Self-adjoint unbounded polynomial interaction at fixed regulator

By [F1], $R$ is compact and self-adjoint. Applying [F2] and using $\ker R=\{0\}$ shows that the closed span of its nonzero eigenspaces is all of $H_N$. If $R\psi =\mu \psi$ with $\psi \mathrm{\ne }0$, then $\mu \mathrm{\parallel }\psi {\mathrm{\parallel }}^2=⟨R\psi ,\psi ⟩\ge 0$ by positivity, while injectivity rules out $\mu =0$; hence every nonzero eigenvalue is positive.

The domains defined by the squared coefficient sums for $E_n$ and ${\lambda }_n=E_n-(C_0+1)$ coincide. Indeed ${\lambda }_n^2\le 2E_n^2+2(C_0+1{)}^2,E_n^2\le 2{\lambda }_n^2+2(C_0+1{)}^2\mathrm{.}$ Multiplying by $\mathrm{\mid }{c}_n{\mathrm{\mid }}^2$ and summing preserves finiteness in both directions because ${\sum }_n\mathrm{\mid }{c}_n{\mathrm{\mid }}^2=\mathrm{\parallel }u{\mathrm{\parallel }}_2^2<\mathrm{\infty }$ by [F4]. This proves the displayed coefficient-domain description of $D(h)$.

Each eigenspace is finite-dimensional by [F2], and so has an orthonormal basis by [F3]. The set of nonzero eigenvalues is at most countable, and [A1] selects these finite bases; their union is an orthonormal family whose closed span is $H_N$ by step 1.1. The space is infinite-dimensional: the indicators of the pairwise disjoint unit cubes $[r,r+1)\times [0,1{)}^{N-1}$, $r\in \mathbb{N}$, are nonzero mutually orthogonal $L^2$ vectors. Thus the family can be indexed by $n\ge 1$. The bound $\mathrm{\parallel }R\mathrm{\parallel }\le \frac{1}{2}$ gives $0<{\mu }_n\le \frac{1}{2}$ and $E_n\ge 1$. The compact spectral theorem says only finitely many eigenvalues satisfy $\mu \ge \epsilon$ for each $\epsilon >0$; recursively taking a largest remaining eigenvalue (which exists because above half its positive supremum there are only finitely many) orders them nonincreasingly with multiplicity and gives ${\mu }_n\to 0$, hence $E_n\to \mathrm{\infty }$.

The family $({\psi }_n)$ is complete by step 1.1, so [F4] gives $u={\sum }_n⟨u,{\psi }_n⟩{\psi }_n$ in norm for every $u\in H_N$ and Parseval. In particular $D(S)$ is dense because it contains all finite linear combinations of the ${\psi }_n$. Since each $E_n$ is real, the diagonal formula for $S$ gives $⟨Su,v⟩=⟨u,Sv⟩$ for finite sums, and then for all $u,v\in D(S)$ by [F7] and convergence of the coefficient series; thus $S$ is symmetric. For $u\in D(S)$ with coefficients $c_n$, Cauchy--Schwarz [F7] makes ${\sum }_n{E}_n\mathrm{\mid }{c}_n{\mathrm{\mid }}^2$ absolutely convergent, and step 2.1 gives $E_n\ge 1$, so $⟨Su,u⟩={\sum }_n{E}_n\mathrm{\mid }{c}_n{\mathrm{\mid }}^2\ge {\sum }_n\mathrm{\mid }{c}_n{\mathrm{\mid }}^2=\mathrm{\parallel }u{\mathrm{\parallel }}_2^2$.

To compute the adjoint domain, let $v={\sum }_n{d}_n{\psi }_n\in D(S^{\ast })$ and write $S^{\ast }v=z={\sum }_n{z}_n{\psi }_n$. Testing the adjoint identity on each ${\psi }_n$ gives $z_n=E_n{d}_n$. Because $z\in H_N$, Parseval gives ${\sum }_n{E}_n^2\mathrm{\mid }{d}_n{\mathrm{\mid }}^2<\mathrm{\infty }$, so $v\in D(S)$. Conversely, if this sum is finite, then $z:={\sum }_n{E}_n{d}_n{\psi }_n\in H_N$ and the coefficient identity $⟨Su,v⟩=⟨u,z⟩$ holds first for finite sums and then for every $u\in D(S)$ by [F7]; hence $v\in D(S^{\ast })$ and $S^{\ast }v=Sv$. Therefore $D(S^{\ast })=D(S)$ and $S^{\ast }=S$ as operators; by the unbounded self-adjointness definition in [F5], $S$ is self-adjoint.

