Triple Products of Eigenfunctions and Schrodinger/Witten Operators

Author

Joe Schaefer

Dedication

To Autumn

Abstract

Using elementary techniques from Geometric Analysis, Partial Differential Equations, and Abelian $C^*$Algebras, we uncover a novel, yet familiar, global geometric discriminant – namely the indexed set of integrals of triple products of eigenfunctions of the Schrodinger/Witten Operator, to precisely characterize which isospectral Schrodinger/Witten Operators are isometric.

Introduction

Let $(M,g)$ be a smooth, connected, complete Riemannian manifold (possibly non-compact, possibly of infinite volume). Fix a real potential $V\in C^\infty(M)$ which is bounded below and proper in the sense

$$V(x)\to +\infty\qquad\text{as }x\to\infty$$

(i.e., ${x:V(x)\le C}$ is compact for every $C$). Let

$$H=-\Delta_g+V$$

be the associated Schrodinger operator, essentially self-adjoint on $C_c^\infty(M)$. Under these hypotheses it is classical that $\sigma(H)=\sigma_{\mathrm{disc}}(H)={\lambda_0\lt\lambda_1\le\lambda_2\le\cdots}\to+\infty$, each eigenvalue of finite multiplicity; there is an orthonormal basis ${e^i}_{i=0}^\infty\subset L^2(M,d\mathrm{vol}_g)$ of smooth eigenfunctions,

$$He^i=\lambda_i e^i.$$

(The same conclusions hold for $-\Delta_g$ itself on some complete manifolds of finite volume with no continuous spectrum, but the confining potential makes the analysis cleaner.)
Define the triple products exactly as in the compact case:

$$M^{i,j,k}:=\langle e^i e^j,e^k\rangle_{L^2}=\int_M e^i e^j\ \overline{e^k}\ d\mathrm{vol}_g.$$

Because the eigenfunctions are Schwartz-class at infinity in the sense that they decay faster than any polynomial in $V$ (Agmon estimates [Agmon1982]/ weighted Sobolev [TaylorPDE1]), the products $e^i e^j$ lie in $L^2$ and the integrals converge absolutely.

Results

Theorem

Let $(M,g,V)$ and $(N,h,W)$ be two such data, with isospectral Schrödinger operators $H_M$ and $H_N$. Suppose there exist orthonormal eigenbases ${e^i}$ of $H_M$ and ${f^i}$ of $H_N$, ordered by non-decreasing eigenvalues, such that

$$M_M^{i,j,k}=M_N^{i,j,k}\qquad\forall\qquad i,j,k.$$

Then there is a diffeomorphism $F:N\to M$ with $F^*g=h$ and $V\circ F=W$.

Corollary

Let $(M,g)$ be a complete connected Riemannian manifold and $\varphi\in C^\infty(M)$ a weight such that the Witten operator

$$L_\varphi u=-\Delta_g u+\langle\nabla\varphi,\nabla u\rangle$$

is essentially self-adjoint and bounded below on $C_c^\infty(M)$, with

$$\sigma(L_\varphi)=\sigma_{\mathrm{disc}}(L_\varphi)={\nu_0=0<\nu_1\le\nu_2\le\cdots}\to+\infty$$

in $L^2(M,e^{-\varphi}\ d\mathrm{vol}_g)$. (This holds whenever $\varphi$ is confining in the Bakry–Émery sense: e.g. $\operatorname{Ric}_g+\operatorname{Hess}\varphi\ge K$ and $\varphi(x)\to+\infty$ as $x\to\infty$.) Let $\set{f^i} _{i=0}^\infty$ be an orthonormal eigenbasis,

$$L_\varphi f^i=\nu_i f^i,\qquad\int_M f^i\overline{f^j}\ e^{-\varphi}\ d\mathrm{vol}_g=\delta^{ij},$$

ordered by nondecreasing eigenvalues, and set

$$W^{i,j,k}:=\int_M f^i f^j\overline{f^k}\ e^{-\varphi}\ d\mathrm{vol}g=\langle f^if^j,f^k\rangle{L^2(e^{-\varphi}\mathrm{vol})}.$$

Let $(N,h,\psi)$ be a second such triple, with Witten operator $L_\psi$ isospectral to $L_\varphi$, and suppose there is an eigenbasis ${u^i}$ of $L_\psi$ with the same eigenvalues and

$$\int_N u^i u^j\overline{u^k}\ e^{-\psi}\ d\mathrm{vol}_h=W^{i,j,k}\quad\text{for all }i,j,k.$$

Then there exists a diffeomorphism $F:N\to M$ such that
$F^*g=h,\qquad \psi=\varphi\circ F+\log \frac{\mu(M)}{\mu(N)}.$
In particular the weighted Riemannian manifolds $(M,g,e^{-\varphi}\mathrm{vol}_g)$ and $(N,h,e^{-\psi}\mathrm{vol}_h)$ are isometric.

