M. Yu. Kokurin, “Completeness of asymmetric products of harmonic functions and uniqueness of the solution to the Lavrent'ev equation in inverse wave sounding problems”, Izv. RAN. Ser. Mat., 86:6 (2022), 101–122; Izv. Math., 86:6 (2022), 1123

Let $\mathcal{H}(D)$ be the class of all functions from $C^2(\overline{D})$ harmonic in a bounded domain $D \subset \mathbb{R}^n$, $n \geqslant 2$, and $\mathcal{H}_j = \mathcal{H}_j (D)\subset \mathcal{H}(D)$ be some families of functions from $\mathcal{H}(D)$, $j=1,2$. Below, by a domain we will always mean an open connected set.

The study of uniqueness of solutions to various classes of coefficient inverse problems for equations of mathematical physics is generally facilitated by the ability to answer (to some extent) the following approximation theory problem: under which conditions the family of all pairwise products

Let us briefly recall some known results in this direction. The first result here is due to A. P. Calderón (1980), who considered the simplest case where each of two families of harmonic functions is just the entire class $\mathcal{H}(D)$.

From the point of view of approximation theory, it seems natural in this result to reduce the families $\mathcal{H}_1$, $\mathcal{H}_2$ without impairing the completeness property of pairwise products. Below, we will see that such a reduction plays an important role in establishing uniqueness of solutions for a number of applied inverse problems with minimal requirements to the dimension of their input data set. We note the following result pertaining to harmonic functions with fixed values on a part of the boundary. We set

There are generalizations of Proposition 1 in which the Laplace equation $\Delta u=0$ is replaced by more general equations $P(-i\partial)u+c(x)u=0$, $x\in D$. In this regard, we mention the following result for elliptic second-order equations. Let $c_j \in L_{\infty}(D)$ and let $\mathcal{H}_j$ be the family of all generalized solutions to the equation $\Delta u+c_j(x) u=0$ that lie in $L_2(D)$, $j=1,2$.

It is worth pointing out that in Proposition 3 and in its generalizations from [2], § 5.3, the families $\mathcal{H}_1$, $\mathcal{H}_2$ are formed by the solutions of generally different equations.

It is easily seen that, in the above examples, the families $\mathcal{H}_1$, $\mathcal{H}_2$ coincide with the set of all solutions of the equation under consideration in the domain (possibly, with a boundary condition on a part of the boundary $\partial D$). In the case of interest, in which both families are generated by the same Laplace equation, these families are equal. In this sense, Proposition 1 establishes completeness of the symmetric products of harmonic functions. It is also known that the completeness property in Proposition 1 remains valid also in the case where one of the involved families $\mathcal{H}_1$, $\mathcal{H}_2$ is much narrower than the other one. In these cases, we will speak about asymmetric products of functions harmonic in the domain. The next result, which strengthens Proposition 1, established completeness of the asymmetric products of harmonic functions in the above sense.

In Proposition 4, the class $\mathcal{H}_2$ is formed only by Newtonian potentials of points of an open one-dimensional subset of a straight line. Note that a similar result also holds for the Helmholtz equation [5]. The motivation of the present paper is to remove the condition $L \cap \overline{D}=\varnothing$, which played an important role in the proof of Proposition 4, and which, at the same time, does not seem essential. Below, we will consider the case $n=3$, which has direct applications to inverse wave sounding problems in $\mathbb{R}^3$.

The key result of the present paper is the following theorem, which strengthens Proposition 4 for $n=3$.

The scheme of our further exposition is as follows. Theorem 1 will be proved in § 2. In subsequent sections, we will consider applications of Theorem 1 to the study of uniqueness of solutions to inverse wave sounding problems in various settings. In these applications, Theorem 1 will be used in analysis of uniqueness of the solution to the Lavrent’ev linear integral equation, which is naturally related to the inverse problems under considerations. In § 3, we will briefly recall the derivation of this equation, and then employ Theorem 1 to verify uniqueness of the solution of the coefficient inverse problem for a hyperbolic equation in the spatially non-overdetermined setting. In § 4, an inverse problem of wave sounding by means of time-harmonic fields in the phaseless setting is investigated. We will show that Theorem 1 (in contrast to other results in this field) is capable of reducing the dimension of the data support, and also provides a non-overdetermined statement of the inverse problem. In § 5, for the inverse problems under consideration, we will establish theorems on the relation of axial symmetry of input data with that of the probed inhomogeneity. The corresponding results are similar to S. N. Karp’s theorem and its variants, which link the spherical symmetry of the scattering amplitude with that of the inhomogeneity under study.

