The Bott operator, suspension, and reduction of the index to Euclidean space

Written by Claude Opus 5.5 (Anthropic), September 2026. Self-checked by the writing AI. Public domain (CC0).

Edited and supplemented by Codex, September 2026. The additions and editorial corrections are also public domain (CC0).

This lesson moves the index of an elliptic operator on a compact manifold to an operator on a Euclidean space. The route has four steps.

1. We build one operator of index one on $\mathbb R^n$, the Bott operator. It uses only the Euclidean structure of $\mathbb R^n$, so it commutes with the orthogonal group. Its index is computed from an energy identity for the harmonic oscillator (Sections 2–8).
2. We combine an elliptic symbol on a compact manifold $Y$ with the Bott symbol along the fibres of a Euclidean vector bundle over $Y$, and we prove that the index does not change. This is the suspension theorem (Sections 9–11).
3. We embed $Y$ in some $\mathbb R^\nu$. Its normal bundle is then diffeomorphic to a tubular neighbourhood of $Y$, and every vector bundle over a compact manifold becomes trivial after adding a suitable complement (Section 12).
4. We apply the suspension theorem to the normal bundle, trivialize the bundles, and move the operator into the tubular neighbourhood. The result is a square system on $\mathbb R^\nu$ that equals the identity outside a compact set and has the same index as the operator we started with (Section 13).

After this reduction, an index formula for systems on Euclidean space that are trivial at infinity gives an index formula on every compact manifold. The same strategy, carried out in K-theory, is the embedding proof of the Atiyah–Singer index theorem.

The lesson assumes the index theory of elliptic pseudodifferential operators on a compact manifold, including the index of a continuous symbol and its homotopy invariance, from the lesson Symbols, finite defects, and the index on a closed manifold. It also uses Fredholm theory in Banach spaces (Finite defects under perturbation), the Euclidean symbol calculus (From symbol estimates to operators on every Sobolev scale), symbols for a slowly varying metric (Localizing symbols when the measuring scale moves, Two measuring scales, one Weyl product and When a moving symbol scale controls an operator), and pseudodifferential operators on manifolds (Detecting regularity without choosing coordinates). The facts we use are stated in full in Section 1.

The proof inputs are the linked course lessons; Section 17 writes out the additional receiving arguments.

1. Conventions and background

Conventions

• $D_j=-i\partial_j$. The Fourier transform is $\widehat u(\xi)=\int e^{-ix\cdot\xi}u(x)\,dx$, and a symbol $a(x,\xi)$ acts by left quantization, $$a(x,D)u(x)=\operatorname{Op}(a)u(x)=(2\pi)^{-n}\int e^{ix\cdot\xi}a(x,\xi)\widehat u(\xi)\,d\xi .$$ This is the convention of the lessons From symbol estimates to operators on every Sobolev scale and Symbols, finite defects, and the index on a closed manifold.
• Inner products are linear in the first argument: $(u,v)=\int\langle u(x),v(x)\rangle\,dx$. Norms without a subscript are $L^2$ norms.
• $\langle x\rangle=(1+|x|^2)^{1/2}$. For $X=(x,\xi)\in\mathbb R^{2n}$ we write $R(X)=(1+|x|^2+|\xi|^2)^{1/2}$.
• A metric on $\mathbb R^{2n}$ is a field $X\mapsto g_X$ of positive definite quadratic forms. For a metric $g$ and a positive function $m$ (a weight), the class $S(m,g)$ consists of the smooth functions $a$ on $\mathbb R^{2n}$ such that, for every $k$, $|D^ka(X)[T_1,\dots,T_k]|\le C_k\,m(X)\prod_{i=1}^kg_X(T_i)^{1/2}$ for all $X,T_1,\dots,T_k\in\mathbb R^{2n}$. The best constants $C_k$ are the seminorms of $a$.
• The classes $S^m_{1,0}$ and $S^m_{\mathrm{phg}}$ on $\mathbb R^n\times\mathbb R^n$ have bounds uniform in $x$: $|\partial_\xi^\alpha\partial_x^\beta a(x,\xi)|\le C_{\alpha\beta}\langle\xi\rangle^{m-|\alpha|}$. “Classical” means polyhomogeneous: for large $|\xi|$, $a$ has an asymptotic expansion in terms that are homogeneous in $\xi$ of degrees $m,m-1,m-2,\dots$. Symbols may take values in linear maps between fixed finite-dimensional Hermitian spaces; products keep their order.
• Manifolds are Hausdorff and second countable. A compact manifold has no boundary. On a compact manifold, $\Psi^m_{\mathrm{cl}}(X;E,F)$ is the space of classical pseudodifferential operators of order $m$ from sections of $E$ to sections of $F$. Usually $E$ and $F$ contain the factor $\Omega^{1/2}$, the bundle of half-densities, so that the $L^2$ pairing of sections needs no choice of measure. The geometric adjoint $P^*$ of $P$ is its adjoint for this pairing and the Hermitian metrics. The symbol index $\operatorname{s-ind}$ of a continuous symbol is recalled in Fact 1.13. By a homotopy of continuous symbols we mean a jointly continuous family that is invertible outside one compact set; by Fact 1.13(a) it preserves $\operatorname{s-ind}$.
• Two metrics on $\mathbb R^{2n}$ recur: $$g=\frac{|dx|^2+|d\xi|^2}{1+|x|^2+|\xi|^2},\qquad G=\frac{|dx|^2}{1+|x|^2}+\frac{|d\xi|^2}{1+|\xi|^2}. \tag{1.1}$$ The first treats $x$ and $\xi$ alike; it is the natural metric for the harmonic oscillator. The second is a product metric.

