Bounded fixed-N operator

[Self-adjointness of $K$] The function $\kappa$ is real, measurable and finite-valued, so $M_{\kappa }$ is self-adjoint on $D(M_{\kappa })=\{v:\kappa v\in L^2\}$ by [F1]; conjugating by the unitary ${\mathcal{F}}_2$ ([F1]) gives that $K={\mathcal{F}}_2^{\ast }{M}_{\kappa }{\mathcal{F}}_2$ is self-adjoint with $D(K)={\mathcal{F}}_2^{-1}D(M_{\kappa })=\{u:\kappa {\mathcal{F}}_{2}u\in L^2\}$ ([F2]). For $u\in D(K)$, $⟨Ku,u⟩=⟨M_{\kappa }{\mathcal{F}}_{2}u,{\mathcal{F}}_{2}u⟩=\int \kappa \mathrm{\mid }{\mathcal{F}}_{2}u{\mathrm{\mid }}^2\ge 0$, so $K\ge 0$.

[Finitely many components] Let ${\mathfrak{K}}_1,\dots ,{\mathfrak{K}}_r$ be Hilbert spaces and let $T_a$ be self-adjoint on ${\mathfrak{K}}_a$ for $a=1,\dots ,r$. On the direct sum ${⨁}_a{\mathfrak{K}}_a$ with domain $\{(u_a):u_a\in D(T_a)\}$ define $T((u_a)):=(T_a{u}_a)$. Then $T$ is symmetric (componentwise), and if $(u^{(k)})$ is in the domain with $u^{(k)}\to u$ and $T{u}^{(k)}\to \eta$, then componentwise $u_a^{(k)}\to u_a$ and $T_a{u}_a^{(k)}\to {\eta }_a$; closedness of $T_a$ gives $u_a\in D(T_a)$, $T_a{u}_a={\eta }_a$ and ${\sum }_a\mathrm{\parallel }{T}_a{u}_a{\mathrm{\parallel }}^2=\mathrm{\parallel }\eta {\mathrm{\parallel }}^2<\mathrm{\infty }$, so $T$ is closed. Testing the adjoint componentwise shows $D(T^{\ast })=D(T)$ and $T^{\ast }=T$; a symmetric densely defined operator whose adjoint has the same domain is self-adjoint. If all $T_a=T_0$ and ${\mathfrak{K}}_a={\mathfrak{K}}_0$, then $T\ge -\mathrm{\parallel }V{\mathrm{\parallel }}_{\mathrm{\infty }}$ whenever $T_0\ge -\mathrm{\parallel }V{\mathrm{\parallel }}_{\mathrm{\infty }}$, because the quadratic form is the sum of the components.

[Identification with $W^{2,2}$ and the core] For $\xi \in {\mathbb{R}}^{3N}$ one has $c_{\min }(1+\mathrm{\mid }\xi {\mathrm{\mid }}^2)\le 1+\kappa (\xi )\le c_{\max }(1+\mathrm{\mid }\xi {\mathrm{\mid }}^2)$ for suitable positive constants $c_{\min },c_{\max }$ depending only on $N,m_i,\mathrm{\hslash }$; hence $\kappa {\mathcal{F}}_{2}u\in L^2$ if and only if $(1+\mathrm{\mid }\xi {\mathrm{\mid }}^2){\mathcal{F}}_{2}u\in L^2$, i.e. $u\in W^{2,2}({\mathbb{R}}^{3N};\mathbb{C})$, with equivalent norms by [F3]. Since $C_c^{\mathrm{\infty }}$ is dense in $W^{2,2}$ ([F3]) and the graph norm of $K$ is equivalent to the $W^{2,2}$ norm, $C_c^{\mathrm{\infty }}$ is a core for $K$, and $K$ is the unique self-adjoint extension of $K{\mathrm{\mid }}_{C_c^{\mathrm{\infty }}}$ ([F5]).

[Bounded perturbation] Since $\mathrm{\parallel }V{\mathrm{\parallel }}_{\mathrm{\infty }}<\mathrm{\infty }$, the operator $M_V$ is bounded with $\mathrm{\parallel }{M}_V\mathrm{\parallel }\le \mathrm{\parallel }V{\mathrm{\parallel }}_{\mathrm{\infty }}$; it is symmetric as multiplication by a real function, and on $D(K)$ the bound $\mathrm{\parallel }{M}_{V}u\mathrm{\parallel }\le 0\cdot \mathrm{\parallel }Ku\mathrm{\parallel }+\mathrm{\parallel }V{\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{\parallel }u\mathrm{\parallel }$ exhibits the admissible pair $(a,b)=(0,\mathrm{\parallel }V{\mathrm{\parallel }}_{\mathrm{\infty }})$. By [F4] with $A=K$ and $\gamma =0$, $H=K+M_V$ is self-adjoint on $D(K)$ and bounded below by $-\mathrm{\parallel }V{\mathrm{\parallel }}_{\mathrm{\infty }}$. Moreover $\mathrm{\parallel }Hu\mathrm{\parallel }\le \mathrm{\parallel }Ku\mathrm{\parallel }+\mathrm{\parallel }V{\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{\parallel }u\mathrm{\parallel }$ and $\mathrm{\parallel }Ku\mathrm{\parallel }\le \mathrm{\parallel }Hu\mathrm{\parallel }+\mathrm{\parallel }V{\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{\parallel }u\mathrm{\parallel }$, so the graph norms of $H$ and $K$ are equivalent on $D(K)$; hence the core $C_c^{\mathrm{\infty }}$ of step 2.1 is also a core for $H$. This proves (a) and (b).

[The scalar case with $r$ components] The Hilbert space $L^2({\mathbb{R}}^{3N};\mathbb{C})\otimes {\mathbb{C}}^r$ is identified with ${⨁}_{a=1}^r{L}^2({\mathbb{R}}^{3N};\mathbb{C})$ componentwise; under this identification $K\otimes I_r+M_V\otimes I_r$ is the direct sum of $r$ copies of $H$ with domain $D(K{)}^{\oplus r}=D(K)\otimes {\mathbb{C}}^r$. By step 3.1 and step 1.2 it is self-adjoint and bounded below by $-\mathrm{\parallel }V{\mathrm{\parallel }}_{\mathrm{\infty }}$, which is (c).

[Conclusion and scope] Steps 1.1-3.1 prove (a), (b) and (c); the displayed reduction for external and pair potentials only uses the triangle inequality for essential suprema, $\mathrm{\parallel }V{\mathrm{\parallel }}_{\mathrm{\infty }}\le {\sum }_i\mathrm{\parallel }{V}_i{\mathrm{\parallel }}_{\mathrm{\infty }}+{\sum }_{i<j}\mathrm{\parallel }{W}_{ij}{\mathrm{\parallel }}_{\mathrm{\infty }}<\mathrm{\infty }$. No HVZ threshold, stability, thermodynamic-limit or physical-interpretation claim is made.