Partial differential equations, property C for

Property $C$ stands for "completeness" of the set of products of solutions to homogeneous linear partial differential equations. It was introduced in [a1] and used in [a2], [a3], [a4], [a5], [a6], [a7], [a8], [a9], [a10], [a11], [a12], [a13] as a powerful tool for proving uniqueness results for many multi-dimensional inverse problems, in particular, inverse scattering problems (cf. also Inverse scattering, multi-dimensional case).

Let $D$ be a bounded domain in $\textbf{R}^n$, $n\geq 2$, let $L_mu(x):=\sum^{J}_{|j|=0}a_{jm}(x)D^ju(x)$, where $j$ is a multi-index, $D^ju=\partial^{|j|}u/\partial x_1^{j_{1}}...\partial x_n^{j_{n}}$, derivatives being understood in the distributional sense, the $a_{jm}(x)$, $m=1,2$, are certain $L^{\infty}(D)$ functions, $N_m:=\{w:L_mw=0\text{in}D\}$ is the null-space of the formal differential operator $L_m$, and the equation $L_mw=0$ is understood in the distributional sense.

Consider the subsets $\widetilde{N}_1\in N_2$ and $\widetilde{N}_2\in N_n$ for which the products $w_1w_2$ are defined, $w_1\in\widetilde{N}_1$, $w_2\in\widetilde{N}_2$.

The pair $\{L_1,L_2\}$ has property $C_p$ if and only if the set $\{w_1w_2\}_{\forall w_{m}\in\widetilde{N}_{m}}$ is total (complete) in $L^p(D)$, ($p\geq 1$ is fixed), that is, if $f(x)\in L^p(D)$ and

\begin{equation}\int_{D}f(x)w_1(x)w_2(x)dx=0,\\\forall w_1\in\widetilde{N}_1,\forall w_2\in\widetilde{N}_2,\end{equation}

then $f(x)\equiv 0$.

By property $C$ one often means property $C_2$ or $C_p$ with any fixed $p\geq1$.

Is property $C$ generic for a pair of formal partial differential operators $L_1$ and $L_2$?

For the operators with constant coefficients, a necessary and sufficient condition is given in [a10] for a pair $\{L_1,L_2\}$ to have property $C$. For such operators it turns out that property $C$ is generic and holds or fails to hold simultaneously for all $p\in[1,\infty)$: Assume $a_{jm}(x)=a_{jm}=\text{const}$. Denote $L_m(z):=\sum^{J}_{|j|=0}a_{jm}z^j$, $z\in\textbf{C}^n$. Note that $L_m(e^{zx})=e^{zx}L_m(z)$, $z.x:=\sum^n_{j=1}z_jx_j$.

Therefore $e^{zx}\in\widetilde{N}_m$ if and only if $L_m(z)=0$.

Define the algebraic varieties (cf. also Algebraic variety)

\begin{equation}\mathcal{L}_m:=\{z:z\in\mathbf{C}^n,L_m(z)=0\}.\end{equation}

One says that $\mathcal{L}_1$ is transversal to $\mathcal{L}_2$, and writes $\mathcal{L}_1\nparallel\mathcal{L}_2$, if and only if there exist a point $\zeta\in\mathcal{L}_1$ and a point $\xi\in\mathcal{L}_2$ such that the tangent space $T_1$ to $\mathcal{L}_1$ (in $\mathbf{C}^n)$ at the point $\zeta$ and the tangent space $T_2$ to $\mathcal{L}_2$ at the point $\xi$ are transversal (cf. Transversality).

The following result is proved in [a1]: The pair $\{L_1,L_2\}$ of formal partial differential operators with constant coefficients has property $C$ if and only if $\mathcal{L}_1\nparallel\mathcal{L}_2$.

Thus, property $C$ fails to hold for a pair $\{L_1,L_2\}$ of formal differential operators with constant coefficients if and only if the variety $\mathcal{L}_1\cup\mathcal{L}_2$ is a union of parallel hyperplanes in $\mathbf{C}^n$.

Therefore, property $C$ for partial differential operators with constant coefficients is generic.

If $L_1=L_2=L$ and the pair $\{L,L\}$ has property $C$, then one says that $L$ has property $C$.

Examples.

