| Russian Mathematical Surveys |
|
|
|
|
|
Brief Communications Spectral problem for the vector Stieltjes string A. I. Aptekareva, V. A. Kalyaginb a Keldysh Institute of Applied Mathematics of Russian Academy of Sciences b HSE University
Russian version:
Uspekhi Matematicheskikh Nauk, 2022, Volume 77, Issue 5(467), DOI: https://doi.org/10.4213/rm10067 
Bibliographic databases:
    1. The Stieltjes stringThe system of difference equations
$$
\begin{equation}
\begin{cases} \theta_{k+1}-\theta_k=m_{k+1}z\eta_k, & k\in \mathbb{Z}_+, \ \theta_0=0, \\ \eta_k-\eta_{k-1}=l_k\theta_k, & k\in \mathbb{N}, \ \hphantom{{}_{+}} \eta_0=1, \end{cases}
\end{equation}
\tag{1}
$$
determines sequentially, from the parameters $\{m_k,l_k\}$ of a string, the characteristics $\{\eta_k, \theta_k\}$ of its oscillations, in their dependence on the spectral parameter $z$. On the other hand, the solutions $\{\theta_k,\eta_k\}$ of (1) and the solutions $\{\theta^{(1)}_k,\eta^{(1)}_k\}$ of the same system corresponding to the initial data $\theta^{(1)}_0=1$ and $\eta^{(1)}_0=0$ form convergents of a classical Stieltjes continued fraction [1], [2]:
$$
\begin{equation}
S_n(z)=\begin{cases} \theta^{(1)}_k/\theta_k, & n=2k-1,\\ \eta^{(1)}_k/\eta_k, & n=2k, \end{cases}\ \ k\in \mathbb{N};\quad S(z)=\frac{1|}{|m_1 z\,+}\,\,\frac{1|}{|l_1\,+}\,\, \frac{1|}{|m_2z\,+}\,\,\frac{1|}{|l_2+\dotsb}\,.
\end{equation}
\tag{2}
$$
The equations of the Stieltjes string (1) are prototypical for many remarkable systems of operators, such as the Krein string [2], [3] or canonical de Branges systems [4], [5]. Another continued fraction equivalent to (2) and the corresponding recurrence relations has the following form:
$$
\begin{equation}
S(z)=\frac{1|}{|z\,+}\,\frac{a_1|}{|1\,+}\,\frac{a_2|}{|z\,+}\, \frac{a_3|}{|1{}+{}\cdots}\,,\quad A_{n+1}=\epsilon_nA_n+a_nA_{n-1},
\end{equation}
\tag{3}
$$
where $\epsilon_n=z$ for $n=2k$ and $\epsilon_n=1$ for $n \ne 2k$, $k\in \mathbb{N}$; here $A_{-1}=0$ and $A_0=1$ for the numerators, while $A^{(1)}_{-1}=1$ and $A^{(1)}_0=1$ for the denominators of convergents. After cancellations, convergents of (2) coincides with convergents of (3). If $m_k,l_k>0$ ($a_k>0$) and the associated moment problem is determinate, then the Stieltjes fraction converges locally uniformly in $\mathbb{C} \setminus [0,+\infty)$ to a Stieltjes function $S(z)$ generated by some measure $\sigma>0$:
$$
\begin{equation}
S_n(z) \to S(z)=\int_0^{+\infty}\frac{d\sigma(x)}{z+x}\,, \qquad \operatorname{supp} \sigma \in \mathbb{R}_+.
