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Property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p1300302.png" /> stands for  "completeness"  of the set of products of solutions to homogeneous linear partial differential equations. It was introduced in [[#References|[a1]]] and used in [[#References|[a2]]], [[#References|[a3]]], [[#References|[a4]]], [[#References|[a5]]], [[#References|[a6]]], [[#References|[a7]]], [[#References|[a8]]], [[#References|[a9]]], [[#References|[a10]]], [[#References|[a11]]], [[#References|[a12]]], [[#References|[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|Inverse scattering, multi-dimensional case]]).
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Property $C$ stands for  "completeness"  of the set of products of solutions to homogeneous linear partial differential equations. It was introduced in [[#References|[a1]]] and used in [[#References|[a2]]], [[#References|[a3]]], [[#References|[a4]]], [[#References|[a5]]], [[#References|[a6]]], [[#References|[a7]]], [[#References|[a8]]], [[#References|[a9]]], [[#References|[a10]]], [[#References|[a11]]], [[#References|[a12]]], [[#References|[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|Inverse scattering, multi-dimensional case]]).
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p1300303.png" /> be a bounded domain in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p1300304.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p1300305.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p1300306.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p1300307.png" /> is a multi-index, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p1300308.png" />, derivatives being understood in the distributional sense, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p1300309.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003010.png" />, are certain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003011.png" /> functions, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003012.png" /> is the null-space of the formal [[Differential operator|differential operator]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003013.png" />, and the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003014.png" /> is understood in the distributional sense.
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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|differential operator]] $L_m$, and the equation $L_mw=0$ is understood in the distributional sense.
  
Consider the subsets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003016.png" /> for which the products <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003017.png" /> are defined, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003019.png" />.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003020.png" /> has property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003022.png" /> if and only if the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003023.png" /> is total (complete) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003024.png" />, (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003025.png" /> is fixed), that is, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003026.png" /> and
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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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003027.png" /></td> </tr></table>
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\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}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003028.png" /></td> </tr></table>
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then $f(x)\equiv 0$.
  
then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003029.png" />.
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By property $C$ one often means property $C_2$ or $C_p$ with any fixed $p\geq1$.
  
By property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003031.png" /> one often means property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003032.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003033.png" /> with any fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003034.png" />.
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Is property $C$ generic for a pair of formal partial differential operators $L_1$ and $L_2$?
  
Is property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003035.png" /> generic for a pair of formal partial differential operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003036.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003037.png" />?
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For the operators with constant coefficients, a necessary and sufficient condition is given in [[#References|[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 $L_m(e^{zx})=e^{zx}L_m(z)$, $z.x:=\sum^n_{j=1}z_jx_j$.
  
For the operators with constant coefficients, a necessary and sufficient condition is given in [[#References|[a10]]] for a pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003038.png" /> to have property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003039.png" />. For such operators it turns out that property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003040.png" /> is generic and holds or fails to hold simultaneously for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003041.png" />: Assume <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003042.png" />. Denote <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003043.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003044.png" />. Note that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003045.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003046.png" />.
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Therefore $e^{zx}\in\widetilde{N}_m$ if and only if $L_m(z)=0$.
 
 
Therefore <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003047.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003048.png" />.
 
  
 
Define the algebraic varieties (cf. also [[Algebraic variety|Algebraic variety]])
 
Define the algebraic varieties (cf. also [[Algebraic variety|Algebraic variety]])
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003049.png" /></td> </tr></table>
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\begin{equation}\mathcal{L}_m:=\{z:z\in\mathbf{C}^n,L_m(z)=0\}.\end{equation}
  
One says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003050.png" /> is transversal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003051.png" />, and writes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003052.png" />, if and only if there exist a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003053.png" /> and a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003054.png" /> such that the tangent space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003055.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003056.png" /> (in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003057.png" />) at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003058.png" /> and the tangent space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003059.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003060.png" /> at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003061.png" /> are transversal (cf. [[Transversality|Transversality]]).
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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|Transversality]]).
  
The following result is proved in [[#References|[a1]]]: The pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003062.png" /> of formal partial differential operators with constant coefficients has property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003063.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003064.png" />.
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The following result is proved in [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003065.png" /> fails to hold for a pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003066.png" /> of formal differential operators with constant coefficients if and only if the variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003067.png" /> is a union of parallel hyperplanes in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003068.png" />.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003069.png" /> for partial differential operators with constant coefficients is generic.
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Therefore, property $C$ for partial differential operators with constant coefficients is generic.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003070.png" /> and the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003071.png" /> has property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003072.png" />, then one says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003073.png" /> has property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003075.png" />.
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If $L_1=L_2=L$ and the pair $\{L,L\}$ has property $C$, then one says that $L$ has property $C$.
  
 
==Examples.==
 
==Examples.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003076.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003077.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003078.png" />. It is easy to check that there are points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003079.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003080.png" /> at which the tangent hyperplanes to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003081.png" /> are not parallel. Thus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003082.png" /> has property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003083.png" />. This means that the set of products of harmonic functions in a bounded domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003084.png" /> is complete in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003085.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003086.png" /> (cf. also [[Harmonic function|Harmonic function]]). Similarly one checks that the operators
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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|Harmonic function]]). Similarly one checks that the operators
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003087.png" /></td> </tr></table>
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\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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003088.png" />.
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have property $C$.
  