Proofs used from earlier lessons

The following facts are proved in the preceding course lessons linked after each statement. Their hypotheses and conclusions are stated here so that each application can be checked at its full generality. Operators between Banach spaces are bounded and linear. An operator is Fredholm if its kernel and its cokernel are finite-dimensional; its range is then closed, and its index is $\dim\ker-\dim\operatorname{coker}$.

Fact 1.1 (Compactness test). An operator $T:X\to Y$ between Banach spaces has finite-dimensional kernel and closed range if and only if every bounded sequence $(x_j)$ for which $(Tx_j)$ converges has a convergent subsequence. In that case, for every closed complement $M$ of $\ker T$ there is $a>0$ with $\|Tm\|\ge a\|m\|$ for all $m\in M$. See Finite defects under perturbation.

Fact 1.2 (Small perturbations). If $T:X\to Y$ is Fredholm, there is $\varepsilon>0$ such that for every $E$ with $\|E\|<\varepsilon$ the operator $T+E$ is Fredholm, $\dim\ker(T+E)\le\dim\ker T$, and $\operatorname{ind}(T+E)=\operatorname{ind}T$. See Finite defects under perturbation.

Fact 1.3 (Products, sums and parametrices). If $A:X\to Y$ and $B:Y\to Z$ are Fredholm, then $BA$ is Fredholm and $\operatorname{ind}(BA)=\operatorname{ind}A+\operatorname{ind}B$. A finite direct sum of Fredholm operators is Fredholm, and its index is the sum of the indices. A bounded bijection has index $0$. If $T:X\to Y$ and there are $L,R:Y\to X$ such that $LT-I$ and $TR-I$ are compact, then $T$ is Fredholm. See Finite defects under perturbation.

Fact 1.4 (Strongly continuous families). Let $I$ be a compact space, and let $T_t:X\to Y$ and $S_t:Y\to X$, $t\in I$, be strongly continuous: $t\mapsto T_tx$ and $t\mapsto S_ty$ are continuous for each $x$ and each $y$. Suppose that the families $S_tT_t-I$ and $T_tS_t-I$ are collectively compact: the set of all $(S_tT_t-I)x$ with $t\in I$ and $\|x\|\le1$ has compact closure, and likewise for the other family. Then every $T_t$ is Fredholm, the function $t\mapsto\dim\ker T_t$ is upper semicontinuous, and $t\mapsto\operatorname{ind}T_t$ is locally constant, hence constant when $I$ is connected. See Finite defects under perturbation.

Fact 1.5 ($L^2$ boundedness). There is an integer $L_n$, depending only on $n$, with the following property. If $a(x,\xi)$ is a smooth matrix-valued function on $\mathbb R^{2n}$ whose derivatives of order at most $L_n$ are all bounded by $M$, then $\|\operatorname{Op}(a)u\|\le C_nM\|u\|$ for $u\in\mathcal S(\mathbb R^n)$, so $\operatorname{Op}(a)$ extends to a bounded operator on $L^2$. This is a form of the Calderón–Vaillancourt theorem. See From symbol estimates to operators on every Sobolev scale.

Fact 1.6 (Kernels of operators of low order). If $a\in S^q_{1,0}$ with $q<-n$, the Schwartz kernel $K(x,y)$ of $\operatorname{Op}(a)$ is a continuous function, and $|K(x,y)|\le C_N\langle x-y\rangle^{-N}$ for every $N$. See From symbol estimates to operators on every Sobolev scale.

Fact 1.7 (Composition and asymptotic sums). If $a\in S^{m_1}_{1,0}$ and $b\in S^{m_2}_{1,0}$, then $\operatorname{Op}(a)\operatorname{Op}(b)=\operatorname{Op}(a\circ b)$ with $a\circ b\in S^{m_1+m_2}_{1,0}$, and $a\circ b-\sum_{|\alpha|<N}(\partial_\xi^\alpha a)(D_x^\alpha b)/\alpha!\in S^{m_1+m_2-N}_{1,0}$ for every $N$. If $a_j\in S^{m_j}_{1,0}$ and $m_j\to-\infty$, there is $a\in S^{\max_jm_j}_{1,0}$ with $a-\sum_{j<k}a_j\in S^{\max_{j\ge k}m_j}_{1,0}$ for every $k$, and with $\operatorname{supp}a\subset\bigcup_j\operatorname{supp}a_j$. See From symbol estimates to operators on every Sobolev scale.

