Triple Products of Eigenfunctions and Schrödinger/Witten Operators

Author

Joe Schaefer

Dedication

To Autumn

Abstract

For a second-order operator $P_{\alpha,V}=-\Delta_g+\langle\nabla\alpha,\nabla\cdot\rangle+V$ on a complete Riemannian manifold, acting in $L^2(e^{-\alpha}\ d\mathrm{vol})$ with discrete spectrum, the eigenvalues determine the eigenspaces only up to independent unitaries on each eigenspace. We prove that a spectral identification which also preserves the triple products

$$\Phi_\mu(u,v,w)=\int uvw\ d\mu,\qquad d\mu=e^{-\alpha}\ d\mathrm{vol},$$

is implemented by a diffeomorphism $F$ with $F^*g=h$, $\beta=\alpha\circ F+c$, and $W=V\circ F$. Schrödinger operators are the case $\alpha=0$; Witten operators are the case $V=0$. The geometric invariant is the multiplication table on the spectral subspaces, not the spectrum alone.

Keywords

spectrum, triple products, Schrödinger, Witten, Laplacian, eigenfunctions, inverse problem

Introduction

Let $(M,g)$ be a complete connected Riemannian manifold, $\alpha,V\in C^\infty(M)$, and

$$P_{\alpha,V}u=-\Delta_g u+\langle\nabla\alpha,\nabla u\rangle+Vu$$

on $C_c^\infty(M)$, acting in $L^2(M,\mu)$ with $d\mu=e^{-\alpha}\ d\mathrm{vol}_g$. Assume $P _ {\alpha,V}$ is essentially self-adjoint and bounded below, with empty essential spectrum

$$\sigma(P_{\alpha,V})=\sigma_{\mathrm{disc}}(P_{\alpha,V})=\set{\lambda_0<\lambda_1\le\lambda_2\le\cdots}\to+\infty.$$

Write

$$\mathcal{V} _ M:=\bigoplus_{n\ge0}\ker(P_{\alpha,V}-\lambda_n)\subset L^2(M,\mu)\cap C^\infty(M),$$

and assume $\mathcal{V}_M\subset L^3(M,\mu).$ Hölder then gives a bounded symmetric trilinear form

$$\Phi_\mu:\mathcal{V}_M\times\mathcal{V}_M\times\mathcal{V}_M\to\mathbb{C},\qquad\Phi_\mu(u,v,w)=\int_M uvw\ d\mu.$$

The same data $(N,h,\beta,W)$ produce $Q=P_{\beta,W}$, $d\nu=e^{-\beta}d\mathrm{vol}_h$, $\mathcal{V}_N\subset L^3(N,\nu)$, and $\Phi_\nu$.

A linear map $U:\mathcal{V}_M\to\mathcal{V}_N$ is a spectral identification if it is unitary for the $L^2$ inner products and

$$U\bigl(\ker(P_{\alpha,V}-\lambda_n)\bigr)=\ker(Q-\lambda_n)\qquad\text{for every }n.$$

Results

Theorem 1

Suppose $P_{\alpha,V}$ and $P_{\beta,W}$ are isospectral and $U:\mathcal{V}_M\to\mathcal{V}_N$ is a spectral identification with

$$\Phi_\nu(Uu,Uv,Uw)=\Phi_\mu(u,v,w)\qquad\text{for all }u,v,w\in\mathcal{V}_M.$$

Then there is a diffeomorphism $F:N\to M$ such that

$$F^*g=h,\qquad \beta=\alpha\circ F+c,\qquad W=V\circ F$$

for a constant $c\in\mathbb{R}$.