Numerous applications of property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003089.png" /> to inverse problems can be found in [[#References|[a1]]].
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Numerous applications of property $C$ to inverse problems can be found in [[#References|[a1]]].
  
Property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003090.png" /> holds for a pair of Schrödinger operators with potentials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003091.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003092.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003093.png" /> is the set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003094.png" /> functions with compact support{} (cf. also [[Schrödinger equation|Schrödinger equation]]).
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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|Schrödinger equation]]).
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003095.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003096.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003097.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003098.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p13003099.png" /> is the unit sphere in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p130030100.png" />, are the scattering solutions corresponding to the Schrödinger operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p130030101.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p130030102.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p130030103.png" />, then the set of products <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p130030104.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p130030105.png" /> is fixed, is complete in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p130030106.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p130030107.png" /> is an arbitrary fixed bounded domain [[#References|[a1]]]. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p130030108.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p130030109.png" /> is fixed, is total in the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p130030110.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p130/p130030/p130030111.png" /> is the [[Sobolev space|Sobolev space]] [[#References|[a1]]].
+
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 [[#References|[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|Sobolev space]] [[#References|[a1]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.G. Ramm,  "Multidimensional inverse scattering problems" , Longman/Wiley  (1992)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.G. Ramm,  "Scattering by obstacles" , Reidel  (1986)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A.G. Ramm,  "Completeness of the products of solutions to PDE and uniqueness theorems in inverse scattering"  ''Inverse Probl.'' , '''3'''  (1987)  pp. L77–L82</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  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)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  A.G. Ramm,  "Recovery of the potential from fixed energy scattering data"  ''Inverse Probl.'' , '''4'''  (1988)  pp. 877–886  (Also: 5 (1989), 255)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  A.G. Ramm,  "Multidimensional inverse problems: Uniqueness theorems"  ''Appl. Math. Lett.'' , '''1''' :  4  (1988)  pp. 377–380</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  A.G. Ramm,  "Property C and uniqueness theorems for multidimensional inverse spectral problem"  ''Appl. Math. Lett.'' , '''3'''  (1990)  pp. 57–60</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  A.G. Ramm,  "Completeness of the products of solutions of PDE and inverse problems"  ''Inverse Probl.'' , '''6'''  (1990)  pp. 643–664</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  A.G. Ramm,  "Necessary and sufficient condition for a PDE to have property C"  ''J. Math. Anal. Appl.'' , '''156'''  (1991)  pp. 505–509</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  A.G. Ramm,  "Property C and inverse problems" , ''ICM-90 Satellite Conf. Proc. Inverse Problems in Engineering Sci.'' , Springer  (1991)  pp. 139–144</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  A.G. Ramm,  "Stability estimates in inverse scattering"  ''Acta Applic. Math.'' , '''28''' :  1  (1992)  pp. 1–42</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  A.G. Ramm,  "Stability of solutions to inverse scattering problems with fixed-energy data"  ''Rend. Sem. Mat. e Fisico''  (2001)  pp. 135–211</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.G. Ramm,  "Multidimensional inverse scattering problems" , Longman/Wiley  (1992)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.G. Ramm,  "Scattering by obstacles" , Reidel  (1986)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A.G. Ramm,  "Completeness of the products of solutions to PDE and uniqueness theorems in inverse scattering"  ''Inverse Probl.'' , '''3'''  (1987)  pp. L77–L82</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  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)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  A.G. Ramm,  "Recovery of the potential from fixed energy scattering data"  ''Inverse Probl.'' , '''4'''  (1988)  pp. 877–886  (Also: 5 (1989), 255)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  A.G. Ramm,  "Multidimensional inverse problems: Uniqueness theorems"  ''Appl. Math. Lett.'' , '''1''' :  4  (1988)  pp. 377–380</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  A.G. Ramm,  "Property C and uniqueness theorems for multidimensional inverse spectral problem"  ''Appl. Math. Lett.'' , '''3'''  (1990)  pp. 57–60</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  A.G. Ramm,  "Completeness of the products of solutions of PDE and inverse problems"  ''Inverse Probl.'' , '''6'''  (1990)  pp. 643–664</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  A.G. Ramm,  "Necessary and sufficient condition for a PDE to have property C"  ''J. Math. Anal. Appl.'' , '''156'''  (1991)  pp. 505–509</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  A.G. Ramm,  "Property C and inverse problems" , ''ICM-90 Satellite Conf. Proc. Inverse Problems in Engineering Sci.'' , Springer  (1991)  pp. 139–144</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  A.G. Ramm,  "Stability estimates in inverse scattering"  ''Acta Applic. Math.'' , '''28''' :  1  (1992)  pp. 1–42</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  A.G. Ramm,  "Stability of solutions to inverse scattering problems with fixed-energy data"  ''Rend. Sem. Mat. e Fisico''  (2001)  pp. 135–211</TD></TR></table>

Latest revision as of 08:38, 27 April 2021

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
How to Cite This Entry:
Partial differential equations, property C for. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Partial_differential_equations,_property_C_for&oldid=17893
This article was adapted from an original article by A.G. Ramm (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article