Jacobi operator factorization

On the surface associated to $u^{\prime }=\varphi (x,u)$, the operator acting on normal geodesic deviations, the operator $A^2+\mathcal{K}$ corresponding to a Schrodinger equation (general family), decomposes into two first-order factors. This factorization is the algebraic engine behind the Jacobi-field integration method of @panalvarez2024integration, and it is the $ϵ=0$ case of the deformation operators.

Definition. Let $A={\partial }_x+\varphi {\partial }_u$ be the associated vector field and $h\in {\mathcal{C}}^{\mathrm{\infty }}(U)$. Define the two first-order operators

where ${\varphi }_u={\partial }_u\varphi =\mathrm{div}(A)$.

Proposition. (@panalvarez2024integration)

• (a) $\mathfrak{T}(h)=0⟺h$ is an integrating factor of $-\varphi dx+du$.
• (b) $\mathfrak{S}(h)=0⟺h^{-1}$ is an integrating factor.
• (c) Factorization: $(\mathfrak{T}\circ \mathfrak{S})(h)=(A^2+\mathcal{K})(h)$, where $\mathcal{K}$ is the Gaussian curvature of the associated surface.

Proof sketch of (c). $(\mathfrak{T}\circ \mathfrak{S})(h)=A^2(h)-(A({\varphi }_u)+{\varphi }_u^2)h=A^2(h)-({\varphi }_{ux}+\varphi {\varphi }_{uu}+{\varphi }_u^2)h=A^2(h)-{\partial }_u(A\varphi )h=(A^2+\mathcal{K})(h)$. $◼$

Interpretation. $\mathfrak{S}$ is the variational operator: along a solution $f(x)$, the equation $\mathfrak{S}(\delta )=0$ reads ${\delta }^{\prime }={\varphi }_u(x,f(x))\delta$, the variational equation, so its kernel consists of the infinitesimal separations between nearby solutions (i.e. the Jacobi fields). $\mathfrak{T}$ is the integrating-factor operator: $\ker \mathfrak{T}$ is exactly the condition $\mathrm{div}(hA)=0$.

Decomposition of the Schrödinger-type equation

For a relative Jacobi field $J=\sigma A+\delta {\partial }_u$, the Jacobi equation decomposes into

The second equation is a Schrödinger-type equation, and by (c) it factorizes as

This produces the integrating factors of the integration procedure: if $\mathfrak{S}(\delta )=0$ then ${\delta }^{-1}$ is an integrating factor; otherwise $\mathfrak{T}(\mathfrak{S}(\delta ))=0$, so $\mathfrak{S}(\delta )$ is an integrating factor.

The decomposition is the SUSY/Darboux structure with superpotential ${\varphi }_u$: indeed ${\varphi }_u$ satisfies the Riccati equation (curvature-dependent Riccati connection)

which is precisely the condition making $A^2+\mathcal{K}=(A+{\varphi }_u)(A-{\varphi }_u)$ factor. The deformed surfaces $S_{ϵ}$ generalize this by replacing the superpotential with ${\mathrm{\Delta }}_{ϵ}=A(ϵ)+{\varphi }_u$; the original surface corresponds to $ϵ=0$.

Related notes: relative Jacobi field, curvature-dependent Riccati connection, surface deformation of ODEs, Gauss curvature of an ODE-surface, u-independent curvature ODEs, integrating factor, Riccati equation

Reference: @panalvarez2024integration