Fact 1.8 (Elliptic parametrices). Let $a\in S^m_{1,0}$ take values in square matrices and be uniformly elliptic: $\|a(x,\xi)v\|\ge c\langle\xi\rangle^m\|v\|$ for all $x$, all $v$ and $|\xi|\ge C$. Choose $b_0\in S^{-m}_{1,0}$ with $ab_0=b_0a=I$ for large $|\xi|$, and put $r=I-a\circ b_0$ and $\ell=I-b_0\circ a$, which lie in $S^{-1}_{1,0}$. Asymptotic sums of the series $\sum_jb_0\circ r^{\circ j}$ and $\sum_j\ell^{\circ j}\circ b_0$ give $b\in S^{-m}_{1,0}$ with $a\circ b-I$ and $b\circ a-I$ in $S^{-\infty}$. Consequently, if $u\in H^s$ for some $s$ and $\operatorname{Op}(a)u\in H^\infty=\bigcap_tH^t$, then $u\in H^\infty$; in particular $u$ is smooth. See From symbol estimates to operators on every Sobolev scale.

Fact 1.9 (Composition for a slowly varying metric). Let $\sigma((x,\xi),(y,\eta))=\xi\cdot y-x\cdot\eta$ be the symplectic form on $\mathbb R^{2n}$. For a metric $g$, let $g^\sigma_X(T)=\sup_{S\ne0}\sigma(T,S)^2/g_X(S)$ be its dual metric, and $h(X)^2=\sup_{T\ne0}g_X(T)/g^\sigma_X(T)$ its Planck function. Assume: - $g$ is slowly varying: there are $c,C>0$ such that $g_X(Y-X)\le c$ implies $C^{-1}g_X\le g_Y\le Cg_X$; - $g$ is symplectically temperate: $g\le g^\sigma$, and $g^\sigma_X(T)\le Cg^\sigma_Y(T)\bigl(1+g^\sigma_Y(X-Y)\bigr)^N$ for all $X,Y,T$; - $g_{(x,\xi)}(t,\tau)=g_{(x,\xi)}(t,-\tau)$.

Let the weights $m_1,m_2$ be $g$-continuous ($g_X(Y-X)\le c$ implies $C^{-1}\le m(Y)/m(X)\le C$) and temperate ($m(Y)\le Cm(X)(1+g^\sigma_Y(X-Y))^N$). If $a\in S(m_1,g)$ and $b\in S(m_2,g)$, then $\operatorname{Op}(a)\operatorname{Op}(b)=\operatorname{Op}(a\circ b)$ on $\mathcal S$, and $a\circ b-ab\in S(hm_1m_2,g)$. Each seminorm of $a\circ b-ab$ is bounded by a constant times a product of finitely many seminorms of $a$ and of $b$, and the constant depends only on $n$, on the seminorm and on the constants in the hypotheses. See When a moving symbol scale controls an operator.

The complete split-metric receiving proof is Section 17.1, (P261.1)–(P261.7). It combines Quadratic Fourier multipliers at a moving scale, Theorems 7.1 and 8.1 and (G24)–(G26), Two measuring scales, one Weyl product, (W31)–(W34), and From Weyl symbols to operators and changes of coordinates, (A21), (A25), (A33)–(A43).

In Facts 1.10–1.16, $X$ is a compact manifold and $E,F$ are Hermitian vector bundles over it.

Fact 1.10 (Rellich compactness). For $\delta>0$ the inclusion $H^{s+\delta}(X;E)\to H^s(X;E)$ is compact. We also use the local form: a set of distributions on $\mathbb R^n$ that are supported in one fixed compact set and bounded in $H^1(\mathbb R^n)$ is relatively compact in $L^2(\mathbb R^n)$. See Symbols, finite defects, and the index on a closed manifold.

Fact 1.11 (The elliptic alternative). Let $P\in\Psi^m_{\mathrm{cl}}(X;E\otimes\Omega^{1/2},F\otimes\Omega^{1/2})$ be elliptic. Then $P:H^s\to H^{s-m}$ is Fredholm for every $s$. The kernel of $P$ and the kernel of its geometric adjoint $P^*$ consist of smooth sections and do not depend on $s$, and $\operatorname{ind}P=\dim\ker P-\dim\ker P^*$. The kernel and cokernel have the same dimensions when $P$ acts on smooth sections. The index of a product of elliptic operators is the sum of their indices, and a formally self-adjoint elliptic operator has index $0$. See Symbols, finite defects, and the index on a closed manifold.

Fact 1.12 (Building operators). If $U$ is a coordinate chart of $X$ over which $E$ and $F$ are trivial, then a classical operator of order $m$ on $\mathbb R^n$ whose Schwartz kernel has compact support in $U\times U$ defines, after extension by zero, an element of $\Psi^m_{\mathrm{cl}}(X;E,F)$. For every smooth section $s$ of $\operatorname{Hom}(\pi^*E,\pi^*F)$ over $T^*X\setminus0$ that is homogeneous of degree $m$, there is $P\in\Psi^m_{\mathrm{cl}}(X;E,F)$ with principal symbol $s$. See Detecting regularity without choosing coordinates.

