The zero-range (delta) model: self-adjointness and the unique attractive bound-state preparation

Adopt the postulates P1, P2, P4, P5, the declared line model The declared one-dimensional line model: state space, Hamiltonian, bound states and generalized stationary solutions and the declared zero-range model of The declared interval models with Dirichlet and Robin boundary conditions, and the declared zero-range (delta) model clause (d) with mass parameter $m>0$, spin label $s=0$, coupling $g\in \mathbb{R}$ (in joule metres) and ${\alpha }_g=2mg\mathrm{/}{\mathrm{\hslash }}^2$. Let $H_g=({\mathrm{\hslash }}^2\mathrm{/}2m)A_{{\alpha }_g}$ be the assigned energy operator of the zero-range model on the domain $D(H_g)=D(A_{{\alpha }_g})$ of the interface operator. Assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The Axiom of Choice). Then:

(a) $D(H_g)$ consists exactly of the classes $u\in H^1(\mathbb{R})$ whose restrictions to $(0,\mathrm{\infty })$ and $(-\mathrm{\infty },0)$ are of class $H^2$, subject to the interface condition $u^{\mathrm{\prime }\ast }(0+)-u^{\mathrm{\prime }\ast }(0-)=\frac{2mg}{{\mathrm{\hslash }}^2}{u}^{\ast }(0),$ On that domain, $(H_{g}u){\mathrm{\mid }}_{{\mathbb{R}}_{\pm }}=-\frac{{\mathrm{\hslash }}^2}{2m}{D}^2(u{\mathrm{\mid }}_{{\mathbb{R}}_{\pm }})$, where ${\mathbb{R}}_{+}=(0,\mathrm{\infty })$, ${\mathbb{R}}_{-}=(-\mathrm{\infty },0)$ and each second weak derivative is taken on its open half-line; these two $L^2$ classes are glued across the null point $0$. The action is not the global weak second derivative of $u$ on $\mathbb{R}$. $(H_g,\mathrm{J})$ is an assigned sharp observable: $H_g$ is densely defined and self-adjoint on $D(H_g)$.

(b) Repulsive and free couplings. If $g\ge 0$, then $H_g$ has no negative eigenvalue, and for every $g\in \mathbb{R}$ the value $0$ is not an eigenvalue of $H_g$.

(c) Attractive coupling. If $g<0$, then $H_g$ has exactly one negative eigenvalue, namely $E_g=-\frac{m{g}^2}{2{\mathrm{\hslash }}^2}<0,$ and its eigenspace is one-dimensional, spanned by the unit class ${\psi }_g(x)=\sqrt{\frac{m\mathrm{\mid }g\mathrm{\mid }}{{\mathrm{\hslash }}^2}}\exp \text{\hspace{0.17em}⁣}(-\frac{m\mathrm{\mid }g\mathrm{\mid }}{{\mathrm{\hslash }}^2}\mathrm{\mid }x\mathrm{\mid })\mathrm{.}$ For $g<0$ this is the unique bound-state preparation of the zero-range model: every eigenvector of $H_g$ with negative eigenvalue is a nonzero multiple of ${\psi }_g$, the preparation $[{\psi }_g]$ is stationary with energy $E_g$, and the energy outcome law assigns probability one to $\{E_g\}$.

(d) Scope. The statement is conditional on P1, P2, P4, P5, the declared interface realization and the cited mathematics; the coupling $g$ is a declared model parameter, not a measured quantity. No scattering coefficient, no transmission or reflection statement, no material impurity and no measured spectrum is asserted; the formal writing $g\delta (x)$ is not used as a function and no potential in $L^2$ is involved.

• Sign conventions. The interface condition is $u^{\mathrm{\prime }\ast }(0+)-u^{\mathrm{\prime }\ast }(0-)={\alpha }_g{u}^{\ast }(0)$ with ${\alpha }_g=2mg\mathrm{/}{\mathrm{\hslash }}^2$, the convention of the cited interface operator; with $g<0$ the jump is negative and the single bound state appears. The statement makes no claim about any other sign convention or normalization of the formal coupling.
• What is not asserted. The theorem proves the self-adjointness of the declared interface operator and its exact negative spectrum on the line. It does not compute transmission or reflection through the interface, does not treat the delta interaction on an interval or in higher dimension, and makes no empirical claim.