§ 2. Proof of Theorem 1

Let us formulate the result to be proved in a form convenient for further analysis. In view of (1.1), it will be sufficient to verify that the relation

Let us now proceed with the proof of Theorem 1 with due account of the above remarks.

Proof of Theorem 1. We need to prove that equality (2.2) implies $h(x)=0$ almost everywhere in $D$. We equip $\mathbb{R}^3=(x_1,x_2,x_3)$ with the spherical coordinates $x_1=\rho\sin \theta \cos\varphi$, $x_2=\rho\sin \theta \sin\varphi$, $x_3=\rho\cos\theta$, where $\theta \in [0,\pi]$, $\varphi \in [0,2\pi)$. Consider the spherical functions $\{Y_{lm} \}$, $l=0,1,\dots$, $|m|\leqslant l$, defined by (see [7], § 23, [8], § 5.2)

$$ \begin{equation} \begin{aligned} \, Y_{lm}(\theta,\varphi)&=e^{i m\varphi} \sqrt{\frac{2l+1}{4\pi}\, \frac{(l-m)!}{(l+m)!}}\, P_l^m (\cos\theta), \qquad 0\leqslant m \leqslant l, \\ Y_{l, -m}(\theta,\varphi)&=(-1)^m Y_{lm}(\theta,-\varphi). \end{aligned} \end{equation} \tag{2.3} $$

Here,

$$ \begin{equation*} P_l^m(t)=(1-t^2)^{m/2}\, \biggl( \frac{d}{dt} \biggr)^m P_l(t) \end{equation*} \notag $$

is the associated Legendre function, $P_l(t)=P_l^0(t)$ is the Legendre polynomial of degree $l$ (see [7], § 23). From (2.3) we have

$$ \begin{equation} P_l (\cos \theta)= \sqrt{ \frac{4\pi}{2l+1}}\, Y_{l0}(\theta,\varphi), \qquad \varphi\in [0,2\pi). \end{equation} \tag{2.4} $$

Given $y \in L_1^{+}$, we set $\tau=|y|^{-1}$. Then $\tau$ assumes arbitrary values in $(0,1]$. Since $L_1^{+} \subset L_R^+$, equality (2.2) holds for all $y\in L_1^{+}$. For arbitrary points $x\in D$, $y\in L_1^{+}$, we get

$$ \begin{equation*} \frac{1}{|x-y|}=\frac{\tau}{|\tau x- y^0|}, \qquad y^0=(0,0,1). \end{equation*} \notag $$

$$ \begin{equation} \frac{1}{|x-y|}=\frac{\tau}{\sqrt{1-2\tau\rho \cos\theta+(\tau \rho)^2}}=\tau \sum_{l=0}^{\infty} P_l(\cos\theta) (\tau \rho)^l. \end{equation} \tag{2.5} $$

Substituting (2.5) into (2.2), we find that

$$ \begin{equation} \sum_{l=0}^{\infty} \tau^l \biggl( \int_D h(x) u(x) \rho^l P_l(\cos\theta)\, dx \biggr)=0 \end{equation} \tag{2.6} $$

for all $\tau \in (0,1]$. Here, the change in the order of integration and summation is legitimate because of the uniform convergence in (2.5) with respect to $x \in D$, $\tau\in [0,1]$. Since $|P_l(t)|\leqslant 1$ for $t\in [-1,1]$ (see [7], § 7) and $\rho\leqslant R<1$ for $x\in D$, the power series in (2.6) has positive radius of convergence with respect to $\tau$. As a result,

$$ \begin{equation} \int_D h(x) u(x) \rho^l P_l(\cos\theta)\, dx =0 \end{equation} \tag{2.7} $$

for all integer $l \geqslant 0$ and all functions $u\in \mathcal{H}(D)$. Using (2.4), we rewrite equality (2.7) as

$$ \begin{equation} \int_D h(x) u(x) \rho^l Y_{l0}(\theta,\varphi)\, dx =0, \qquad l\geqslant 0. \end{equation} \tag{2.8} $$

As a regular harmonic function $u$ in (2.8), consider the harmonic polynomial

$$ \begin{equation*} u(x)=\rho^L Y_{LM}(\theta,\varphi) \end{equation*} \notag $$

with arbitrary integer $L\geqslant 0$ and $|M|\leqslant L$. So, for all integer $l, L \geqslant 0$, and $|M|\leqslant L$,

$$ \begin{equation} \int_D h(x) \rho^{L+l} Y_{LM}(\theta,\varphi) Y_{l0}(\theta,\varphi)\, dx =0. \end{equation} \tag{2.9} $$