Remark

The Witten Laplacian on $p$-forms is the Hodge Laplacian of the weighted manifold plus

$$t\operatorname{Hess}\varphi\#_p+t^2\lvert\nabla\varphi\rvert^2.$$

Once $(M,g,\varphi)$ is reconstructed from the scalar array $W^{i,j,k}$, the operators $L_{\varphi}^{(p)}$ are determined, so their spectra and eigenform pairings against scalar eigenfunctions are invariants, not additional moduli.

Preliminaries

Let $\mathcal{A}_0\subset L^2(M)$ be the algebraic span of the eigenfunctions. Finite linear combinations

$$\phi=\sum_{i=0}^N\hat\phi(i)\ e^i$$

are dense in $L^2$ and, by elliptic regularity plus the decay of eigenfunctions, lie in $C^\infty\cap C_0(M)$. Pointwise multiplication is

$$\phi\psi=\sum_k\Bigl(\sum_{i,j}\hat\phi(i)\hat\psi(j)M^{i,j,k}\Bigr)e^k.$$

Thus the structure constants determine the product of any two finite eigenfunction expansions. The same identity holds on $N$. Consequently the linear map

$$\vec F:\mathcal{A}_0(M)\to\mathcal{A}_0(N),\qquad e^i\mapsto f^i$$

is a unital algebra homomorphism for pointwise multiplication (the constant function is a multiple of the ground state only if $V$ is constant; in general $\vec F$ need not send $1$ to $1$, but it preserves the product of any two functions that can be so expanded). Characteristic functions of sublevel sets of $V$ can be approximated in $L^2$ by functional calculus of $H$, so $\vec F$ also preserves characteristic functions of sets of finite measure in the $L^2$ sense.

This is identical to the compact calculation; the only extra ingredient is that the eigenfunction expansion of a compactly supported smooth function still converges rapidly enough for the triple-product formula to be justified (which follows from the same elliptic estimates that give $\lambda_i^{N}|e^i|_\infty\to 0$ locally, plus Agmon decay at infinity).

Proof of Theorem

(Following [JS24]) For sufficiency, we now consider the linear, bijective orthonormal eigenfunction basis map $\vec{F}$ from $C^\infty(M)$ to $C^\infty(N)$ and note that from the calculations in Preliminaries section above, $\vec{F}$ preserves pointwise products for smooth functions (and preserves characteristic functions when extended to $L^2(M,g)$) by the premise that $\set{M^{i,j,k}}$ is invariant under this map.

Lemma

$\vec F$ preserves the uniform norm on the span of the eigenfunctions, hence extends to an isometric $*$-homomorphism

$$\vec F:C_0(M)\supset\overline{\mathcal{A}_0}^{|\cdot|_\infty}\longrightarrow C_0(N).$$

Proof of Lemma

On a compact manifold one uses a smooth partition of unity and the fact that $|a|_\infty=1$ for a partition function $a$ is detected by $|a^p|_\infty=1$ for all $p$, which is visible in the Fourier coefficients because the limiting characteristic function of ${a=1}$ has $L^2$-mass.

In the non-compact setting replace a global partition of unity by a sequence of cutoffs $\chi_R$ that equal $1$ on ${V\le R}$ and vanish outside ${V\le R+1}$. On each such “almost-compact” region the same argument applies: if $|\phi|_\infty=c$ is attained on a set of positive measure inside ${V\le R}$, the powers

$$\phi^p/|\phi|_\infty^p$$

converge in $L^2_{\mathrm{loc}}$ to a characteristic function, and $\vec F$ preserves that limit because it preserves all triple products. Sending $R\to\infty$ and using that every function in $C_0(M)$ is uniformly small outside a large sublevel set of $V$ gives $|\vec F(\phi)|_\infty=|\phi|_\infty$.

