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Mathematical Physics and Computer Simulation, 2017, Volume 20, Issue 4, Pages 6–17 (Mi vvgum192)  

Mathematics

The asymptotic of eigenvalues for difference operator with growing potentia

G. V. Garkavenkoa, N. B. Uskovab

a Voronezh State Pedagogical University
b Voronezh State Technical University

Abstract: 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) $0<d_i=\inf_{i\ne j}|a(i)-a(j)|\to\infty$, $|i|\to\infty$.
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
$$ \sigma(\mathcal{A})=\sigma_{(k)}\bigcup(\bigcup_{|i|>k}\sigma_i), $$
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
$$ \widetilde{T}^{(k)}(t)x=\sum_{|n|>k}e^{i\mu_nt}P_nx, \quad x\in\mathcal{H}^{(k)}, \quad t\in\mathbb{R}. $$


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

Funding Agency Grant Number
Russian Foundation for Basic Research 16-01-00197


DOI: https://doi.org/10.15688/mpcm.jvolsu.2017.4.1

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Citation: G. V. Garkavenko, N. B. Uskova, “The asymptotic of eigenvalues for difference operator with growing potentia”, Mathematical Physics and Computer Simulation, 20:4 (2017), 6–17

Citation in format AMSBIB
\Bibitem{GarUsk17}
\by G.~V.~Garkavenko, N.~B.~Uskova
\paper The asymptotic of eigenvalues for difference operator with growing potentia
\jour Mathematical Physics and Computer Simulation
\yr 2017
\vol 20
\issue 4
\pages 6--17
\mathnet{http://mi.mathnet.ru/vvgum192}
\crossref{https://doi.org/10.15688/mpcm.jvolsu.2017.4.1}


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