We set

$$ \begin{equation*} S=\{ x\in \mathbb{R}^3\colon |x|=1 \}. \end{equation*} \notag $$

For any integrable compactly supported function $F=F(x)$, we have (see [9], Ch. I, § 4.14)

$$ \begin{equation} \int_{\mathbb{R}^3} F(x)\, dx=\int_S \biggl( \int_0^{\infty} \rho^2 F(\rho\eta)\, d\rho \biggr)\, d\sigma =\int_0^{\infty} \rho^2 \biggl( \int_S F(\rho\eta)\, d\sigma \biggr) \, d\rho, \end{equation} \tag{2.10} $$

where $\eta$ varies over the sphere $S$, and $d\sigma$ is the Lebesgue measure on $S$. Assuming that $h$ is extended by zero outside $D$ and using (2.10), we can write equality (2.9) as

$$ \begin{equation} \int_0^{\infty} \rho^{L+l+2} \biggl( \int_S h(\rho\eta) Y_{LM}(\theta,\varphi) Y_{l0}(\theta,\varphi)\, d\sigma \biggr)\, d\rho=0. \end{equation} \tag{2.11} $$

In (2.11), the vector $\eta=x/|x|\in S$ has spherical coordinates $(1,\theta,\varphi)$. Note that by the above assumptions the outer integral in (2.11) is taken in fact over the interval $[0,R].$ The orthonormal system of spherical functions $\{ Y_{lm} \}$, $l=0,1,\dots$, $|m|\leqslant l$, is complete in $L_2(S)$, and hence, for almost all $\rho\geqslant 0$,

$$ \begin{equation} h(\rho\eta) =\sum_{\lambda=0}^{\infty} \sum_{\mu=-\lambda}^{\lambda} a_{\lambda \mu}(\rho) Y_{\lambda\mu}(\theta,\varphi), \qquad \sum_{\lambda=0}^{\infty} \sum_{\mu=-\lambda}^{\lambda}|a_{\lambda \mu}(\rho)|^2 <\infty. \end{equation} \tag{2.12} $$

Indeed, $h(r\eta)$ lies in $L_2$ qua a function of $(r,\theta,\varphi)$. Hence its restriction to the plane $r=\rho$ belongs to the space $L_2(S)$ for almost all $\rho \geqslant 0$ (see, for example, Lemma 1.3.8 in [10]).

From (2.11) and (2.12) it follows that, for all integer $L,l \geqslant 0$, and $|M|\leqslant L$,

$$ \begin{equation} \int_0^{\infty} \rho^{L+l+2} \biggl\{ \sum_{\lambda=0}^{\infty} \sum_{\mu=-\lambda}^{\lambda} a_{\lambda \mu}(\rho) \biggl( \int_S Y_{\lambda\mu}(\theta,\varphi) Y_{LM}(\theta,\varphi) Y_{l0}(\theta,\varphi)\, d\sigma \biggr) \biggr\}\, d\rho=0. \end{equation} \tag{2.13} $$

Here, the change in the order of integration and summation is legitimate inasmuch as the system of spherical functions is closed in $L_2(S)$ (see [11], Ch. VII, § 3).

In our further analysis, we will require certain properties of the inner integral in (2.13). Integrals of this kind appear in various fields of quantum mechanics for evaluation of matrix elements of vector physical quantities (see [12], Ch. XIV). Let us recall some properties of this integral (see, for example, [8], § 5.9, [12], § 107). First of all, this integral can be explicitly written as

$$ \begin{equation*} \begin{aligned} \, &\int_S Y_{\lambda\mu}(\theta,\varphi) Y_{LM}(\theta,\varphi) Y_{l0}(\theta,\varphi)\, d\sigma \\ &\qquad=\sqrt{\frac{(2\lambda+1)(2L+1)(2l+1)}{4\pi}}\, \begin{pmatrix} \lambda & L & l \\ 0 & 0 & 0 \end{pmatrix}\begin{pmatrix} \lambda & L & l \\ \mu & M & 0 \end{pmatrix}. \end{aligned} \end{equation*} \notag $$

