UDC 517.948.35:513.88

MATHEMATICS

V. E. LYANTSE

A NON-SELF-ADJOINT DIFFERENCE OPERATOR

(Presented by Academician I. N. Vekua on 10 VI 1966)

In the present note a difference analogue of the non-self-adjoint Sturm—Liouville operator on the half-line is studied.

1. We shall consider the difference expression \(l\), which transforms a sequence of complex numbers \(y=(y_0,y_1,y_2,\ldots)\) into the sequence \(ly=((ly)_1,(ly)_2,\ldots)\) by the formulas

\[ (ly)_j=\frac12(y_{j-1}+y_{j+1})-b_jy_j,\qquad j=1,2,\ldots; \tag{1} \]

here \((b_1,b_2,\ldots)\) is a given sequence of complex numbers. We shall be interested in certain solutions \(y\) of the difference equation

\[ (ly)_j=\lambda y_j=0\qquad \bigl(\lambda=\tfrac12(\rho^{-1}+\rho),\ |\rho|\geqslant 1\bigr), \tag{2} \]

containing the complex parameter \(\lambda\), which it is convenient to regard as the function of \(\rho\) indicated in (2).

1. Let \(s_j(\lambda)\), \(c_j(\lambda)\) be solutions of equation (2) satisfying the initial conditions \(s_0(\lambda)=0,\ s_1(\lambda)=1,\ c_0(\lambda)=1,\ c_1(\lambda)=0\)*. These solutions are polynomials in \(\lambda\) of degrees \(j-1\) and \(j-2\), respectively. For each \(\delta>0\) denote by \(W_\delta\) the following domain in the complex \(\lambda\)-plane: \(W_\delta=\{\rho:\ |\rho|\geqslant 1,\ |\rho^2-1|\geqslant\delta\}\). There exists a \(C>0\), and for each \(\delta>0\) there exists a \(C_\delta>0\), such that for \(\rho\in W_\delta\) and for all \(j=2,3,\ldots\)

\[ |s_j(\lambda)|\leqslant C|\rho|^{j-1}\exp C_\delta |\rho|^{-1}\sum_{l=1}^{j-1}|b_l|, \qquad |c_j(\lambda)|\leqslant C|\rho|^{j-2}\exp C_\delta |\rho|^{-1}\sum_{l=1}^{j-1}|b_l|. \]

1. Put

\[ \sigma_j=\sum_{l=j+1}^{\infty}|b_l|,\qquad \sigma_{1j}=\sum_{l=j+1}^{\infty}l|b_l|. \]

If \(\sigma_{1j}<\infty\), then equation (2) has a solution \(e_j(\rho)\), which can be represented in the form

\[ e_j(\rho)=\rho^{-j}+\sum_{l=j+1}^{\infty}k_{jl}\rho^{-l},\qquad |\rho|\geqslant 1,\qquad j=0,1,2,\ldots, \]

and moreover

\[ |k_{jl}|\leqslant C e^{\sigma_{1j}}\sigma_{[(j+l)/2]},\qquad l>j=0,1,2,\ldots; \]

here \(C\) is a certain number, and \([x]\) is the integer part of \(x\). The assertion just formulated is analogous to the well-known theorem on the transformation operator for solutions of differential equations.

It is easy to see that the solution \(e_j(\rho)\) is continuous in \(\rho\) for \(|\rho|\geqslant 1\) and holomorphic for \(|\rho|>1,\ j=0,1,2,\ldots\).

For the existence of a solution \(e_j(\rho)\) equal to \(\rho^{-j}(1+o(1))\) as \(j\to\infty\), it is sufficient that \(\sigma_j<\infty\).

1. If \(\sigma_j<\infty\), then for each \(\delta>0\) there exists a natural number \(h_\delta\) such that equation (2) has a solution \(\hat e_j(\rho)\), satisfying asymptoti-

* If \(b_1=b_2=\cdots=0\), then \(s_j(\lambda)=(\rho^{-j}-\rho^j)(\rho^{-1}-\rho)^{-1}\) and is a difference analogue of the function \(\lambda^{-1/2}\sin x\lambda^{1/2}\); \(c_j(\lambda)\) is a difference analogue of \(\cos x\lambda^{1/2}\).

to the asymptotic equality

\[ \hat e_j(\rho)=\rho^j[1+O(1/\rho)],\qquad |\rho|\to\infty \]

uniformly for \(j>h_\delta,\ \rho\in W_\delta\). Moreover, for every \(\alpha>0\)

\[ \hat e_j(\rho)=\rho^{(j)}[1+o(1)],\qquad j\to\infty, \]

uniformly with respect to \(\rho\) for \(\rho\in W_\delta,\ |\rho|>1+\alpha\).

