The asymptotic of eigenvalues for difference operator with growing potentia

G. V. Garkavenko ; N. B. Uskova

Matematičeskaâ fizika i kompʹûternoe modelirovanie, Tome 20 (2017) no. 4, pp. 6-17

Résumé

We consider $A: D(A)\subset l_2(\mathbb{Z})\to l_2(\mathbb{Z})$, $(Ax)(n)=a(n)x(n)$, $n\in\mathbb{Z}$, $x\in D(A)$, and $(Bx)(n)=-2x(n)+x(n-1)+x(n+1)$. Let $a: \mathbb{Z}\to\mathbb{C}$ be a sequence with property: 1) $a(i)\ne a(j)$, $i\ne j$; 2) $\lim\limits_{|n|\to\infty}|a(n)|=\infty$; 3) $0$, $|i|\to\infty$. By $\mathcal{A}$ we denote the operator $A-B$. By $P_n$ we denote $P_n=P(a(n), A)$, $n\in\mathbb{Z}$, and by $Q_k$ denote the operator $Q_k=\sum\limits_{|i|\leqslant k}P_i$. Theorem 1. There exists a number $k\geqslant 0$, such that the spectrum $\sigma(\mathcal{A})$ of operator $\mathcal{A}$ has form $$ \sigma(\mathcal{A})=\sigma_{(k)}\bigcup\bigg(\bigcup_{|i|>k}\sigma_i\bigg), $$ where $\sigma_{(k)}$ is a finite set with number of points not exceeding $2k+1$ and $\sigma_i=\{\mu_i\}$, $|i|>k$, are singleton sets. The asymptotic formulas of eigenvalues have the following form: $$ \mu_i=a(i)+2+O(d_i^{-1}), $$ $$ \mu_i=a(i)+2-\frac{a(i+1)-2a(i)+a(i-1)}{(a(i+1)-a(i))(a(i-1)-a(i))}+O(d_i^{-3}), \quad |i|>k. $$ Theorem 2. Let the sequence $a:\mathbb{Z}\to\mathbb{C}$ satisfies the condition $\mathrm{Re}\,a(n)\leqslant\beta$ for all $n\in\mathbb{Z}$ and a $\beta\in\mathbb{R}$. Then the operator $\mathcal{A}$ is the generator of the semigroup operators $T: \mathbb{R}_+\to\mathrm{End}\,l_2(\mathbb{Z})$ and this semigroup is similar to $\widetilde{T}: \mathbb{R}_+\to\mathrm{End}\,l_2(\mathbb{Z})$ type $$ \widetilde{T}(t)=\widetilde{T}_{(k)}(t)\oplus \widetilde{T}^{(k)}(t), \quad t\in\mathbb{R}_+, $$ acting in $l_2(\mathbb{Z})=\mathcal{H}_{(k)}\oplus\mathcal{H}^{(k)}$, where $\mathcal{H}_{(k)}=\mathrm{Im}\,Q_k$ and $\mathcal{H}^{(k)}=\mathrm{Im}\,(I-Q_k)$. The semigroup $\widetilde{T}^{(k)}: \mathbb{R}_+\to\mathrm{End}\,\mathcal{H}^{(k)}$ determined by the formula $$ \widetilde{T}^{(k)}(t)x=\sum_{|n|>k}e^{\mu_nt}P_nx, \quad x\in\mathcal{H}^{(k)}, \quad t\in\mathbb{R}_+, $$ where the numbers $\mu_n$, $|n|>k$, are defined by Theorem 1. Theorem 3. Let $\alpha\leqslant \mathrm{Re}\,a(n)\leqslant\beta$, $\alpha$, $\beta\in\mathbb{R}$, for every $n\in\mathbb{Z}$. Then the operator $\mathcal{A}: D(\mathcal{A})\subset l_2(\mathbb{Z})\to l_2(\mathbb{Z})$ is generator of group $T: \mathbb{R}\to \mathrm{End}\,l_2(\mathbb{Z})$. This group is similar to $\widetilde{T}: \mathbb{R}\to \mathrm{End}\,l_2(\mathbb{Z})$, where $\widetilde{T}(t)=\widetilde{T}_{(k)}(t)\oplus \widetilde{T}^{(k)}(t)$, $t\in\mathbb{R}$ and $$ \widetilde{T}^{(k)}(t)x=\sum_{|n|>k}e^{\mu_nt}P_nx, \quad x\in\mathcal{H}^{(k)}, \quad t\in\mathbb{R}. $$ Theorem 4. Let the operator $\mathcal{A}: D(\mathcal{A})\subset l_2(\mathbb{Z})\to l_2(\mathbb{Z})$ be self-adjoint. Then $i\mathcal{A}$ is a generator of isometric group $T: \mathbb{R}\to \mathrm{End}\,l_2(\mathbb{Z})$. This group is similar to $$ \widetilde{T}(t)=\widetilde{T}_{(k)}(t)\oplus \widetilde{T}^{(k)}(t), \quad t\in\mathbb{R}. $$ and $$ \widetilde{T}^{(k)}(t)x=\sum_{|n|>k}e^{i\mu_nt}P_nx, \quad x\in\mathcal{H}^{(k)}, \quad t\in\mathbb{R}. $$

Détail

Keywords: method of similar operators, difference operator, eigenvalues, semigroup of operators, generator of operator semigroup.