So, using (2.13) we obtain

$$ \begin{equation} \int_0^{\infty} \rho^{L+l+2} \biggl\{ \sum_{\lambda=0}^{\infty} \sum_{\mu=-\lambda}^{\lambda} b_{\lambda \mu}(\rho)\begin{pmatrix} \lambda & L & l\\ 0 & 0 & 0 \end{pmatrix}\begin{pmatrix} \lambda & L & l\\ \mu & M & 0 \end{pmatrix} \biggr\}\, d\rho=0, \end{equation} \tag{2.14} $$

where we set

$$ \begin{equation*} b_{\lambda \mu}(\rho)=\sqrt{2\lambda+1}\, a_{\lambda \mu}(\rho). \end{equation*} \notag $$

Here and in what follows,

$$ \begin{equation} \begin{pmatrix} j_1 & j_2 & j_3 \\ m_1 & m_2 & m_3 \end{pmatrix}=\begin{pmatrix} j_3 & j_1 & j_2 \\ m_3 & m_1 & m_2 \end{pmatrix}=\begin{pmatrix} j_2 & j_3 & j_1 \\ m_2 & m_3 & m_1 \end{pmatrix} \end{equation} \tag{2.15} $$

denotes the $3jm$-symbol (the Wigner coefficient), whose explicit representation for arbitrary $m_1$, $m_2$, $m_3$ is too cumbersome and hence is not given (for more details, see [8], § 8.1, § 8.2, [12], § 106, and also (2.18)). Coefficients (2.15) are real functions of the integer arguments $(j_1,\dots,m_3)$. We will also need the Clebsch–Gordan coefficients $C_{j_1 m_1 \, j_2 m_2}^{j_3 m_3}$, which are related to the Wigner coefficients (2.15) as follows:

$$ \begin{equation} C_{j_1 m_1 \, j_2 m_2}^{j_3 m_3}=(-1)^{j_1-j_2+m_3} \sqrt{2j_3+1}\, \begin{pmatrix} j_1 & j_2 & j_3 \\ m_1 & m_2 & -m_3 \end{pmatrix}. \end{equation} \tag{2.16} $$

Here, the integer $j_k$, $m_k$ satisfy $j_k \geqslant 0$, $|m_k|\leqslant j_k$, $1\leqslant k\leqslant 3$. According to [8], pp. 235–237, the Wigner coefficient (2.15) vanishes if at least one of the constraints

$$ \begin{equation} |j_1-j_2|\leqslant j_3 \leqslant j_1+j_2, \qquad m_1+m_2+m_3=0 \end{equation} \tag{2.17} $$

is violated. In addition (see [8], § 8.5.2, [12], § 106), for integer $p\geqslant 0$,

$$ \begin{equation} \begin{aligned} \, &\begin{pmatrix} j_1 & j_2 & j_3 \\ 0 & 0 & 0 \end{pmatrix} \nonumber \\ &=\begin{cases} \biggl( \dfrac{(j_1+j_2-j_3)!\, (j_1-j_2+j_3)!\, (-j_1+j_2+j_3)!}{(2p+1)!} \biggr)^{1/2} & \\ \quad\times\dfrac{(-1)^p p!}{(p-j_1)!\, (p-j_2)!\, (p-j_3)!}, &j_1+j_2+j_3=2p, \\ 0, &j_1+j_2+j_3=2p+1. \end{cases} \end{aligned} \end{equation} \tag{2.18} $$

For reference, we also recall the integral representation of the Clebsch–Gordan coefficients (see [8], § 8.3)

$$ \begin{equation} \begin{aligned} \, &C_{a\alpha\, b\beta}^{c\gamma}=\frac{(-1)^{a-c+\beta}}{2^{J+1}} \nonumber \\ &\times \biggl( \frac{(c+\gamma)!\, (J-2c)!\, (J+1)!\, (2c+1)}{(a-\alpha)!\, (a+\alpha)!\, (b-\beta)!\, (b+\beta)!\, (c-\gamma)!\, (J-2a)!\, (J-2b)!} \biggr)^{1/2} \nonumber \\ &\times\int_{-1}^1 (1-t)^{a-\alpha} (1+t)^{b-\beta} \biggl( \frac{d}{dt} \biggr)^{c-\gamma} [(1-t)^{J-2a} (1+t)^{J-2b}]\, dt, \qquad J=a+b+c. \end{aligned} \end{equation} \tag{2.19} $$

We will use (2.16) and (2.19) to verify that

$$ \begin{equation} \begin{pmatrix} N-2s & N-s & s\\ \mu & -\mu & 0 \end{pmatrix} \neq 0 \end{equation} \tag{2.20} $$

for any integer $N, s\geqslant 0$ and $\mu$ such that

$$ \begin{equation*} |\mu|\leqslant N-2s. \end{equation*} \notag $$

From (2.16) it follows that (2.20) differs from the Clebsch–Gordan coefficient $C_{N-2s \,\, \mu \,\, N-s \,\, -\mu}^{s 0}$ by a nonzero factor. In turn, by (2.19) this factor is equal, up to a nonzero factor, to the integral

$$ \begin{equation*} \int_{-1}^1 (1-t)^{N-s-\mu} (1+t)^{N-s+\mu}\, dt >0. \end{equation*} \notag $$

Now the required result (2.20) follows.

