A model and characterization of a class of symmetric semibounded operators
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910901012004864 |
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| author | Belishev, M. I. Simonov, S. A. |
| author_facet | Belishev, M. I. Simonov, S. A. |
| contents | Let $\mathcal G$ be a Hilbert space and $\mathfrak B(\mathcal G)$ the algebra of bounded operators, $\mathcal H=L_2([0,\infty);\mathcal G)$. An operator-valued function $Q\in L_{\infty,\rm loc}\left([0,\infty);\mathfrak B(\mathcal G)\right)$ determines a multiplication operator in $\mathcal H$ by $(Qy)(x)=Q(x)y(x)$, $x\geqslant0$. We say that an operator $L_0$ in a Hilbert space is a Schrödinger type operator, if it is unitarily equivalent to $-d^2/dx^2+Q(x)$ on a relevant domain. The paper provides a characterization of a class of such operators. The characterization is given in terms of properties of an evolutionary dynamical system associated with $L_0$. It provides a way to construct a functional Schrödinger model of $L_0$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_01000 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A model and characterization of a class of symmetric semibounded operators Belishev, M. I. Simonov, S. A. Mathematical Physics Functional Analysis 47A45, 47A46, 47A68, 47B25, 47B93, 35R30 Let $\mathcal G$ be a Hilbert space and $\mathfrak B(\mathcal G)$ the algebra of bounded operators, $\mathcal H=L_2([0,\infty);\mathcal G)$. An operator-valued function $Q\in L_{\infty,\rm loc}\left([0,\infty);\mathfrak B(\mathcal G)\right)$ determines a multiplication operator in $\mathcal H$ by $(Qy)(x)=Q(x)y(x)$, $x\geqslant0$. We say that an operator $L_0$ in a Hilbert space is a Schrödinger type operator, if it is unitarily equivalent to $-d^2/dx^2+Q(x)$ on a relevant domain. The paper provides a characterization of a class of such operators. The characterization is given in terms of properties of an evolutionary dynamical system associated with $L_0$. It provides a way to construct a functional Schrödinger model of $L_0$. |
| title | A model and characterization of a class of symmetric semibounded operators |
| topic | Mathematical Physics Functional Analysis 47A45, 47A46, 47A68, 47B25, 47B93, 35R30 |
| url | https://arxiv.org/abs/2504.01000 |