\end{equation}
\tag{4}
$$
2. The vector Stieltjes fractionThe concept of a vector continued fraction is based on the Jacobi–Perron procedure of taking the inverse of a vector $\mathbf{c}=(c_1,c_2,\dots,c_d) \in \mathbb{C}^d$, namely, $\mathbf{c}^{-1}=\boldsymbol{1}/\mathbf{c}= (1/c_d,c_1/c_d,\dots,c_{d-1}/c_d)$. In this paper we consider a vector generalization of the Stieltjes fraction (2) and the related system of difference equations, which is similar to (1). Theorem 1. Given a vector continued fraction $\mathbf{S}(z)$, assume that $m_k,l_{1,k},l_{2,k} \ne 0$: Note that in [6] a vector generalization of the equivalent Stieltjes fraction (3) was considered. For the numerators $A^{(1)}_n$ and $A^{(2)}_n$ and denominators $A_n$ of convergents we obtained
$$
\begin{equation}
A_{n+1}=\epsilon_nA_n+a_{n-1}A_{n-2}, \quad \epsilon_n=z \;\,\text{for } n=3k \quad\text{and}\quad \epsilon_n=1 \;\,\text{for }n \ne 3k,\quad k\in \mathbb{N};
\end{equation}
\tag{9}
$$
here
$$
\begin{equation*}
a_{3k-2}\,{=}\,(m_kl_{1,k}l_{2,k})^{-1},\ a_{3k-1}\,{=}\,(m_{k+1}l_{1,k}l_{2,k})^{-1},\text{ and } a_{3k}\,{=}\,(m_{k+1}l_{1,k+1}l_{2,k+1})^{-1}.
\end{equation*}
\notag
$$
3. The spectral problemIt is more convenient to solve the direct spectral problem of finding $(\sigma_1,\sigma_2)$ from (8) in terms of the coefficients $\{a_n\}$ in (9). It is known [7] that the transformation $f_j(z)=z^{j+1}S_j(z^3)$, $j=1,2$, takes the Stieltjes system of functions to the Nikishin system $f=(f_1,f_2)$ generated by the measures $d\mu_j(x)=d\sigma_j(x^3)$. A Nikishin system if one of the basic systems of functions for which the denominators of Hermite–Padé approximants $\{Q_n(z)\}$ are multiply orthogonal polynomials with respect to a system of measures $(\mu_1,\mu_2)$ [8]. For applications of Nikishin systems to the spectral theory of Schrödinger operators on graphs, see [9] and [10]. We have
$$
\begin{equation}
Q_{n+1}(x)=x Q_n(x)-a_{n-2}Q_{n-2}(x),\text{ where } Q_0(x)=1,\ Q_1(x)=x,\ Q_2(x)=x^2.
\end{equation}
\tag{10}
$$
Thus, the direct spectral problem for the vector Stieltjes string reduces to finding the orthogonality measures $(\mu_1,\mu_2)$ from the coefficients of (10). Let $W(z)$ be the algebraic function satisfying $W^3-zW^2+1=0$, with branches such that $W_0(\infty)=\infty$ and $\operatorname{Im} W_2=0$ on $[0,\alpha)$, where $\alpha:=(27/4)^{1/3}$. Let $\{Q_n^{(j)}\}_{j=1,2}$ denote the other two solutions of (10) corresponding to the initial data $Q^{(1)}_0(x)=0$, $Q^{(1)}_1(x)=1$, $Q^{(1)}_2(x)=x$, and $Q^{(2)}_0(x)=0$, $Q^{(2)}_1(x)=0$, $Q^{(2)}_2(x)=1$, and let
$$
\begin{equation*}
D_n(x):=\begin{vmatrix} Q_n(x) & Q_{n+1}(x)& W_2^2(x) \\ Q_{n-1}(x) & Q_n(x) & W_2(x) \\ Q_{n-2}(x) & Q_{n-1}(x) & 1 \end{vmatrix}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
D_n^{(j)}(x):=\begin{vmatrix} Q_n(x) & Q^{(j)}_n(x)& W_2^2(x) \\ Q_{n-1}(x) & Q^{(j)}_{n-1}(x) & W_2(x) \\ Q_{n-2}(x) & Q^{(j)}_{n-2}(x) & 1 \end{vmatrix}.
\end{equation*}
\notag
$$
Theorem 2. If $\sum_{n=1}^{\infty} |a_n-1|<\infty$, then $d\mu_j(x)=\rho_j(x)\,dx$ and, uniformly in $x$, 
Citation in format AMSBIB
\Bibitem{AptKal22} |
|
||||||||||||||||||||||||||||||||||||||||||||||||||||
|
Contact us: |
Terms of Use
|
Registration to the website |
Logotypes |
|