Fact 1.13 (The index of a continuous symbol). Let $p$ be a continuous section of $\operatorname{Hom}(\pi^*E,\pi^*F)$ over $T^*X$ that is invertible outside a compact set $K$. Let $h$ be the length function of a Riemannian metric on the fibres of $T^*X$, and choose $R>0$ with $K\subset\{h<R\}$. The radial restriction $p_R(x,\xi)=p(x,R\xi/h(x,\xi))$ is a continuous invertible symbol of degree $0$ on $T^*X\setminus0$. A smooth symbol that approximates it closely enough, uniformly on the unit cosphere bundle, is elliptic; the index of any of its quantizations is the symbol index $\operatorname{s-ind}p$, and it does not depend on the choices. It has these properties. - (a) Homotopy invariance. If $p_t$, $0\le t\le1$, is jointly continuous in $(t,x,\xi)$ and every $p_t$ is invertible outside one compact set, then $\operatorname{s-ind}p_t$ does not depend on $t$. - (b) Agreement with the index. If $P\in\Psi^m_{\mathrm{cl}}$ is elliptic, then $\operatorname{ind}P=\operatorname{s-ind}p$, where $p$ is the principal symbol of $P$, extended in any continuous way across a neighbourhood of the zero section. - (c) Smoothing. A continuous section of a vector bundle over the compact cosphere bundle can be approximated uniformly by smooth sections. - (d) Reduction to degree one. If every component of $X$ has positive dimension, then $p$ is homotopic, through continuous symbols invertible outside one compact set, to a symbol that is homogeneous of degree $1$, smooth and invertible off the zero section. - (e) Zero-dimensional components. If $X$ is a finite set, then $\operatorname{s-ind}p=\sum_{x\in X}(\operatorname{rank}E_x-\operatorname{rank}F_x)$. In general the symbol index is the sum of its values on the components of $X$.

See Symbols, finite defects, and the index on a closed manifold.

Fact 1.14 (Norm limits). Let $P_j\in\Psi^m_{\mathrm{cl}}(X;E,F)$, and let $P$ be a linear map on smooth sections that extends to bounded maps $H^s\to H^{s-m}$ with $\|P_j-P\|_{H^s\to H^{s-m}}\to0$ for every real $s$. Suppose that the principal symbols of the $P_j$ converge, uniformly on the unit cosphere bundle, to a continuous symbol $p$, homogeneous of degree $m$ and invertible on $T^*X\setminus0$. Then $P:H^s\to H^{s-m}$ is Fredholm for every $s$. The kernels of $P$ and of its geometric adjoint $P^*$ consist of smooth sections and do not depend on $s$, the range of $P$ is the annihilator of $\ker P^*$, and $\operatorname{ind}P=\dim\ker P-\dim\ker P^*=\operatorname{s-ind}p$. See Symbols, finite defects, and the index on a closed manifold.

Fact 1.15 (Partial operators). Let $z=(x,y)\in\mathbb R^n\times\mathbb R^{n'}$, with dual variables $(\xi,\eta)$. Let $a(z,\xi)$ be a classical symbol of order $m>0$ in $\xi$, with symbol bounds uniform in $z$ and principal part $a_m$, and let $A=a(z,D_x)$ act in $x$ with $y$ as a parameter. Then $A$ maps $H^{s+m}(\mathbb R^{n+n'})$ to $H^s(\mathbb R^{n+n'})$ for every real $s$. Moreover there are classical operators $A_\varepsilon$, $0<\varepsilon\le1$, of order $m$ on $\mathbb R^{n+n'}$ with $\|A_\varepsilon-A\|_{H^{s+m}\to H^s}\le C_s\varepsilon^m$ for every real $s$, whose principal symbols differ from $a_m(z,\xi)$ (extended by $0$ at $\xi=0$) by at most $C\varepsilon^m$ on the unit sphere $|\xi|^2+|\eta|^2=1$. See Symbols, finite defects, and the index on a closed manifold.

Fact 1.16 (Products of symbols). Let $X$ and $Y$ be compact manifolds. Let $p$ be a continuous symbol on $T^*X$ from $E_X$ to $F_X$, and $q$ one on $T^*Y$ from $E_Y$ to $F_Y$, both invertible outside compact sets. On $T^*(X\times Y)=T^*X\times T^*Y$, the block symbol $d(p,q)=\begin{pmatrix}p\otimes I&-I\otimes q^*\\ I\otimes q&p^*\otimes I\end{pmatrix}$, from $(E_X\otimes E_Y)\oplus(F_X\otimes F_Y)$ to $(F_X\otimes E_Y)\oplus(E_X\otimes F_Y)$, is invertible outside a compact set, and $\operatorname{s-ind}d(p,q)=\operatorname{s-ind}p\cdot\operatorname{s-ind}q$. This holds also when $X$ or $Y$ is a finite set. See Symbols, finite defects, and the index on a closed manifold.