Now we apply the Gelfand-Naimark-Segal Representation Theorem (in contravariant functor form) for Abelian $C^*$ algebras $C_0(N), C_0(M)$ [JC19] to represent this isomorphism $\vec{F}$ by a homeomorphism $F$ between $N$ and $M$. Since $\vec{F}$ is bijective on smooth functions, $F$ too must be smooth.

As this now diffeomorphism preserves eigenvalues and eigenfunctions (by hypothesis on $\vec{F}(f) = f \circ F$), it must preserve the Schrodinger operator on smooth functions. By construction

$$\vec F(\phi)=\phi\circ F$$

on the eigenbasis, hence on all finite combinations. Therefore

$$H_N(\phi\circ F)=(H_M\phi)\circ F$$

for every eigenfunction $\phi$, and by density for every Schwartz-class function. In particular $F$ pulls eigenfunctions of $H_M$ back to eigenfunctions of $H_N$ with the same eigenvalues.

Writing $H=-\Delta+V$ in local coordinates, the principal symbol of $H$ is the same as that of $-\Delta$, namely the cometric $g^{ij}\xi_i\xi_j$. An elliptic operator that is intertwined by a diffeomorphism has its principal symbol pulled back. Hence

$$F^*g=h.$$

The zeroth-order terms then give $V\circ F=W$.

This completes the proof of the Theorem. As the proof of the Corollary is redundant, it is omitted.

Example

On $M=\mathbb{R}$ with the Euclidean metric, take

$$H=-\frac{d^2}{dx^2}+x^2.$$

The potential $V(x)=x^2$ is smooth, bounded below, and $V(x)\to+\infty$ as $|x|\to\infty$. Hence $\sigma_{\mathrm{ess}}(H)=\emptyset$. The spectrum is

$$\lambda_n=2n+1,\qquad n=0,1,2,\dots$$

(each of multiplicity one). The corresponding $L^2(\mathbb{R})$-normalized eigenfunctions are the Hermite functions

$$\psi_n(x)=\bigl(2^n n!\ \sqrt{\pi}\bigr)^{-1/2}H_n(x)\ e^{-x^2/2},$$

where $H_n$ are the physicist’s Hermite polynomials. They form an orthonormal basis of $L^2(\mathbb{R})$, lie in the Schwartz class $\mathcal{S}(\mathbb{R})\subset C_0(\mathbb{R})$, and satisfy the Agmon decay that makes every pointwise product $\psi_i\psi_j$ square-integrable.

Triple Products

Because the eigenfunctions are real,

$$M^{i,j,k}=\langle\psi_i\psi_j,\psi_k\rangle=\int_{-\infty}^{\infty}\psi_i(x)\psi_j(x)\psi_k(x)\ dx.$$

These integrals vanish unless $i+j+k$ is even (parity) and the three indices satisfy a triangle-type constraint coming from the linearization of Hermite polynomials. The linearization formula

$$H_n(x)H_m(x)=\sum_{r=0}^{\min(n,m)}\binom{n}{r}\binom{m}{r}r!\ 2^r\ H_{n+m-2r}(x)$$

together with the Gaussian weight converts, after normalizing, into an explicit formula for $M^{i,j,k}$. Equivalently, the generating-function identity

$$\int_{-\infty}^{\infty}e^{-x^2}H_a(x)H_b(x)H_c(x)\ dx=\sqrt{\pi}\ 2^N\frac{a!\ b!\ c!}{(N-a)!(N-b)!(N-c)!}$$

(when $a+b+c=2N$ is even and each factorial in the denominator is defined and non-negative, and $0$ otherwise) yields $M^{i,j,k}$ after inserting the normalization constants of the $\psi_n$. In particular:

$M^{n,n,0}=\int\psi_n^2\psi_0\ dx>0$ (the ground state $\psi_0=\pi^{-1/4}e^{-x^2/2}$ is a strictly positive Gaussian);
the map $\psi_i\psi_j=\sum_k M^{i,j,k}\psi_k$ is the exact multiplication rule in the Hermite basis.

So the structure constants of pointwise multiplication on the dense subspace $\mathrm{span}\ {\psi_n}\subset C_0(\mathbb{R})$ are known in closed form.