Corollary 2

The same conclusion holds if there exist $\mu$-orthonormal and $\nu$-orthonormal eigenbases $\set{e^i}$ and $\set{f^i}$, ordered by their common nondecreasing eigenvalues, with

$$M^{i,j,k} := \int_M e^ie^j\bar e^k\ d\mu=\int_N f^if^j\bar f^k\ d\nu\qquad\text{for all }i,j,k.$$

Remark 3

Corollary 2 is the matrix form of Theorem 1. Given such bases, the map $Ue^i=f^i$ yields a spectral identification via its linearity, and

$$M^{i,j,k}=\Phi_\mu(e^i,e^j,\bar e^k) = \Phi_\nu(f^i, f^j, \bar f^k).$$

Conversely, any spectral identification that intertwines $\Phi_\mu$ and $\Phi_\nu$ becomes Corollary 2 after choosing orthonormal bases of each eigenspace and transporting them by $U$. The residual freedom in a spectral identification is $\prod_n\mathrm{U}(m_n)$; intertwining $\Phi$ is the condition that cuts that product down to an isometry of $(M,g,\alpha,V)$ with $(N,h,\beta,W)$.

Remark 4

The spectrum of $P_{\alpha,V}$ does not determine $(g,\alpha,V)$. On $\mathbb{R}$ this is the McKean–Trubowitz class of the oscillator [MT82]. On closed manifolds there are isospectral potentials that are not isometric [GS03]. Theorem 1 is the statement that the missing invariant is the multiplication table $\Phi_\mu$.

Proofs

Let $U:\mathcal{V}_M\to\mathcal{V}_N$ be a spectral identification intertwining $\Phi_\mu$ and $\Phi_\nu$ (of course, $U$ naturally extends to a unitary isometry on $L^2$). Choose $\mu$-orthonormal and $\nu$-orthonormal real-valued eigenbases $\set{e^i}$ and $\set{f^i}$ with $Ue^i=f^i$, ordered by nondecreasing eigenvalues. Hölder and $\mathcal{V}_M\subset L^3(\mu)$ make

$$M^{i,j,k}:=\Phi_\mu(e^i,e^j,e^k)=\int_M e^ie^j e^k\ d\mu$$

finite, and likewise on $N$.

Finite spectral projectors of $P_{\alpha,V}$ send $C_c^\infty(M)$ into $\mathcal{V}_M$. Essential self-adjointness and smooth coefficients put

$$C_c^\infty\subset\bigcap_k\mathrm{Dom}(P_{\alpha,V}^k),$$

so those projectors converge in $C^\infty_{\mathrm{loc}}$. Thus $\mathcal{V}_M$ is dense in $C^1$ near every $p\in M$. If $\mathrm{span}\set{de^i(p):i\ge0}\ne T_p^*M$, some $0\neq v\in T_pM$ would annihilate every $d\phi$ for $\phi\in\mathcal{V}_M$, and a $C^1$-limit would annihilate every element of $C_c^\infty$, which is absurd. So some $d$-tuple satisfies

$$\det\bigl(de^{i_a}\otimes de^{i_b}(p)\bigr)\ne0.$$

The joint evaluation

$$\mathrm{Ev} _ M:M\to\mathbb{R}^{\mathbb{N}},\qquad p\mapsto\bigl(e^n(p)\bigr)_{n\ge0}$$

is therefore an immersion. It is injective because $\mathcal{V}_M$ separates points ($C^0 _ {\mathrm{loc}}$ density; on a connected manifold the ground state of $P _ {\alpha,V}$ may be taken strictly positive). So $\mathrm{Ev}_M$ is an embedding onto its image. The same holds for $\mathrm{Ev}_N$ and $\mathcal{V}_N$.

Claim 4.1

The images of the joint evaluation maps coincide in $\mathbb{R}^{\mathbb{N}}$.

Let $\mathcal{A}_M$ be the algebra generated by $\set{e^i}$ under pointwise product. The generators of $\mathcal{A}_M$ need not lie in $C_0(M)$. They lie in $C(M)$. The algebra $\mathcal{A}_M$ separates points and vanishes nowhere, so Stone–Weierstrass [Sto48] (as naturally extended to the compact-open topology) gives density of $\mathcal{A}_M$ in $C(M)$, and likewise for $\mathcal{A}_N$ in $C(N)$. Matching structure constants $M^{i,j,k}$ convert $U| _ {\mathcal{A}_M}$ into an algebra isomorphism $\Psi:\mathcal{A}_M\to\mathcal{A}_N$[JS24].