Fact 1.17 (Differential topology). The inverse function theorem. Smooth partitions of unity exist subordinate to every open cover, with the support of each function a closed set inside its member of the cover. Green’s formula $\int_M\langle\Delta u,u\rangle\,dV=\int_M|du|^2\,dV$ holds for the nonnegative Laplace–Beltrami operator $\Delta$ of a compact Riemannian manifold $M$, applied to each component of a vector-valued function.

The complete inverse, subordinate-support partition and Green proofs are Section 17.2, (P261.8)–(P261.15). Their independent calculus, compactness, cutoff and inversion entries are Metric and topological foundations, Sections 5, 9, 12.4–12.8, 13.4–13.6 and 14.2.

Fact 1.18 (Hilbert spaces). Riesz representation: every bounded linear functional on a Hilbert space $H$ is $u\mapsto(u,v)$ for a unique $v\in H$. An integral operator whose kernel lies in $L^2(\mathbb R^k\times\mathbb R^k)$ is compact on $L^2(\mathbb R^k)$, and its norm is at most the $L^2$ norm of the kernel.

The complete linear-first Hilbert representation and compact-kernel receiver are Section 17.3, (P261.16)–(P261.20). See also Lower-bounded spectral calculus, Section 2.3, (PR37)–(PR38), and Banach and Hilbert foundations, (LP3)–(LP4), (LP10)–(LP12), for product integration, completeness and compact smooth density.

Fact 1.19 (Integration and compactness). Dominated convergence. The Arzelà–Ascoli theorem: a set of continuous functions on a compact metric space that is uniformly bounded and equicontinuous is relatively compact in the uniform norm.

The complete dominated-convergence and Arzelà–Ascoli receivers are Section 17.4, (P261.21)–(P261.23). Their independent integration and metric compactness providers are Banach and Hilbert foundations, Section 15.1, (LP1)–(LP2), and Section 16.1, and Metric and topological foundations, Sections 7 and 14.1.

Fact 1.20 (Distributions with zero gradient). A distribution on a connected open subset of $\mathbb R^n$ whose first partial derivatives all vanish is a constant function.

The complete compact primitive decomposition and connected-domain distribution proof are Section 17.5, (P261.24)–(P261.27). Their cutoff, coordinate integration and derivative entries are Metric and topological foundations, Sections 5 and 13.4–13.6, and Banach and Hilbert foundations, Sections 15.1 and 15.5.

2. The model $x+d/dx$ on the line

We need an operator of index one on $\mathbb R^n$ that uses only the Euclidean structure. In one dimension there is a simple candidate: $x+d/dx$, which is $\sqrt2$ times the annihilation operator of the harmonic oscillator. First we fix the space on which it acts.

Definition 2.1. Let $$\mathcal B=\mathcal B(\mathbb R^n)=\{u\in L^2(\mathbb R^n):\ x_ju\in L^2,\ D_ju\in L^2,\ j=1,\dots,n\}, \qquad \|u\|_{\mathcal B}^2=\|u\|^2+\sum_j\bigl(\|x_ju\|^2+\|D_ju\|^2\bigr), \tag{2.1}$$ with derivatives in the sense of distributions. For a finite-dimensional Hermitian space $W$ we write $\mathcal B\otimes W$ for the $W$-valued functions whose coefficients lie in $\mathcal B$.

Lemma 2.2. (1) $\mathcal B$ is a Hilbert space, and $C_c^\infty(\mathbb R^n)$ is dense in it. (2) The inclusion $\mathcal B\to L^2$ is compact.

Proof. (1) If $u_k$ is Cauchy in $\mathcal B$, then $u_k$, $x_ju_k$ and $D_ju_k$ converge in $L^2$ to some $u,f_j,h_j$. Limits in $L^2$ are limits in distributions, so $f_j=x_ju$ and $h_j=D_ju$. For density, fix $\chi\in C_c^\infty$ with $0\le\chi\le1$ and $\chi=1$ for $|x|\le1$, and put $\chi_R(x)=\chi(x/R)$. Fix $R_\chi\ge2$ with $\operatorname{supp}\chi\subset\{|x|\le R_\chi\}$. For $u\in\mathcal B$, dominated convergence gives $\chi_Ru\to u$ and $\chi_Rx_ju\to x_ju$ in $L^2$. Also $D_j(\chi_Ru)=\chi_RD_ju+R^{-1}(D_j\chi)(x/R)u\to D_ju$. So compactly supported elements are dense. If $u\in\mathcal B$ vanishes outside the ball of radius $r$, let $u_\epsilon=\rho_\epsilon*u$ with a mollifier supported in $|x|\le\epsilon\le1$. Then $u_\epsilon\in C_c^\infty$, $u_\epsilon\to u$ and $D_ju_\epsilon=\rho_\epsilon*D_ju\to D_ju$ in $L^2$. All supports lie in the ball of radius $r+1$, so $\|x_j(u_\epsilon-u)\|\le(r+1)\|u_\epsilon-u\|\to0$.

