Spectral analysis of a matrix-valued quantum-difference operator (Q2825624)

The purpose of the paper is to extend some results for matrix difference equations obtained by the first author and \textit{E. Bairamov} [Appl. Math. Comput. 218, 9676--9681 (2012; Zbl 1246.39003)] to the case of \(q\)-difference equations. Let \(I\) be the identity matrix, \(q>1\) and \(\mu(t)=(q-1)t\). Denote by \(\mathbb N_0=\{0\}\cup \mathbb N\) the set of non-negative integers. The authors assume that two matrix sequences \(\{A(t)\}\), \(t\in q^{\mathbb N_0}\), \(\{B(t)\}\), \(t\in q^{\mathbb N}\), satisfy the hypothesis \(\sum_{t\in q^ {\mathbb N}}\left(\left\|I-A(t)\right\| + \left\|B(t)\right\|\right)<\infty\), where by \(\left\|\cdot\right\|\) the matrix norm in \(\mathbb C^m\) is meant. Define the Jost solution as the bounded matrix solution \(E(\cdot,z)\) of the matrix-valued \(q\)-difference equation NEWLINE\[NEWLINEqA(t)y(qt)+B(t)y(t)+A\left(\frac{t}{q}\right)y\left(\frac{t}{q}\right)=\sqrt{q}(z+z^{-1})y(t), \, t\in q^{\mathbb N},NEWLINE\]NEWLINE satisfying the condition \(\lim_{t\rightarrow\infty} E(t,z) z^{\frac{-\ln t}{\ln q}}\sqrt{\mu(t)}=I,\) with \(z\in \mathbb S^1=\{z\in\mathbb C:|z|=1\}\). The first main results establish the existence of the Jost solution and give the representation of \(E(t,z)\), \(t\in q^{\mathbb N_0}\), \(z\in \mathbb S^1\), in terms of \(\{A(t)\}, \{B(t)\}\). If, additionally, it is assumed that Condition \((C):\, \sum_{t\in q^ {\mathbb N}} \frac{\ln t}{\ln q}\left(\left\|I-A(t)\right\| + \left\|B(t)\right\|\right)<\infty\) is fulfilled, then the authors prove that the Jost solution \(E(\cdot,z)\) has an analytic continuation from \(\mathbb S^1\) to the punctured open unit disk. Even more, it satisfies the asymptotic equation \(E(t,z)= \frac{z^{\frac{\ln t}{\ln q}}}{\sqrt{\mu(t)}}(I+o(1))\), where \(z\) belongs to the punctured closed unit disk. In the second part of the paper, the authors investigate the spectral properties of the operator \(L\) generated by the above \(q\)-difference expression (\(L\) is defined in the Hilbert space \(\ell_2(q^{\mathbb N},\mathbb C^m)\) of all vector sequences \(\{y(t)\in\mathbb C^m: t\in q^{\mathbb{N}}\}\) satisfying \(\sum_{t\in q^{\mathbb{N}}}\left\|y(t)\right\|^2_{\mathbb C^m}\mu(t)<\infty\), with inner product given by \(\left\langle y,z\right\rangle_q=\sum_{t\in q^{\mathbb{N}}}\left\langle y(t),z(t)\right\rangle_{\mathbb C^m}\mu(t)\)). In particular, under the assumption of Condition \((C),\) they obtain that the continuous spectrum of \(L\) is the closed interval \([-2\sqrt{q},2\sqrt{q}]\), and \(L\) has at most finitely many real eigenvalues which are simple in the sense that the multiplicity of any zero of the function \(\det E(1,z)\) is simple.