The declared interval models with Dirichlet and Robin boundary conditions, and the declared zero-range (delta) model

(a) Framework data. Adopt the framework postulates and units of clause (a) of The declared one-dimensional line model: state space, Hamiltonian, bound states and generalized stationary solutions, and the vocabulary of its clauses (b)-(f): the effective one-dimensional state space, the position and momentum assignments, the assigned energy operator $H_V=H_0^{\mathrm{l}\mathrm{i}\mathrm{n}}+M_V$ for a real bounded measurable potential, bound states, stationary solutions and declared generalized stationary solutions. Fix $m>0$, $\mathrm{\hslash }>0$ and the units of Dimensions, SI unit conventions and quantity types of the framework.

(b) Dirichlet interval model. Fix $0<L<\mathrm{\infty }$ and a real $V\in C([0,L];\mathbb{R})$, read as a potential energy in joules, and put $\tilde{V}:=(2m\mathrm{/}{\mathrm{\hslash }}^2)V$, a real continuous function on $[0,L]$ with the dimension of an inverse square length under the conventions of the framework; the factor $2m\mathrm{/}{\mathrm{\hslash }}^2$ has unit ${\mathrm{J}}^{-1}{\mathrm{m}}^{-2}$, so the products in clause (c) of the next item are dimensionally consistent. The Dirichlet interval model declares the state space $L^2(0,L;\mathbb{C})$ of almost-everywhere classes with the pairing linear in the first argument (One-dimensional Sobolev spaces on intervals) and the assigned energy operator $H_{D,V}:=-\frac{{\mathrm{\hslash }}^2}{2m}{D}^2+M_V,D(H_{D,V}):=\{u\in H^2(0,L):u^{\ast }(0)=u^{\ast }(L)=0\},$ where $D^2$ is the second weak derivative and $u^{\ast }$ the continuous representative of Absolutely continuous representative of an L2 function with L2 weak derivative; in the notation of Identification of the Dirichlet Laplacian with the sine diagonal operator this operator is $({\mathrm{\hslash }}^2\mathrm{/}2m)(A_D+M_{\tilde{V}})$ with $D(A_D)=\{u\in H^2(0,L):u^{\ast }(0)=u^{\ast }(L)=0\}$, and for $V=0$ its declared free eigenpairs are the pairs $(\frac{{\mathrm{\hslash }}^2}{2m}(\frac{n\pi }{L}{)}^2,e_n)$ with $e_n(x)=\sqrt{2\mathrm{/}L}\sin (n\pi x\mathrm{/}L)$, $n\ge 1$, read from Identification of the Dirichlet Laplacian with the sine diagonal operator and Completeness of the Dirichlet sine system on an interval. The vanishing endpoint condition is a declared idealization of an infinitely high wall at each endpoint; the pair does not derive it from the line model, and no reflection coefficient or wall model of a material system is asserted.

(c) Robin interval model. Fix $0<L<\mathrm{\infty }$, real $\alpha ,\beta$, read as declared boundary parameters in ${\mathrm{m}}^{-1}$, a real $V\in C([0,L];\mathbb{R})$ and $\tilde{V}:=(2m\mathrm{/}{\mathrm{\hslash }}^2)V$ as in (b). The Robin interval model declares the state space $L^2(0,L;\mathbb{C})$ and the assigned energy operator $H_{\alpha \beta ,V}:=-\frac{{\mathrm{\hslash }}^2}{2m}{D}^2+M_V,D(H_{\alpha \beta ,V}):=\{u\in H^2(0,L):u^{\mathrm{\prime }\ast }(0)=\alpha u^{\ast }(0),u^{\mathrm{\prime }\ast }(L)=\beta u^{\ast }(L)\},$ the operator $({\mathrm{\hslash }}^2\mathrm{/}2m)A$ of Green resolvent for separated Robin boundary conditions with potential $\tilde{V}$; the boundary parameters are declared model parameters, not derived quantities, and no physical interpretation of a particular value of $\alpha$ or $\beta$ is asserted. The Dirichlet model of (b) is not the limit $\alpha ,\beta \to \mathrm{\infty }$ in this definition; the Dirichlet condition is a separate declared boundary condition.

(d) Zero-range (delta) interface model. Fix a real coupling $g$, read in joule metres, and put ${\alpha }_g:=2mg\mathrm{/}{\mathrm{\hslash }}^2$, a real number in ${\mathrm{m}}^{-1}$. Let $A_{{\alpha }_g}$ be the delta interface operator of The one-dimensional delta interface operator on $L^2(\mathbb{R};\mathbb{C})$, $D(A_{{\alpha }_g})=\{u\in H^1(\mathbb{R}):u{\mathrm{\mid }}_{(0,\mathrm{\infty })},u{\mathrm{\mid }}_{(-\mathrm{\infty },0)}\in H^2,u^{\mathrm{\prime }\ast }(0+)-u^{\mathrm{\prime }\ast }(0-)={\alpha }_g{u}^{\ast }(0)\},A_{{\alpha }_g}u=-D^{2}u,$ where the displayed action is understood piecewise: $(A_{{\alpha }_g}u){\mathrm{\mid }}_{(0,\mathrm{\infty })}=-D^2(u{\mathrm{\mid }}_{(0,\mathrm{\infty })})$ and $(A_{{\alpha }_g}u){\mathrm{\mid }}_{(-\mathrm{\infty },0)}=-D^2(u{\mathrm{\mid }}_{(-\mathrm{\infty },0)})$, with each weak derivative taken on that open half-line. The resulting two $L^2$ classes are glued across the null point $0$; $D^{2}u$ here is not the global weak second derivative on $\mathbb{R}$, which need not be an $L^2$ class when the first derivative jumps. Declare the zero-range model with coupling $g$ to be the model whose assigned energy operator is $H_g:=\frac{{\mathrm{\hslash }}^2}{2m}{A}_{{\alpha }_g},D(H_g):=D(A_{{\alpha }_g})\mathrm{.}$ The interface condition $u^{\mathrm{\prime }\ast }(0+)-u^{\mathrm{\prime }\ast }(0-)=(2mg\mathrm{/}{\mathrm{\hslash }}^2)u^{\ast }(0)$ is the declared realization of a zero-range interaction at the origin: the pair reads it as the model counterpart of a formally written term $g\delta (x)$ in the energy expression, while recording that $\delta$ is not a function and that the model is defined by the interface condition and not by an $L^2$ potential. The scaling of the operator is $({\mathrm{\hslash }}^2\mathrm{/}2m)$ times the mathematical operator, by the conventions of clause (c) of The declared one-dimensional line model: state space, Hamiltonian, bound states and generalized stationary solutions. No identification of a concrete impurity or material with the model is asserted.

(e) Units, scope and status. All three assignments are declared energy operators with the joule as declared unit. The interval-spectra theorem proves self-adjointness, compact resolvent, the complete discrete eigenbases and the stated spectra for $H_{D,V}$ and $H_{\alpha \beta ,V}$. For $H_g$, the zero-range theorem proves self-adjointness and the stated classification of negative eigenvalues and bound states; it does not assert a complete description of the spectrum or an eigenbasis for $H_g$. These properties are established by the theorems of this page, not presupposed here. The boundary conditions, boundary parameters and coupling are declared assumptions of the pair; nothing in this definition asserts that a laboratory system realizes them, and no count, rate or measured value is recorded.