If, for some \(\gamma>0\), the series \(\sum l^{1+\gamma}|b_l|\) converges, then for \(\lambda=1\) and \((\lambda=-1)\) equation (2) has a solution \(\hat e_j(\rho)\) for which

\[ \hat e_j(1)=j[1+O(j^{-\gamma})],\qquad j\to\infty \]

\[ \bigl(\hat e_j(-1)=(-1)^j j[1+O(j^{-\gamma})],\qquad j\to\infty\bigr). \]

1. The Wronskian \(w(\alpha_j,\beta_j)=\alpha_j\beta_{j+1}-\alpha_{j+1}\beta_j\) of any pair of solutions \(\alpha_j,\beta_j\) of equation (2) does not depend on \(j\). We have

\[ w\bigl(e_j(\rho),\hat e_j(\rho)\bigr)=\rho-\rho^{-1},\qquad \rho\in W_\delta . \]

1. Denote by \(H=l_2[1,\infty)\) the Hilbert space of sequences \(y=(y_1,y_2,\ldots)\) with norm

\[ \|y\|=\left(\sum_{n=1}^{\infty}|y_n|^2\right)^{1/2}. \]

For each sequence \(y\in H\) set

\[ y_0=\theta y_1, \tag{3} \]

where \(\theta\) is a fixed complex number. In what follows, when computing \((ly)_1\), we shall take into account the “boundary” condition (3), and by \(Ly=L_\theta y\) we shall denote the sequence \(((ly)_1,(ly)_2,\ldots)\). If \(\sup |b_j|<\infty\), in particular if \(\sigma_j<\infty\), then \(L\) is a continuous linear operator mapping all of \(H\) into itself.

1. Let \(\sigma_j<\infty\). Then the operator \(L\) has no eigenvalues \(\lambda\) in the interval \((-1,+1)\). The spectrum of the operator \(L\) consists of the segment \([-1,+1]\) and eigenvalues \(\lambda\) determined by the formula \(\lambda=\tfrac12(\rho^{-1}+\rho)\), where \(\rho\) is a root of the equation \(\theta e_1(\rho)-e_0(\rho)=0\) and \(|\rho|>1\). The eigenvalues of \(L\) form a bounded (at most countable) set, whose limit points may lie only on the segment \([-1,+1]\). All points \(\lambda\in(-1,+1)\) belong to the continuous spectrum of \(L\). If the series \(\sum l^{1+\gamma}|b_l|\) converges for some \(\gamma>0\), then the points \(\lambda=\pm1\) also belong to the continuous spectrum of \(L\).

2. Let \(\sigma_j<\infty\) and let \(\lambda\) be a point of the resolvent set, and \(R_\lambda=(L-\lambda)^{-1}\) the resolvent of the operator \(L\). Put

\[ a(\rho)=\theta e_1(\rho)-e_0(\rho),\qquad \omega_j(\lambda)=s_j(\lambda)+\theta c_j(\lambda); \]

\[ R_{jl}(\lambda)= \begin{cases} 2e_j(\rho)\omega_l(\rho)/a(\rho), & \text{for } l=1,\ldots,j-1,\\ 2\omega_j(\rho)e_l(\rho)/a(\rho), & \text{for } l=j,j+1,\ldots . \end{cases} \]

Then for all \(f\in H\)

\[ (R_\lambda f)_j=\sum_{l=1}^{\infty}R_{jl}(\lambda)f_l,\qquad j=1,2,\ldots . \]

There exists \(C>0\), and for every \(\delta>0\) there exists \(C_\delta\) such that

\[ C/|a(\rho)|\sqrt{|\rho|-1}\le \|R_\lambda\|\le C_\delta/|a(\rho)|(|\rho|-1), \]

where the second inequality holds only for \(\rho\in W_\delta\).

