+O\biggl(\frac{p}{n}\biggr). \end{equation*}$$

This shows that (4.8) corresponds to the first term on the right-hand side of (4.7) in the case when the ratio $p/n$ is close to zero. Otherwise the representation (4.8) can give a large error: the corresponding numerical examples were presented in [50].

Remark 4.3.  Our asymptotic expansion (4.7) holds for $\lambda_{j,n}$ outside a small neighbourhood $W_0$ of zero and for large $p$. For small $p$, for instance, $p=0,1,\dots,m-1$, $m\ll n$, the values of $b_p^{(j,n)}$ can be calculated using the relation between the Fourier coefficients of $b^{(j,n)}=1/(h(t)-\lambda_{j,n} t)$ and $h(t)=(1-t)^\alpha f(t)$. Here we obtain a triangular system of equations, which can be solved by $O(m^2)$ operations, and taking $m=[\sqrt{n}\,]$ we need $O(n)$ operations. We can calculate the remaining components using asymptotic formulae (4.7), which also have order of complexity $O(n)$ from a numerical standpoint.