2. Let $u_k$ be bounded in $\mathcal B$, say $\|u_k\|_{\mathcal B}\le M$. For each $R\ge1$, the functions $\chi_Ru_k$ are bounded in $H^1(\mathbb R^n)$ and vanish outside the ball of radius $R_\chi R$. By the local form of Rellich’s theorem (Fact 1.10), $\chi_Ru_k$ has an $L^2$-convergent subsequence. The tails are small uniformly: $\|(1-\chi_R)u_k\|\le R^{-1}\||x|u_k\|\le M/R$. Take successive subsequences for $R=1,2,\dots$ and then the diagonal subsequence. For it, $\|u_k-u_l\|\le\|\chi_R(u_k-u_l)\|+2M/R$, so it is Cauchy in $L^2$. $\square$

Proposition 2.3 (The model). Let $n=1$, $H_1=\mathcal B(\mathbb R)$, $H_0=L^2(\mathbb R)$, and $P=x+iD=x+d/dx$. Then for $u\in H_1$ $$\|Pu\|^2=\|xu\|^2+\|Du\|^2-\|u\|^2 . \tag{2.2}$$ $P:H_1\to H_0$ is bounded and surjective, its kernel is spanned by $e^{-x^2/2}$, and $P$ is Fredholm of index $1$.

Proof. Identity (2.2). For $u\in C_c^\infty$, $\|u'+xu\|^2=\|xu\|^2+\|u'\|^2+2\operatorname{Re}(xu,u')$, and $2\operatorname{Re}(xu,u')=\int x\,(|u|^2)'\,dx=-\int|u|^2\,dx$. Both sides of (2.2) are continuous in the norm of $H_1$, so Lemma 2.2(1) extends the identity to $H_1$. In particular $\|Pu\|\le\|u\|_{\mathcal B}$.

Kernel. If $Pu=0$ as a distribution, then $(e^{x^2/2}u)'=e^{x^2/2}(u'+xu)=0$. A distribution on $\mathbb R$ with zero derivative is constant (Fact 1.20), so $u=Ce^{-x^2/2}$, and this function lies in $H_1$.

Closed range and finite kernel. We use the compactness test (Fact 1.1). Let $u_k$ be bounded in $H_1$ with $Pu_k$ convergent. By Lemma 2.2(2) a subsequence converges in $L^2$. Apply (2.2) to differences: $\|x(u_k-u_l)\|^2+\|D(u_k-u_l)\|^2=\|P(u_k-u_l)\|^2+\|u_k-u_l\|^2\to0$. So the subsequence is Cauchy in $H_1$. Hence $P$ has finite-dimensional kernel and closed range.

Dense range. Let $v\in L^2$ be orthogonal to $PH_1$. Then $\int(\varphi'+x\varphi)\overline v\,dx=0$ for every $\varphi\in C_c^\infty$. As a distribution this says $x\overline v-(\overline v)'=0$, so $(e^{-x^2/2}\overline v)'=0$ and $\overline v=Ce^{x^2/2}$. This is in $L^2$ only if $C=0$.

A closed dense range is everything, so $P$ is surjective. The index is $1-0=1$. $\square$

The adjoint direction behaves in the opposite way: $x-d/dx$ is injective on $H_1$ and its range has codimension one (Exercise 14.1). Proposition 2.3 is the case $n=1$ of Proposition 4.4 below: for $n=1$ the operator $p(x+iD)$ of Section 4 is exactly $x+d/dx$.

3. The exterior algebra and the odd–even symbol $p(w)$

To go from one dimension to $n$, we need a matrix-valued symbol $p(w)$, linear over $\mathbb R$ in $w\in\mathbb C^n$, whose values are invertible for $w\ne0$. The exterior algebra provides one.

Let $\Lambda=\Lambda(\mathbb C^n)=\bigoplus_{q=0}^n\Lambda^q$ be the exterior algebra of $\mathbb C^n$. We give $\mathbb C^n$ the inner product $\langle z,w\rangle=\sum_jz_j\overline{w_j}$, and $\Lambda^q$ the inner product $$\langle u_1\wedge\dots\wedge u_q,\ v_1\wedge\dots\wedge v_q\rangle=\det\bigl(\langle u_i,v_k\rangle\bigr)_{i,k=1}^q,$$ with different degrees orthogonal. For $J=\{j_1<\dots<j_q\}$ put $e_J=e_{j_1}\wedge\dots\wedge e_{j_q}$. These vectors form an orthonormal basis. Write $\Lambda^e=\bigoplus_q\Lambda^{2q}$ and $\Lambda^o=\bigoplus_q\Lambda^{2q+1}$. For $w\in\mathbb C^n$ let $\Lambda(w)v=w\wedge v$ and let $\Lambda(w)^*$ be its adjoint. Put $\varepsilon_j=\Lambda(e_j)$, $\iota_j=\varepsilon_j^*$, and $\mathcal N=\sum_j\varepsilon_j\iota_j$.