Elements of $\mathcal{A}_M$ are finite sums of finite products of the $e^i$. Those products expand as $\sum_k M^{i,j,k}e^k$, and the series converge uniformly on compact sets, so $\Psi$ is compact-open continuous on $\mathcal{A}_M$ and extends to a compact-open algebra isomorphism

$$\Psi:C(M)\xrightarrow{\sim}C(N).$$

The compact-open characters of $C(M)$ are the evaluations $\mathrm{ev}_p$, $p\in M$ [GJ60], and likewise for $N$. For each $q\in N$,

$$\chi_q=\mathrm{ev}_q\circ\Psi:C(M)\to\mathbb{R}$$

is a compact-open continuous character, hence $\chi_q=\mathrm{ev}_p$ for a unique $p\in M$. Set $F(q)=p$. Then $e^n\circ F=f^n$ for all $n$, and $F$ is a homeomorphism $N\to M$. The existence of $F$ establishes the claim.

Now fix $q\in N$ and $p=F(q)$. Some $d$-tuple

$$e^{i_1},\ldots,e^{i_d}$$

has independent differentials at $p$, so$x=(e^{i_1},\ldots,e^{i_d})$is a $C^\infty$ chart on a neighborhood of $p$. On a neighborhood of $q$ set

$$y=(f^{i_1},\ldots,f^{i_d})=(e^{i_1}\circ F,\ldots,e^{i_d}\circ F)=x\circ F.$$

Thus $F=x^{-1}\circ y$ wherever $x$ and $y$ are defined. In particular $y$ is a chart on $N$, and in the charts $(x,y)$ the map $F$ is the identity of $\mathbb{R}^d$. Hence $F$ is a $C^\infty$ diffeomorphism. (The same indices work on both sides because they are the coordinate functions of one map.)

By construction

$$Q(u\circ F)=(P_{\alpha,V}u)\circ F,\qquad u\in\mathcal{V}_M.$$

Both sides are second-order scalar differential operators. For $\phi\in\mathcal{V}_M$ the product $\phi^2$ lies in the $C^\infty$ algebra generated by $\mathcal{V}_M$, and

$$P(\phi^2)-2\phi\ P\phi=-2\ d\phi\cdot d\phi - V\phi^2.$$

The principal part is the cometric. Since the differentials $de^i$ span $T^*$ at every point,

$$F^*g=h.$$

The first-order symbols are the drifts $\nabla\alpha$ and $\nabla\beta$, so $d\beta=F^*d\alpha$ and $\beta=\alpha\circ F+c$ for a constant $c\in\mathbb{R}$. The order-zero terms then give $W=V\circ F$.

This proves Theorem 1. Corollary 2 is the same statement in an eigenbasis: $Ue^i=f^i$ and matching $M^{i,j,k}$ are the coordinates of a spectral identification that intertwines $\Phi_\mu$ and $\Phi_\nu$, as in Remark 3.

Remark 5

The same construction applies to compact Riemannian orbifolds: $F$ is a homeomorphism of underlying spaces, an isometry of $P_{\alpha,V}$-data on the regular set, and an isomorphism of local isotropy at singular points. The evaluation map drops rank exactly on the singular strata. Anshul Adve first established this result for unit tangent bundles of compact hyperbolic 2-orbifolds [AA25].

Remark 6

If $\overline{M},\overline{N}$ are compact with smooth boundary and $P,Q$ are Dirichlet realizations, the same argument with $C(\overline{M})$ in place of $C_0(M)$ yields a diffeomorphism of closed manifolds with $F(\partial N)=\partial M$, $F^*g=h$, $\beta=\alpha\circ F+c$, $W=V\circ F$. For Neumann or Robin, $F$ is still a diffeomorphism of closed manifolds; preservation of the boundary condition follows from the isometry of the interior metric.

Remark 7

The same reconstruction applies to graphs, with the amount of analysis matching the function space.

On a finite weighted graph the eigenfunctions of a Schrödinger matrix span $\mathbb{R}^V$. The array $M^{i,j,k}$ is the multiplication table of that algebra, characters are vertices, and $P(\phi^2)-2\phi P\phi$ recovers the edge weights. Spectrum alone does not: Sunada graphs and isospectral trees are distinguished by $\Phi$.