For $f={\sum }_n{f}_n{\psi }_n\in H_N$, boundedness of $R$ and its action on each eigenvector give $Rf={\sum }_n{\mu }_n{f}_n{\psi }_n$. Since $E_n{\mu }_n=1-{\mu }_n\in [0,1]$, we have ${\sum }_n{E}_n^2\mathrm{\mid }{\mu }_n{f}_n{\mathrm{\mid }}^2\le {\sum }_n\mathrm{\mid }{f}_n{\mathrm{\mid }}^2<\mathrm{\infty }$, so $Rf\in D(S)$ and $(S+I)Rf=f$. Conversely, for $u={\sum }_n{c}_n{\psi }_n\in D(S)$, $R(S+I)u={\sum }_n{\mu }_n(E_n+1)c_n{\psi }_n=u$. Thus $S+I:D(S)\to H_N$ is bijective with bounded inverse $R$, which is compact by [F1].

The variational identity for $R$, applied to $f={\psi }_n$, gives $a(v,R{\psi }_n)=⟨v,{\psi }_n⟩$ for every $v\in Q$. Since $R{\psi }_n={\mu }_n{\psi }_n$ and ${\mu }_n>0$, each ${\psi }_n$ lies in $Q$ and $a(v,{\psi }_n)={\mu }_n^{-1}⟨v,{\psi }_n⟩\mathrm{.}$ Thus ${\xi }_n:=\sqrt{{\mu }_n}{\psi }_n$ is an orthonormal family in $Q$. If $v\in Q$ is orthogonal in $a$ to every ${\xi }_n$, the displayed identity makes all $H_N$ coefficients $⟨v,{\psi }_n⟩$ zero. The $H_N$ completeness of $({\psi }_n)$ and injectivity of $Q\hookrightarrow H_N$ force $v=0$ in $Q$. Hence $({\xi }_n)$ is complete in $Q$, and Parseval in $(Q,a)$ gives $a(v,v)={\sum }_n{\mu }_n^{-1}\mathrm{\mid }⟨v,{\psi }_n⟩{\mathrm{\mid }}^2={\sum }_n(E_n+1)\mathrm{\mid }⟨v,{\psi }_n⟩{\mathrm{\mid }}^2\mathrm{.}$

Let $u\in C_c^{\mathrm{\infty }}$ and set $f:=(-{\sum }_j{a}_j{\mathrm{\partial }}_j^2+W+1)u\in H_N$. Integration by parts gives $a(u,v)=⟨f,v⟩$ for $v\in C_c^{\mathrm{\infty }}$; both sides are continuous in $v$ for the $Q$ norm by [F7], and $C_c^{\mathrm{\infty }}$ is dense in $Q$, so the identity holds for all $v\in Q$. It is the defining variational identity for $Rf$, and uniqueness in [F1] gives $Rf=u$. Hence $u\in \mathrm{ran}R=D(S)$ and $(S+I)u=f$, so $Su=-{\sum }_j{a}_j{\mathrm{\partial }}_j^{2}u+Wu$.

Parseval in $H_N$ gives $\mathrm{\parallel }v{\mathrm{\parallel }}_2^2={\sum }_n\mathrm{\mid }⟨v,{\psi }_n⟩{\mathrm{\mid }}^2$. Subtracting $(C_0+2)$ times this identity from step 5.1 yields the claimed formula for $q_h(v)$, because $E_n+1-(C_0+2)=E_n-C_0-1={\lambda }_n$.

If $u\in D(h)=D(S)$ and $c_n=⟨u,{\psi }_n⟩$, then ${\sum }_n{E}_n^2\mathrm{\mid }{c}_n{\mathrm{\mid }}^2<\mathrm{\infty }$ by the definition of $D(S)$. Since $E_n\ge 1$, $E_n+1\le 2E_n^2$, so ${\sum }_n(E_n+1)\mathrm{\mid }{c}_n{\mathrm{\mid }}^2<\mathrm{\infty }$. Step 5.1 therefore gives $u\in Q$. Thus every eigenvector, and indeed every operator-domain vector, belongs to the form domain.

Subtracting the real scalar $(C_0+1)I$ preserves self-adjointness on the same domain, and $E_n\ge 1$ gives eigenvalues ${\lambda }_n=E_n-C_0-1\ge -C_0$. For $u\in D(h)$, the diagonal expansion and Cauchy--Schwarz [F7] make $⟨hu,u⟩={\sum }_n{\lambda }_n\mathrm{\mid }⟨u,{\psi }_n⟩{\mathrm{\mid }}^2$ absolutely convergent; Parseval then gives $⟨hu,u⟩\ge -C_0\mathrm{\parallel }u{\mathrm{\parallel }}_2^2$. The lowest eigenvalue corresponds to the largest eigenvalue ${\mu }_1$ of $R$; its eigenspace is nonzero and finite-dimensional by [F2]. Since $W=V+C_0+1$, step 5.2 shows that $h=S-(C_0+1)I$ extends the specified polynomial differential expression. Finally $h+(C_0+2)I=S+I$, whose inverse is the compact operator $R$ by step 4.2. These statements establish the declared realization and all listed scope limits.