For an arbitrary fixed $\mu_0 \in \mathbb{Z}$, we set $M=-\mu_0$ in (2.14). We will assume that

$$ \begin{equation} L-l \geqslant|\mu_0|. \end{equation} \tag{2.21} $$

From (2.15), (2.17), (2.18) we have

$$ \begin{equation*} \begin{pmatrix} \lambda & L & l \\ \mu & -\mu_0 & 0 \end{pmatrix}= \begin{pmatrix} L & l & \lambda \\ -\mu_0 & 0 & \mu \end{pmatrix}=0 \end{equation*} \notag $$

for any $\lambda$ if $\mu\neq \mu_0$, and with the violated condition $|L-l|\leqslant \lambda \leqslant L+l$ for $\mu=\mu_0$.

Using (2.21), we conclude that in the inner sum in (2.14) only the terms with $\mu=\mu_0$ may nonvanish, and in the outer sum, only the terms for which $L-l\leqslant \lambda \leqslant L+ l$. Hence (2.14) assumes the form

$$ \begin{equation} \int_0^{\infty} \rho^{L+l+2} \biggl\{ \sum_{\lambda=L-l}^{L+l} b_{\lambda \mu_0}(\rho) \begin{pmatrix} \lambda & L & l \\ 0 & 0 & 0 \end{pmatrix}\begin{pmatrix} \lambda & L & l \\ \mu_0 & -\mu_0 & 0 \end{pmatrix} \biggr\}\, d\rho=0. \end{equation} \tag{2.22} $$

Equality (2.22) holds for all nonnegative integers $L$, $l$ satisfying condition (2.21).

We now fix $K=L+l$ so that

$$ \begin{equation} K \geqslant|\mu_0|. \end{equation} \tag{2.23} $$

Let us analyze equalities (2.22) corresponding to various representations of $K$ as the sum $L+l$ under condition (2.21).

We first consider the case $L=K$, $l=0$. In this case, (2.21) holds by (2.23). The interval of variation of $\lambda$ in (2.22) reduces to the single point $\lambda=K$. Therefore, the sum in (2.22) reduces to the single term

$$ \begin{equation*} b_{K\mu_0}(\rho) \begin{pmatrix} K & K & 0 \\ 0 & 0 & 0 \end{pmatrix}\begin{pmatrix} K & K & 0\\ \mu_0 & -\mu_0 & 0 \end{pmatrix}. \end{equation*} \notag $$

Here the first Wigner coefficient is nonzero by (2.18), and the second one is nonzero by (2.20) for $N=K$, $s=0$, $\mu=\mu_0$. So, the function $b_{K \mu_0}(\rho)$ enters (2.22) with nonzero coefficient, and hence

$$ \begin{equation} \int_0^{\infty} \rho^{K+2} b_{K\mu_0}(\rho)\, d\rho=0. \end{equation} \tag{2.24} $$

Now let $L=K-1$, $l=1$. Here, in addition to (2.23), we have

$$ \begin{equation*} K-2 \geqslant|\mu_0|. \end{equation*} \notag $$

This inequality secures (2.21). In this case, the interval of variation of $\lambda$ in (2.22) is $K-2\leqslant \lambda \leqslant K$, and the sum in (2.22) has the form

$$ \begin{equation*} \begin{aligned} \, &b_{K-2 \, \mu_0}(\rho)\begin{pmatrix} K-2 & K-1 & 1 \\ 0 & 0 & 0 \end{pmatrix}\begin{pmatrix} K-2 & K-1 & 1 \\ \mu_0 & -\mu_0 & 0 \end{pmatrix} \\ &\qquad+ b_{K-1 \, \mu_0}(\rho) \begin{pmatrix} K-1 & K-1 & 1 \\ 0 & 0 & 0 \end{pmatrix}\begin{pmatrix} K-1 & K-1 & 1 \\ \mu_0 & -\mu_0 & 0 \end{pmatrix} \\ &\qquad+b_{K\mu_0}(\rho) \begin{pmatrix} K & K-1 & 1 \\ 0 & 0 & 0 \end{pmatrix}\begin{pmatrix} K & K-1 & 1 \\ \mu_0 & -\mu_0 & 0 \end{pmatrix}. \end{aligned} \end{equation*} \notag $$