A compact quantum graph is a one-dimensional Riemannian space with singularities. Discrete spectrum, $\mathcal{V}\subset L^3$, and Claim 4.1 give a homeomorphism of the underlying metric graphs. The principal-symbol identity on each edge recovers lengths; vertex conditions follow from the isometry as in Remark 6. Combinatorial type is not enough: equal graphs with different length vectors are separated by $\Phi$.

An infinite locally finite graph with $\sigma_{\mathrm{ess}}(P)=\emptyset$ is formally in the same class. Compact-open Stone–Weierstrass on the discrete space $V$ produces a bijection of vertices once the generated algebra separates points and the ground state vanishes nowhere (Perron–Frobenius on a connected graph). The same quadratic identity recovers weights. Two hypotheses are not free: eigenfunctions must lie in $\ell^3(\mu)$, which is an Agmon-type decay statement, and they must separate vertices, which fails if a nontrivial automorphism commutes with $P$. An infinite quantum graph with discrete spectrum is the hybrid of the last two paragraphs.

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

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

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 [SA82] that makes every pointwise product $e^ie^j$ square-integrable.

Triple Products

Because the eigenfunctions are real,

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

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 $e^n$. In particular:

$$M^{n,n,0}=\int {e^n}^2e^0\ dx>0$$

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

So the structure constants of pointwise multiplication on the dense subspace $\mathrm{span}\set{e^n}\subset C_0(\mathbb{R})$ 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 $\set{f^n}$ 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

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

Isospectrality alone does not give this. McKean–Trubowitz [MT82] construct an infinite-dimensional family of potentials $q$ on $\mathbb{R}$ with

$$\sigma\Bigl(-\frac{d^2}{dx^2}+q\Bigr)=\sigma(H)={2n+1}_{n\ge0}$$

(up to their normalization of $H$). The parameters are the norming constants of the eigenfunctions; the even potential $x^2$ is the unique even point of the class. Thus $\set{2n+1}$ does not determine $V(x)=x^2$. Matching the table $M^{i,j,k}$ does: it encodes the multiplication of the eigenfunctions, which is strictly more than the spectrum, and Theorem 1 converts that table into $F^*g=dx^2$ and $W\circ F=x^2$.
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

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

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 [T48] 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

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

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

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

and each scaled overlap is a ${}_2F_1$ (or terminates as a short hypergeometric sum).

Harmonic Oscillator to Ornstein-Uhlenbeck

The oscillator $H=-d^2/dx^2+x^2$ is the Schrödinger case $\alpha=0$, $V(x)=x^2$ of $P_{\alpha,V}$, with unweighted measure $dx$ and triple products $M^{i,j,k}=\int e^ie^je^k\ dx$. Ground-state conjugation by $e^0(x)=\pi^{-1/4}e^{-x^2/2}$ produces the Witten operator

$$L={e^0}^{-1}(H-\lambda_0)(e^0\ \cdot)=-\frac{d^2}{dx^2}+2x\frac{d}{dx}$$

on $L^2(\mathbb{R},{e^0}^2\ dx)$. This is $P_{\alpha,0}$ for $\alpha=x^2+\mathrm{const}$: the generator of the Ornstein–Uhlenbeck diffusion. The conjugated basis $f^n=e^n/e^0$ is orthonormal in $L^2({e^0}^2\ dx)$, and the two arrays are related by

$$W^{i,j,k}:=\int_{\mathbb{R}}f^if^jf^k\ {e^0}^2\ dx=\int_{\mathbb{R}}\frac{e^ie^je^k}{e^0}\ dx.$$

Thus $M$ is the multiplication table of Theorem 1 for $H$, and $W$ is the same table for $L$. Matching either determines the line, the Euclidean metric, and the data $(V,\alpha)=(x^2,x^2)+\mathrm{const}$. The explicit values follow.

Titchmarsh [T48] then collapses to

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

and $0$ otherwise. In particular $W^{0,n,n}=1$, which is orthonormality of $\set{e^n}$ 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 $L$.