1. Everywhere in what follows we assume that, for some \(\varepsilon>0\),

\[ \sum_{j=1}^{\infty}(1+\varepsilon)^j|b_j|<\infty . \tag{4} \]

Only under this assumption (which is analogous to the condition of exponential decrease of the “potential,” introduced in the case of a differential operator by M. A. Naimark) shall we construct spectral expansions corresponding to the operator \(L\).

Now the solution \(e_j(\rho)\) (see Sec. 3) admits an analytic continuation from the domain \(|\rho|>1\) to a function holomorphic in the domain \(|\rho|>(1+\varepsilon)^{-1/2}\). Therefore the equation \(\theta e_1(\rho)-e_0(\rho)=0\) has only a finite number of solutions in the domain \(|\rho|\geq 1\), and, in particular, the operator \(L\) has only a finite number of eigenvalues. The roots of the equation \(\theta e_1(\rho)-e(\rho)=0\) such that \(|\rho|\geq 1\) will be called the singular numbers of the operator \(L\). Denote by \(\rho_1,\ldots,\rho_\alpha\) the singular numbers for which \(|\rho_k|>1,\ k=1,\ldots,\alpha\), and by \(\rho_{\alpha+1},\ldots,\rho_\beta\) those singular numbers for which \(|\rho_k|=1,\ k=\alpha+1,\ldots,\beta\). Let \(\lambda_k=\tfrac12(\rho_k^{-1}+\rho_k)\). For \(k=1,\ldots,\alpha\), the number \(\lambda_k\) is an eigenvalue of the operator (and there are no other eigenvalues). The numbers \(\lambda_{\alpha+1},\ldots,\lambda_\beta\) belong to the continuous spectrum; we shall call them spectral singularities of the operator \(L\).

In what follows, by \(m_k\) we denote the multiplicity of the root \(\rho_k\) of the equation \(\theta e_1(\rho)-e_0(\rho)=0\). The numbers \(m_1,\ldots,m_\alpha\) coincide with the multiplicities of the eigenvalues \(\lambda_1,\ldots,\lambda_\alpha\), i.e., with the dimensions of the corresponding root subspaces. The numbers \(m_{\alpha+1},\ldots,m_\beta\) will be called the multiplicities of the spectral singularities \(\lambda_{\alpha+1},\ldots,\lambda_\beta\).

1. Denote by \(\mathcal{B}_{kj}(\lambda)\) any bounded measurable functions on the interval \(-1\leq \lambda\leq +1\), holomorphic in a neighborhood of the spectral singularities \(\lambda_{\alpha+1},\ldots,\lambda_\beta\) and satisfying the conditions

\[ \left\{\left[\left(\frac{d}{d\lambda}\right)^{j'}\mathcal{B}_{kj}(\lambda)\right]_{\lambda=\lambda_{k'}}\right\} = \begin{cases} 1, & \text{if } j=j',\ k=k',\\ 0, & \text{in all other cases.} \end{cases} \]

For an arbitrary function \(\Phi(\lambda)\), differentiable \(m_k-1\) times at the point \(\lambda_k,\ k=\alpha+1,\ldots,\beta\), put

\[ [\mathcal{B}\Phi(\lambda)] = \Phi(\lambda) - \sum_{k=\alpha+1}^{\beta} \sum_{j=0}^{m_k-1} \mathcal{B}_{kj}(\lambda)\Phi^{(j)}(\lambda_k). \]

The point \(\lambda_k\) is a root of the function \([\mathcal{B}\Phi(\lambda)]\) of multiplicity at least \(m_k,\ k=\alpha+1,\ldots,\beta\). The following expansion in eigenfunctions of the operator \(L\) for the “kernel” of the resolvent (see Sec. 8) is valid:

\[ R_{jl}(z) = \frac{2}{\pi} \int_{-1}^{+1} \left[ \mathcal{B}\frac{\omega_j(\lambda)\omega_l(l)}{\lambda-z} \right] \frac{i\sqrt{1-\lambda^2}\,d\lambda} {a(\lambda+i\sqrt{1-\lambda^2})\,a(\lambda-i\sqrt{1-\lambda^2})} + \]

\[ + \sum_{k=1}^{\beta} \left\{ \left[ \left(\frac{d}{d\lambda}\right)^{m_k-1} M_k(\lambda) \frac{\omega_j(\lambda)\omega_l(\lambda)}{\lambda-z} \right]_{\lambda=\lambda_k} \right\}; \tag{5} \]

here \(\sqrt{1-\lambda^2}\geq 0\) for \(\lambda\in[-1,+1]\), \(a(\rho)=\theta e_1(\rho)-e_0(\rho)\), and the functions \(M_k(\lambda)\) for \(k=\alpha+1,\ldots,\beta\) depend on the choice of the functions \(\mathcal{B}_{kj}(\lambda)\). The subintegral function in formula (5) is bounded. If there are no spectral singularities, then the operator \(\mathcal{B}\) in formula (5) is superfluous.