What the theorem says

Suppose $(\mathbb{R},g,W)$ is another complete Riemannian line (so $g=e^{2\phi(x)}\ dx^2$ in some coordinate) with a smooth confining potential $W$, and suppose its Schrödinger operator $-\Delta_g+W$ is isospectral to $H$ and admits an eigenbasis ${f_n}$ with the same triple products $\langle f_i f_j,f_k\rangle=M^{i,j,k}.$

The discrete-spectrum argument then produces a diffeomorphism $F:\mathbb{R}\to\mathbb{R}$ such that

$$F^*g=dx^2,\qquad W\circ F=x^2.$$

In one dimension this is almost tautological once the algebra is identified, but the mechanism is visible by hand:

Finite Hermite combinations are dense in $C_0(\mathbb{R})$ for the uniform norm (Stone–Weierstrass on each compact plus Gaussian decay at infinity). The numbers $M^{i,j,k}$ determine the product of any two such combinations, hence determine a $C^*$-isomorphism

$$C_0(\mathbb{R})\longrightarrow C_0(\mathbb{R}),\qquad \psi_n\mapsto f_n.$$

Gelfand–Naimark supplies a homeomorphism of the underlying locally compact spaces; smoothness of the eigenfunctions upgrades it to a diffeomorphism $F$. Intertwining of $H$ forces the principal symbol (the cometric) and the zeroth-order term (the potential) to match.

Because every eigenspace is one-dimensional, the singular-value packaging collapses to the absolute values $|M^{i,j,k}|$ together with a consistent choice of signs (the “diagonal litmus” of the compact paper: the products $M^{i,i,k}$ determine the even part of the algebra and fix most signs).

Explicit Formula

$$M^{i,j,k}=\langle\psi_i\psi_j,\psi_k\rangle=\int_{\mathbb{R}}\psi_i(x)\psi_j(x)\psi_k(x)\ dx=N_i N_j N_k\int_{\mathbb{R}}H_i(x)H_j(x)H_k(x)\ e^{-3x^2/2}\ dx,$$

where $N_n=(2^n n!\sqrt{\pi})^{-1/2}$.

$M^{i,j,k}=0$ unless both of the following hold – parity: $i+j+k$ is even; triangle: $|i-j|\le k\le i+j$ (and cyclic permutations).

These are the oscillator analogues of the Clebsch–Gordan support conditions.

Change variables $x=\sqrt{2/3}\ y$. Then

$$\int_{\mathbb{R}}H_i(x)H_j(x)H_k(x)\ e^{-3x^2/2}\ dx=\sqrt{\frac{2}{3}}\int_{\mathbb{R}}H_i\Bigl(\sqrt{\tfrac{2}{3}}\ y\Bigr)H_j\Bigl(\sqrt{\tfrac{2}{3}}\ y\Bigr)H_k\Bigl(\sqrt{\tfrac{2}{3}}\ y\Bigr)\ e^{-y^2}\ dy.$$

Each scaled polynomial expands by the finite identity

$$H_n(\gamma y)=\sum_{\ell=0}^{\lfloor n/2\rfloor}\gamma^{n-2\ell}(\gamma^2-1)^\ell\binom{n}{2\ell}\frac{(2\ell)!}{\ell!}\ H_{n-2\ell}(y),$$

with $\gamma=\sqrt{2/3}$. The product of the three expansions is a finite linear combination of products $H_a H_b H_c$. Titchmarsh’s formula then evaluates every remaining integral:

$$\int_{\mathbb{R}}H_a H_b H_c\ e^{-y^2}\ dy = \begin{cases} \dfrac{\sqrt{\pi}\ 2^{s}\ a!\ b!\ c!}{(s-a)!(s-b)!(s-c)!} & \text{if }a+b+c=2s\text{ is even and }s\ge a,b,c,\\ 0 & \text{otherwise.} \end{cases}$$

The result is a finite triple sum (at most $O(ijk)$ terms, typically far fewer). Combining the prefactors $N_i N_j N_k\sqrt{2/3}$ gives $M^{i,j,k}$ exactly.

Equivalently, the product of two normalized functions expands in a scaled Hermite basis

$$\psi_i(x)\psi_j(x)=\sum_{r=0}^{\lfloor(i+j)/2\rfloor}c_{r;ij}\ \psi_{i+j-2r}(\sqrt{2}\ x),$$

with explicit coefficients

$$c_{r;ij}=\frac{(-1)^r}{\sqrt{2^{i+j-2r+1}\pi}}\sqrt{\frac{(i+j-2r)!}{i!\ j!}}\binom{i}{r}\binom{j}{r}r!\ 2^{r}$$

(up to the conventional placement of the $\sqrt{2}$ in the argument; cf. [K16]). Then

$$M^{i,j,k}=\sum_{r}c_{r;ij}\int_{\mathbb{R}}\psi_{i+j-2r}(\sqrt{2}\ x)\ \psi_k(x)\ dx,$$

and each scaled overlap is a ${}_2F_1$ (or terminates as a short hypergeometric sum). This is the form best suited to computation.

