A generalized Birman-Schwinger principle and applications to one-dimensional Schrödinger operators with distributional potentials
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2025
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| author | Gesztesy, Fritz Nichols, Roger |
| author_facet | Gesztesy, Fritz Nichols, Roger |
| contents | Given a self-adjoint operator $H_0$ bounded from below in a complex Hilbert space $\mathcal{H}$, the corresponding scale of spaces $\mathcal{H}_{+1}(H_0) \subset \mathcal{H} \subset \mathcal{H}_{-1}(H_0) = [\mathcal{H}_{+1}(H_0)]^*$, and a fixed $V\in \mathcal{B}(\mathcal{H}_{+1}(H_0),\mathcal{H}_{-1}(H_0))$, we define the operator-valued map $A_V(\,\cdot\,):ρ(H_0)\to \mathcal{B}(\mathcal{H})$ by \[ A_V(z):=-\big(H_0-zI_{\mathcal{H}} \big)^{-1/2}V\big(H_0-zI_{\mathcal{H}} \big)^{-1/2}\in \mathcal{B}(\mathcal{H}),\quad z\in ρ(H_0), \] where $ρ(H_0)$ denotes the resolvent set of $H_0$. Assuming that $A_V(z)$ is compact for some $z=z_0\in ρ(H_0)$ and has norm strictly less than one for some $z=E_0\in (-\infty,0)$, we employ an abstract version of Tiktopoulos' formula to define an operator $H$ in $\mathcal{H}$ that is formally realized as the sum of $H_0$ and $V$. We then establish a Birman-Schwinger principle for $H$ in which $A_V(\,\cdot\,)$ plays the role of the Birman-Schwinger operator: $λ_0\in ρ(H_0)$ is an eigenvalue of $H$ if and only if $1$ is an eigenvalue of $A_V(λ_0)$. Furthermore, the geometric (but not necessarily the algebraic) multiplicities of $λ_0$ and $1$ as eigenvalues of $H$ and $A_V(λ_0)$, respectively, coincide.
As a concrete application, we consider one-dimensional Schrödinger operators with $H^{-1}(\mathbb{R})$ distributional potentials. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_02251 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A generalized Birman-Schwinger principle and applications to one-dimensional Schrödinger operators with distributional potentials Gesztesy, Fritz Nichols, Roger Functional Analysis Mathematical Physics Primary: 34L40, 47A55, Secondary: 46F99, 47A56, 47B07 Given a self-adjoint operator $H_0$ bounded from below in a complex Hilbert space $\mathcal{H}$, the corresponding scale of spaces $\mathcal{H}_{+1}(H_0) \subset \mathcal{H} \subset \mathcal{H}_{-1}(H_0) = [\mathcal{H}_{+1}(H_0)]^*$, and a fixed $V\in \mathcal{B}(\mathcal{H}_{+1}(H_0),\mathcal{H}_{-1}(H_0))$, we define the operator-valued map $A_V(\,\cdot\,):ρ(H_0)\to \mathcal{B}(\mathcal{H})$ by \[ A_V(z):=-\big(H_0-zI_{\mathcal{H}} \big)^{-1/2}V\big(H_0-zI_{\mathcal{H}} \big)^{-1/2}\in \mathcal{B}(\mathcal{H}),\quad z\in ρ(H_0), \] where $ρ(H_0)$ denotes the resolvent set of $H_0$. Assuming that $A_V(z)$ is compact for some $z=z_0\in ρ(H_0)$ and has norm strictly less than one for some $z=E_0\in (-\infty,0)$, we employ an abstract version of Tiktopoulos' formula to define an operator $H$ in $\mathcal{H}$ that is formally realized as the sum of $H_0$ and $V$. We then establish a Birman-Schwinger principle for $H$ in which $A_V(\,\cdot\,)$ plays the role of the Birman-Schwinger operator: $λ_0\in ρ(H_0)$ is an eigenvalue of $H$ if and only if $1$ is an eigenvalue of $A_V(λ_0)$. Furthermore, the geometric (but not necessarily the algebraic) multiplicities of $λ_0$ and $1$ as eigenvalues of $H$ and $A_V(λ_0)$, respectively, coincide. As a concrete application, we consider one-dimensional Schrödinger operators with $H^{-1}(\mathbb{R})$ distributional potentials. |
| title | A generalized Birman-Schwinger principle and applications to one-dimensional Schrödinger operators with distributional potentials |
| topic | Functional Analysis Mathematical Physics Primary: 34L40, 47A55, Secondary: 46F99, 47A56, 47B07 |
| url | https://arxiv.org/abs/2507.02251 |