Lemma 3.1. (1) Let $\sigma(j,J)=(-1)^{\#\{k\in J:\,k<j\}}$. Then $\varepsilon_je_J=\sigma(j,J)e_{J\cup\{j\}}$ if $j\notin J$ and $0$ otherwise; $\iota_je_J=\sigma(j,J)e_{J\setminus\{j\}}$ if $j\in J$ and $0$ otherwise.

2. For all $j,l$: $$\varepsilon_j\varepsilon_l+\varepsilon_l\varepsilon_j=0,\qquad \iota_j\iota_l+\iota_l\iota_j=0,\qquad \varepsilon_j\iota_l+\iota_l\varepsilon_j=\delta_{jl}I . \tag{3.1}$$
3. $\Lambda(w)=\sum_jw_j\varepsilon_j$, $\Lambda(w)^*=\sum_j\overline{w_j}\iota_j$, $\Lambda(w)^2=0$, and $$\Lambda(w)\Lambda(w)^*+\Lambda(w)^*\Lambda(w)=|w|^2I . \tag{3.2}$$ Also $\mathcal Ne_J=|J|\,e_J$: the operator $\mathcal N$ multiplies a form of degree $q$ by $q$.

Proof. (1) To write $e_j\wedge e_J$ in increasing order, move $e_j$ past the elements of $J$ that are smaller than $j$; each move gives a factor $-1$. If $J=K\cup\{j\}$ with $j\notin K$, then $(\varepsilon_je_K,e_J)=\sigma(j,K)=\sigma(j,J)$, and all other inner products vanish; this gives $\iota_j$.

2. The first relation is $e_j\wedge e_l=-e_l\wedge e_j$; the second is its adjoint. For the third, take $j=l$ first. If $j\in J$ then $\varepsilon_j\iota_je_J=e_J$ (the two signs are equal) and $\iota_j\varepsilon_je_J=0$; if $j\notin J$ the roles swap. Now let $j\ne l$. Both $\varepsilon_j\iota_le_J$ and $\iota_l\varepsilon_je_J$ vanish unless $l\in J$ and $j\notin J$. Then both are multiples of $e_K$, $K=(J\setminus\{l\})\cup\{j\}$: $$\varepsilon_j\iota_le_J=\sigma(l,J)\sigma(j,J\setminus\{l\})e_K,\qquad \iota_l\varepsilon_je_J=\sigma(j,J)\sigma(l,J\cup\{j\})e_K .$$ If $j<l$, then $\sigma(j,J\setminus\{l\})=\sigma(j,J)$ and $\sigma(l,J\cup\{j\})=-\sigma(l,J)$. If $j>l$, then $\sigma(j,J\setminus\{l\})=-\sigma(j,J)$ and $\sigma(l,J\cup\{j\})=\sigma(l,J)$. In both cases the two terms cancel.

3. The first two formulas follow from linearity of $w\mapsto w\wedge v$ and conjugate-linearity of the adjoint. $\Lambda(w)^2v=w\wedge w\wedge v=0$. By (3.1), $\Lambda(w)\Lambda(w)^*+\Lambda(w)^*\Lambda(w)=\sum_{j,l}w_j\overline{w_l}(\varepsilon_j\iota_l+\iota_l\varepsilon_j)=\sum_j|w_j|^2I$. Finally $\varepsilon_j\iota_je_J=e_J$ for $j\in J$ and $0$ otherwise. $\square$

Proposition 3.2. Let $w\in\mathbb C^n\setminus\{0\}$.

1. (Koszul complex.) The sequence $0\to\Lambda^0\to\Lambda^1\to\dots\to\Lambda^n\to0$, with every map equal to $\Lambda(w)$, is exact.
2. (Odd–even symbol.) Let $A=\Lambda(w)+\Lambda(w)^*$ and $p(w)=A|_{\Lambda^e}:\Lambda^e\to\Lambda^o$. The adjoint of $p(w)$ is $A|_{\Lambda^o}$, and $$p(w)^*p(w)=|w|^2I_{\Lambda^e},\qquad p(w)p(w)^*=|w|^2I_{\Lambda^o}. \tag{3.3}$$ So $p(w)$ is invertible, $p(w)^{-1}=p(w)^*/|w|^2$, and $p(w)/|w|$ is unitary.
3. If $n\ge1$, then $\dim\Lambda^e=\dim\Lambda^o=2^{n-1}$. If $n=0$, then $\Lambda^e=\mathbb C$ and $\Lambda^o=0$.
4. For $x,\xi\in\mathbb R^n$, $p(x+i\xi)=\sum_j(x_j+i\xi_j)\varepsilon_j+\sum_j(x_j-i\xi_j)\iota_j$ on $\Lambda^e$. So $p$ is real-linear in $(x,\xi)$.
5. For $O\in O(n)$ let $\Lambda(O)$ act by $\Lambda(O)(u_1\wedge\dots\wedge u_q)=Ou_1\wedge\dots\wedge Ou_q$. Then $\Lambda(O)$ is unitary, preserves $\Lambda^e$ and $\Lambda^o$, and $$p(Ow)=\Lambda(O)\,p(w)\,\Lambda(O)^{-1},\qquad Ow=Ox+iO\xi . \tag{3.4}$$

Proof. (1) $\Lambda(w)^2=0$, so each range lies in the next kernel. If $\Lambda(w)v=0$, then (3.2) gives $v=\Lambda(w)\bigl(|w|^{-2}\Lambda(w)^*v\bigr)$, so $v$ is in the range. (The map $|w|^{-2}\Lambda(w)^*$ is a contracting homotopy.)