$\Gamma$ Calculus

By definition [BE85]

$$\Gamma_H(f,g):=\frac12\bigl(H(fg)-f\ Hg-g\ Hf\bigr).$$

On basis vectors this is already diagonal in the spectrum:

$$\Gamma_H(e^i,e^j)=\frac12\sum_k(\lambda_k-\lambda_i-\lambda_j)\ M^{i,j,k}\ e^k.$$

Hence

$$\begin{aligned} \langle\nabla e^i,\nabla e^j\rangle &=-\Gamma_H(e^i,e^j)-\frac12 V e^i e^j\\ &=-\frac12\sum_k[(\lambda_k-\lambda_i-\lambda_j)M^{i,j,k} +\int V e^i e^j \bar e^k\ dx]\ e^k. \end{aligned}$$

Ground-state conjugation repairs the $V$ defect: if $H\psi=\lambda_0\psi$ with $\psi>0$, then

$$L:=\psi^{-1}(H-\lambda_0)(\psi\ \cdot\ )=-\Delta-2\langle\nabla\log\psi,\nabla\ \cdot\ \rangle$$

is a diffusion, and its Bakry–Émery tensor is $\operatorname{Ric}-2\operatorname{Hess}\log\psi$. For the harmonic oscillator the ground state is a Gaussian, $\log\psi_0=-x^2/2+\mathrm{const}$, and one recovers exactly $\mathrm{BE}(2,\infty)$ for this renormalized Ornstein–Uhlenbeck process — the classical diffusion underneath the quantum oscillator.

With $\phi = -2 \log \psi_0$ and $v\cdot w$ the Riemannian inner product on the cotangent bundle, let us write these operators in the eigenbasis of $L$:

$$f_n=\frac{\psi_n}{\psi_0},\qquad M_L^{i,j,k}=\int\frac{\psi_i\psi_j\psi_k}{\psi_0}\ d\mathrm{vol},\\ \begin{aligned} \Gamma_L(f_i,f_j):&=\frac12\sum_k(\lambda_k+\lambda_0-\lambda_i-\lambda_j)M_L^{i,j,k}\ f_k = -df_i \cdot df_j,\\ \Gamma_{2,L}(f_i,f_j):&=\frac14\sum_k(\lambda_k+\lambda_0-\lambda_i-\lambda_j)^2 M_L^{i,j,k}\ f_k\implies\\ \Gamma_{2,L}(f)&=\lvert\operatorname{Hess}f\rvert_{\mathrm{HS}}^2+\operatorname{Ric}(\nabla f,\nabla f) + \operatorname{Hess}\phi(\nabla f, \nabla f). \end{aligned}$$

For a confining Schrödinger operator as in the Theorem, the isometric type of $(M,g,V)$ is encoded in the diffusion data $({\lambda_n-\lambda_0},{M_L^{i,j,k}})$, because $\Gamma_L$ is the cometric and $\psi_0$ recovers $V$ by Riccati.

Returning to Harmonic Oscillator Example

With $f_n=\psi_n/\psi_0$ and $d\mu=\psi_0^2\ dx$,

$$M_L^{i,j,k}=\int_{\mathbb{R}}\frac{\psi_i\psi_j\psi_k}{\psi_0}\ dx=\frac{N_i N_j N_k}{N_0}\int H_i H_j H_k\ e^{-x^2}\ dx.$$

Titchmarsh then collapses to

$$M_L^{i,j,k} = \frac{\sqrt{i!\ j!\ k!}}{(s-i)!(s-j)!(s-k)!} \quad\text{when }i+j+k=2s\text{ and }s\ge i,j,k,$$

and $0$ otherwise. In particular $M_L^{0,n,n}=1$, which is orthonormality of ${\psi_n}$ in $L^2(dx)$.

As these structure constants are non-negative on a multiplicity-1 spectrum, they are the associated singular-value invariants [JS24] for $L$.