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These operators are $H(\\zeta)=U+U^{-1}+V+\\zeta V^{-1}$ and $H_{m,n}=U+V+q^{-mn}U^{-m}V^{-n}$, where $U$ and $V$ are self-adjoint Weyl operators satisfying $UV=q^{2}VU$ with $q=e^{i\\pi b^{2}}$, $b>0$ and $\\zeta>0$, $m,n\\in\\mathbb{N}$. We prove that $H(\\zeta)$ and $H_{m,n}$ are self-adjoint operators with purely discrete spectrum on $L^{2}(\\mathbb{R})$. 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Takhtajan, Lukas Schimmer","submitted_at":"2015-09-30T21:34:50Z","abstract_excerpt":"We investigate Weyl type asymptotics of functional-difference operators associated to mirror curves of special del Pezzo Calabi-Yau threefolds. These operators are $H(\\zeta)=U+U^{-1}+V+\\zeta V^{-1}$ and $H_{m,n}=U+V+q^{-mn}U^{-m}V^{-n}$, where $U$ and $V$ are self-adjoint Weyl operators satisfying $UV=q^{2}VU$ with $q=e^{i\\pi b^{2}}$, $b>0$ and $\\zeta>0$, $m,n\\in\\mathbb{N}$. We prove that $H(\\zeta)$ and $H_{m,n}$ are self-adjoint operators with purely discrete spectrum on $L^{2}(\\mathbb{R})$. 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