2. $A$ is self-adjoint and changes the degree by $\pm1$, so it maps $\Lambda^e$ to $\Lambda^o$ and back. Since $\Lambda^e\perp\Lambda^o$, the adjoint of $A|_{\Lambda^e}$ is $A|_{\Lambda^o}$. By Lemma 3.1(3), $A^2=\Lambda(w)^2+(\Lambda(w)^*)^2+\Lambda(w)\Lambda(w)^*+\Lambda(w)^*\Lambda(w)=|w|^2I$. Restricting to $\Lambda^e$ and to $\Lambda^o$ gives (3.3).

3. If $n\ge1$, $p(e_1)$ is unitary from $\Lambda^e$ onto $\Lambda^o$, so the dimensions agree; they add up to $2^n$. This matches the count $\sum_q(-1)^q\binom nq=(1-1)^n=0$. If $n=0$, $\Lambda=\Lambda^0=\mathbb C$.

4. This is Lemma 3.1(3) with $w_j=x_j+i\xi_j$.

5. $O$ is real and orthogonal, so it preserves $\langle\cdot,\cdot\rangle$ on $\mathbb C^n$. The determinant formula then shows that $\Lambda(O)$ preserves inner products, and it is invertible with inverse $\Lambda(O^{-1})$. It preserves degrees. From $\Lambda(O)(w\wedge v)=Ow\wedge\Lambda(O)v$ we get $\Lambda(O)\Lambda(w)=\Lambda(Ow)\Lambda(O)$. Taking adjoints and using unitarity gives $\Lambda(O)\Lambda(w)^*=\Lambda(Ow)^*\Lambda(O)$. Add the two identities and restrict to $\Lambda^e$. $\square$

Remark 3.3. The invertibility of $p(w)$ is the finite-dimensional case of a general fact about complexes: for an exact complex, the odd–even operator $d+d^*$, from the even to the odd part, has zero kernel, and its index is the Euler characteristic, which is zero. See Traces that survive passage to cohomology. The identity (3.3) gives more: an explicit inverse, with norm $1/|w|$.

Example 3.4 (Two small cases). For $n=1$, $\Lambda^e=\mathbb C\cdot1$, $\Lambda^o=\mathbb C e_1$, and $p(w)1=we_1$; so $p(w)$ is multiplication by $w$. For $n=2$, in the bases $\{1,e_1\wedge e_2\}$ of $\Lambda^e$ and $\{e_1,e_2\}$ of $\Lambda^o$, $$p(w)=\begin{pmatrix}w_1&-\overline{w_2}\\ w_2&\overline{w_1}\end{pmatrix},\qquad \det p(w)=|w|^2 .$$ Indeed $p(w)1=w_1e_1+w_2e_2$, and $p(w)(e_1\wedge e_2)=\Lambda(w)^*(e_1\wedge e_2)=\overline{w_1}e_2-\overline{w_2}e_1$ by Lemma 3.1(1).

4. The Bott oscillator $p(x+iD)$

We now replace $w=x+i\xi$ by the operators $x+iD$. The result is an operator of index one on $\mathbb R^n$ for every $n$.

A form on $\mathbb R^n$ is a function $u=\sum_Ju_Je_J$ with values in $\Lambda$; operators on functions act on each coefficient $u_J$. Put $$a_j=x_j+\partial_j=x_j+iD_j,\qquad a_j^\dagger=x_j-\partial_j,\qquad d_x=\sum_ja_j\varepsilon_j,\quad \delta_x=\sum_ja_j^\dagger\iota_j,\quad \mathcal D=d_x+\delta_x .$$ Thus $d_x$ and $\delta_x$ come from $\Lambda(w)$ and $\Lambda(w)^*$ in Lemma 3.1(3) when $a_j$ replaces $w_j$ and $a_j^\dagger$ replaces $\overline{w_j}$. By Proposition 3.2(4), the symbol of $\mathcal D$ is $\Lambda(x+i\xi)+\Lambda(x+i\xi)^*$. Each term contains either $x_j$ or $\xi_j$ alone, so every quantization gives the same operator. We write $$P=p(x+iD)=\mathcal D\big|_{\text{even forms}},\qquad H_1=\mathcal B\otimes\Lambda^e,\qquad H_0=L^2\otimes\Lambda^o ,$$ and $\mathsf g(x)=e^{-|x|^2/2}$, regarded as a form of degree $0$. Here $\mathcal B$ is the space (2.1).