1. Introduce the notation \(\omega_f(\lambda)=\sum_{j=1}^{\infty} f_j\omega_j(\lambda)\). There exists a number \(C>0\) such that

\[ \int_{-1}^{+1} |\omega_f(\lambda)|^2\sqrt{1-\lambda^2}\,d\lambda \leq C\sum_{j=1}^{\infty}|f_j|^2. \]

Let \(\omega_j^{(m)}(\lambda)=(d/d\lambda)^m\omega_j(\lambda)\); then

\[ w_f^{(m)}(\lambda_k)=\sum_{j=1}^{\infty} f_j\omega_j^{(m)}(\lambda_k), \]

\(k=1,\ldots,\alpha;\ m=0,\ldots,m_k-1\), are continuous functionals of \(f\in H\). The collection of quantities
\(\omega_f(\lambda)\), \(\lambda\in[-1,+1]\), \(\omega_f^{(m)}(\lambda_k)\), \(k=1,\ldots,\alpha\), \(m=0,\ldots,m_k-1\), will be called the \(L\)-Fourier transform of the element \(f\in H\).

There exist linear continuous functionals \(M_{km}: f\to M_{km}(f)\), defined on the space \(H\), such that for every element \(f\in H\) the following expansion in eigenfunctions of the operator \(L\) is valid:

\[ f_j=\frac{2}{\pi}\int_{-1}^{+1} [\mathfrak{B}\omega_j(\lambda)]\,\omega_f(\lambda)\, \frac{\sqrt{1-\lambda^2}\,d\lambda} {a(\lambda+i\sqrt{1-\lambda^2})\,a(\lambda-i\sqrt{1-\lambda^2})} + \sum_{k=1}^{\beta}\sum_{m=0}^{m_k-1} M_{km}(f)\,\omega_j^{(m)}(\lambda_k). \tag{6} \]

For \(k=1,\ldots,\alpha\) the formulas for the functionals \(M_{km}\) are not difficult to write explicitly. For \(k=\alpha+1,\ldots,\beta\) the values of these functionals are uniquely determined by the condition \(f\in H=l_2[1,\infty)\). Formula (6) may be interpreted as the inversion formula for the \(L\)-Fourier transform of a function \(f\in H\). Using methods analogous to those employed in the case of differential operators, the \(L\)-Fourier transform can be extended to functions \(f_j\) growing arbitrarily as \(j\to\infty\). The extended \(L\)-Fourier transform is not uniquely invertible: it is zero on the subspace spanned by the principal functions of the spectral singularities, i.e., on the functions \(\omega_j^{(m)}(\lambda_k)\), \(k=\alpha+1,\ldots,\beta;\ m=0,\ldots,m_k-1\).

1. Denote by \(\mathfrak{S}\) the manifold of those functions \(f\in H\) for which

\[ \int_{-1}^{+1} \left| \frac{\omega_f(\lambda)} {a(\lambda+i\sqrt{1-\lambda^2})} \right|^2 \sqrt{1-\lambda^2}\,d\lambda<\infty. \]

The manifold \(\mathfrak{S}\) is dense in the space \(H\). For any \(f\in\mathfrak{S}\) and \(g\in H\), the following generalized Parseval equality holds:

\[ \sum_{j=1}^{\infty} f_j g_j = \frac{2}{\pi}\int_{-1}^{+1} \omega_f(\lambda)\omega_g(\lambda)\, \frac{\sqrt{1-\lambda^2}\,d\lambda} {a(\lambda+i\sqrt{1-\lambda^2})\,a(\lambda-i\sqrt{1-\lambda^2})} + \sum_{k=1}^{\alpha} \left\{ \left(\frac{d}{d\lambda}\right)^{m_k-1} M_k(\lambda)\omega_f(\lambda)\omega_g(\lambda) \right\}_{\lambda=\lambda_k}. \]

Lviv State University
named after Iv. Franko

Received
8 VI 1966