Let $n\geq 2$, $L=\nabla^2:=\sum^n_{j=1}$. Then $L=\bigg\{z:z\in\mathbf{C}^n,z_1^2+...+z_n^2=0\bigg\}$. It is easy to check that there are points $\zeta\in\mathcal{L}$ and $\xi\in\mathcal{L}$ at which the tangent hyperplanes to $\mathcal{L}$ are not parallel. Thus $L=\nabla^2$ has property $C$. This means that the set of products of harmonic functions in a bounded domain $D\subset\mathbf{R}^n$ is complete in $L^p(D)$, $p\geq 1$ (cf. also Harmonic function). Similarly one checks that the operators

\begin{equation}L=\frac{\partial}{\partial t}-\nabla^2,L=\frac{\partial^2}{\partial t^2}-\nabla^2,L=i\frac{\partial}{\partial t}-\nabla^2\end{equation}

have property $C$.

Numerous applications of property $C$ to inverse problems can be found in [a1].

Property $C=C_2$ holds for a pair of Schrödinger operators with potentials $q_m(x)\in L_0^2(\mathbf{R}^n)$, $n\geq 3$, where $L_0^2(\mathbf{R}^n)$ is the set of $L^2(\mathbf{R}^n)$ functions with compact support{} (cf. also Schrödinger equation).

If $u_m(x,\alpha,k)$, $m=1,2$, $\alpha\in S^{n-1}$, $k=\text{const}>0$, $S^{n-1}$ is the unit sphere in $\mathbf{R}^n$, are the scattering solutions corresponding to the Schrödinger operators $l_m=-\nabla^2+q_m(x)-k^2$, $q_m(x)\in L^{2_{0}}(\mathbf{R}^n)$, $n\geq 3$, then the set of products $\{u_1(x,\alpha,k)u_2(x,\beta,k)\}_{\forall\alpha,\beta\in S^{n-1}}$, $k=\text{const}>0$ is fixed, is complete in $L^2(D)$, where $D\subset\mathbf{R}^n$ is an arbitrary fixed bounded domain [a1]. The set $\{u_m(x,\alpha,k)\}_{\forall\alpha\in S^{n-1}}$, where $k>0$ is fixed, is total in the set $N_m:=\{w:l_mw=0\text{in}D,w\in H^2(D)\}$, where $H^2(D)$ is the Sobolev space [a1].

References

[a1]	A.G. Ramm, "Multidimensional inverse scattering problems" , Longman/Wiley (1992)
[a2]	A.G. Ramm, "Scattering by obstacles" , Reidel (1986)
[a3]	A.G. Ramm, "Completeness of the products of solutions to PDE and uniqueness theorems in inverse scattering" Inverse Probl. , 3 (1987) pp. L77–L82
[a4]	A.G. Ramm, "Multidimensional inverse problems and completeness of the products of solutions to PDE" J. Math. Anal. Appl. , 134 : 1 (1988) pp. 211–253 (Also: 139 (1989), 302)
[a5]	A.G. Ramm, "Recovery of the potential from fixed energy scattering data" Inverse Probl. , 4 (1988) pp. 877–886 (Also: 5 (1989), 255)
[a6]	A.G. Ramm, "Multidimensional inverse problems: Uniqueness theorems" Appl. Math. Lett. , 1 : 4 (1988) pp. 377–380
[a7]	A.G. Ramm, "Multidimensional inverse scattering problems and completeness of the products of solutions to homogeneous PDE" Z. Angew. Math. Mech. , 69 : 4 (1989) pp. T13–T22
[a8]	A.G. Ramm, "Property C and uniqueness theorems for multidimensional inverse spectral problem" Appl. Math. Lett. , 3 (1990) pp. 57–60
[a9]	A.G. Ramm, "Completeness of the products of solutions of PDE and inverse problems" Inverse Probl. , 6 (1990) pp. 643–664
[a10]	A.G. Ramm, "Necessary and sufficient condition for a PDE to have property C" J. Math. Anal. Appl. , 156 (1991) pp. 505–509
[a11]	A.G. Ramm, "Property C and inverse problems" , ICM-90 Satellite Conf. Proc. Inverse Problems in Engineering Sci. , Springer (1991) pp. 139–144
[a12]	A.G. Ramm, "Stability estimates in inverse scattering" Acta Applic. Math. , 28 : 1 (1992) pp. 1–42
[a13]	A.G. Ramm, "Stability of solutions to inverse scattering problems with fixed-energy data" Rend. Sem. Mat. e Fisico (2001) pp. 135–211