Here the coefficient multiplying $b_{K-1 \,\mu_0}(\rho)$ vanishes by (2.18), and the coefficient multiplying $b_{K-2 \, \mu_0}(\rho)$ is nonzero. This follows from (2.18) and (2.20), where in (2.20) one should put $N=K$, $s=1$, $\mu=\mu_0$. Using (2.22) and the already proved equality (2.24), we conclude that

$$ \begin{equation} \int_0^{\infty} \rho^{K+2} b_{K-2 \,\mu_0}(\rho)\, d\rho=0. \end{equation} \tag{2.25} $$

Similarly, for $L=K-2$, $l=2$, under the assumption

$$ \begin{equation*} K-4 \geqslant|\mu_0|, \end{equation*} \notag $$

we get in (2.22) the sum of functions $b_{K-4 \, \mu_0}(\rho)$, $b_{K-3 \, \mu_0}(\rho)$, $b_{K-2 \, \mu_0}(\rho)$, $b_{K-1\, \mu_0}(\rho)$, $b_{K\, \mu_0}(\rho)$, in which the coefficients

$$ \begin{equation*} \begin{gathered} \, \begin{pmatrix} K-3 & K-2 & 2 \\ 0 & 0 & 0 \end{pmatrix}\begin{pmatrix} K-3 & K-2 & 2 \\ \mu_0 & -\mu_0 & 0 \end{pmatrix}, \\ \begin{pmatrix} K-1 & K-2 & 2 \\ 0 & 0 & 0 \end{pmatrix}\begin{pmatrix} K-1 & K-2 & 2 \\ \mu_0 & -\mu_0 & 0 \end{pmatrix} \end{gathered} \end{equation*} \notag $$

multiplying $b_{K-3 \, \mu_0}(\rho)$, $b_{K-1 \, \mu_0}(\rho)$ are all zero, because the first terms in the products vanish in view of (2.18). On the other hand, the coefficient

$$ \begin{equation*} \begin{pmatrix} K-4 & K-2 & 2 \\ 0 & 0 & 0 \end{pmatrix}\begin{pmatrix} K-4 & K-2 & 2 \\ \mu_0 & -\mu_0 & 0 \end{pmatrix} \end{equation*} \notag $$

multiplying $b_{K-4 \, \mu_0}(\rho)$ is nonzero. This follows from (2.18) and (2.20). In view of (2.24) and (2.25), from (2.22) we have

$$ \begin{equation*} \int_0^{\infty} \rho^{K+2} b_{K-4 \, \mu_0}(\rho)\, d\rho=0. \end{equation*} \notag $$

Continuing in this way, we find that

$$ \begin{equation} \int_0^{\infty} \rho^{K+2} b_{K-2j \, \mu_0}(\rho)\, d\rho=0 \end{equation} \tag{2.26} $$

for all $j=0,1,\dots$ such that $K-2j\geqslant| \mu_0|$. In the proof of (2.26) we used the relation

$$ \begin{equation*} \begin{pmatrix} K-2j & K-j & j \\ 0 & 0 & 0 \end{pmatrix}\begin{pmatrix} K-2j & K-j & j \\ \mu_0 & -\mu_0 & 0 \end{pmatrix} \neq 0, \qquad j=0,1,\dots, \biggl[\frac{K-|\mu_0|}2\biggr], \end{equation*} \notag $$

which follows from (2.18) and (2.20). Here the square brackets denote the integral part of a number.

We now fix an integer $\lambda_0 \geqslant 0$ such that $\lambda_0 \geqslant|\mu_0|$. From (2.26) for $K=\lambda_0$, $j=0$ we have

$$ \begin{equation*} \int_0^{\infty} \rho^{\lambda_0+2} b_{\lambda_0\mu_0}(\rho)\, d\rho=0. \end{equation*} \notag $$

Letting $K=\lambda_0+2$ and $j=1$ in (2.26), and then putting $K=\lambda_0+4$ and $j=2$, we conclude that

$$ \begin{equation*} \int_0^{\infty} \rho^{\lambda_0+4} b_{\lambda_0\mu_0}(\rho)\, d\rho=0, \qquad \int_0^{\infty} \rho^{\lambda_0+6} b_{\lambda_0\mu_0}(\rho)\, d\rho=0. \end{equation*} \notag $$

Continuing similarly, we obtain

$$ \begin{equation} \int_0^{\infty} \rho^{\lambda_0+2j+2} b_{\lambda_0\mu_0}(\rho)\, d\rho=0, \qquad j=0,1,\dots\,. \end{equation} \tag{2.27} $$

The series with the terms $(\lambda_0+2j+2)^{-1}$, $j=0,1,\dots$, is divergent, and so, using (2.27) and Müntz’s theorem [13], § 27, we conclude that $b_{\lambda_0\mu_0}(\rho)=0$ for almost all $\rho \geqslant 0$. Hence

$$ \begin{equation*} a_{\lambda_0\mu_0}(\rho)=0. \end{equation*} \notag $$

Since $\lambda_0 \geqslant 0$ and $|\mu_0|\leqslant \lambda_0$ are arbitrary, from (2.12) we have $h(\rho\eta)=0$ almost everywhere on $S$ for almost all $\rho \geqslant 0$, that is, $h(x)=0$ almost everywhere in the domain $D$. This completes the proof of Theorem 1.

Let us now consider applications of this theorem to uniqueness of solutions to some coefficient inverse problems.

§ 3. The Lavrent’ev equation

Let us study the inverse wave sounding problem with bounded inhomogeneity and a set of point sources not contained in this inhomogeneity [14]. The wave field $u(x,t)=u(x,t;y)$ excited by a source at a point $y$ is determined by the solution of the Cauchy problem

For a broad class of partial differential equations, Lavrent’ev [15], [16] proposed an approach to solution of nonlinear coefficient inverse problems capable of reducing such problems to linear integral equations. Let us illustrate his approach for the inverse problem $\{ Y, Z \}$. For an integrable function $f = f(t)$, $t \geqslant 0$, we recall that the Laplace transform is defined by $\widetilde{f}(p) = \int_0^{\infty} e^{-pt} f(t)\, dt$. We will assume that all the functions $u(z,t;y)$, $y\in Y$, and their derivatives with respect to $t$ up to the second order inclusively decay sufficiently rapidly as $t\to +\infty$ for $z\in Z$, so that their Laplace transforms exist. In addition, we will assume that $u(x,t;y)\to 0$ as $|x|\to \infty$ uniformly in $y\in Y$ and $t > 0$. The conditions on $c(x)$ to satisfy these requirements were discussed in [17], [18]. Setting

The magnitude of the wave field $u(z,t;y)$ averaged in time with weight $t^2$ for $(y,z)\in Y\times Z$ serves as input data for evaluation of $\xi$.

The inverse sounding problem of an inhomogeneity by point time-harmonic sources with frequencies $\omega\in(0,\omega_0]$ can also be reduced to equation (3.10); for more details, see [19], Ch. VII, § 1, [20], [21]. In the case of injective operator of equation (3.10), the solution of the original nonlinear inverse problem is unique. This property is satisfied, for example, if $Y$, $Z$ are domains on a plane disjoint from $D$ or bounded surfaces surrounding $D$. It is worth pointing out that, in all above cases, the dimension of the spatial data support $Y\times Z$ in (3.10) is 4, whereas the sought-for function $\xi$ depends only on 3 variables. Thus, the inverse problems in these papers are overdetermined. The above construction also includes the case $y=z$, which corresponds to the problem of reconstruction of an inhomogeneity from inverse scattering data. In this case, uniqueness of the solution can be verified in the non-overdetermined case, where $Y=Z$ is a domain in $\mathbb{R}^3$ disjoint from $\overline{D}$ [21].

Let us clarify how the injectivity of the integral operator (3.10) is related to the approximation problems considered in § § 1, 2. One has to verify that the homogeneous equation

We say that $\mathcal{M} \subset \mathbb{R}^3 \setminus \overline{D}$ is a uniqueness set (for harmonic functions in $\mathbb{R}^3 \setminus \overline{D}$) if, for any regular harmonic (in $\mathbb{R}^3 \setminus \overline{D}$) function $u$ such that

The next proposition justifies the application of Theorem 1 for a proof of unique solvability of the homogeneous equation (3.13).

From Proposition 5 it follows that if $Z$ is a uniqueness set in the above sense, then equality (3.13) implies

If as $Z$ we take a two-dimensional uniqueness set (for example, a domain on the plane, or a closed surface disjoint from $\overline{D}$), then, under the conditions of Theorem 2, the dimension of the spatial data support $Y\times Z$ in problems $\{ Y, Z \}$ and $\{ Z, Y \}$ is 3, which equals the dimension of support $D$ of the sought-for function. In this sense, the inverse problems under consideration turn out to be spatially non-overdetermined. Note that here a complete data set $\{ u(z,t;y)\colon y\in Y$, $z\in Z$, $t>0 \}$ required for evaluation of the function $f_0$ in (3.10) involves the variable $t$, and hence its dimension is one plus $\operatorname{dim} D$. In [23], a related result was established on uniqueness for an equation obtained by transformation of the Lavrent’ev equation to a differential form with subsequent approximation via the Galerkin method.

§ 4. The inverse sounding problem in the phaseless setting

In this section, we will consider the inverse problem for the Helmholtz equation

The classical statement of the inverse scattering problem for (4.1), (4.2) calls for recovery of the function $n(x)$, $x\in D$, from known values of the solution $u=u(x;y,k)$ of problem (4.1), (4.2) for $x=z\in Z$, $y \in Y$, $k\in [k_1,k_2]$, with appropriate sets $Y$, $Z$ and for chosen values $0<k_1\leqslant k_2$. As above, $Z$ is the set of detectors of the scattered wave field generated by a source at $y\in Y$, $Y$ is the set of point sounding sources, where we assume that $(Y \cup Z) \cap \overline{D}=\varnothing$. Inverse scattering problems in this and related settings with various choices of sets of sources and detectors have been extensively studied (see, for example, the books [2], [20], [24] and the references given there). The scattering amplitude in the far-zone, that is, the principal term in the asymptotics of $u(x;y,k)$ as $|x|\to \infty$, frequently plays the role of input data in inverse sounding problems. Here, an incoming plane wave may also serve as a wave field source. At the same time, experimental measurement of a complex function $u(x;y,k)$, for which both modulus (amplitude) $|u(x;y,k)|$ and the phase $\operatorname{arg} u(x;y,k)$ of the wave field should be obtained, is a very restrictive requirement in many applied problems. In many cases, experimental observations provide only the amplitude of a signal scattered by the inhomogeneity, that is, $|u(x;y,k)|$. In this regard, following [25]–[29], we will consider the problem of recovery of the function $n(x)$, $x\in D$, from a known amplitude $|u(z;y,k)|$ of the wave field, where $z\in Z$, $y \in Y$, $k\in [k_1,k_2]$, $0<k_1<k_2$. We will assume that $Y \cap Z=\varnothing$. We are interested in unique recovery of this function from the input data. For brevity, this inverse problem will be denoted by $[Y,Z]$.

We will next show that analysis of unique solvability of $[Y,Z]$ can also be reduced to the study of injectivity of the integral operator from the Lavrent’ev equation (3.10).

According to [24], § 8.2, problem (4.1), (4.2) is equivalent to the integral equation

Under the condition $n\in L_{\infty}(D)$, the integral operator on the right of (4.3) is completely continuous in $L_2(D)$ for any $k \in \mathbb{C}$ (see [30], § 8.5); moreover, the right-hand side of (4.3) depends analytically on the parameter $k$. According to Theorems 8.4, 8.7 in [24], problems (4.3) and (4.1), (4.2) are uniquely solvable for all real $k\geqslant 0$ and for complex $|k|\leqslant \delta$ with sufficiently small $\delta>0$. By theorem on holomorphic families of Fredholm operators (see [24], § 8.5, [17], Ch. VI, § 4), for fixed $x\neq y$, the function $u(x;y,k)$ extends holomorphically with respect to $k$ to a neighbourhood of the positive real axis. Retaining the same notation for the extended function of $k=\kappa_1+i \kappa_2$, we note that the real and imaginary parts ($\operatorname{Re} u(x;y,k)$, $\operatorname{Im} u(x;y,k)$) are regular harmonic functions of the variables $(\kappa_1,\kappa_2)$. Hence, for $x\neq y$, the function

Note that the function $\Phi(z,y;0)=(4\pi|z-y|)^{-1}$ is real-valued. Hence from (4.4), (4.8) it follows that the real-valued function $\xi$ satisfies the Lavrent’ev equation

The function $|u(z;y,k)|^2$ is real analytic in $k>-\delta$, and hence $|u(z;y,k)|^2$ for all $k>-\delta$, $(y,z)\in Y \times Z$ is uniquely determined from the values of $|u(z;y,k)|^2$ for $k \in [k_1,k_2]$ and $(y,z)\in Y \times Z$. So, the function in the right-hand side of equation (4.10) is uniquely defined by the input data of the inverse problem $[Y,Z]$. Hence the solution $[Y,Z]$ is unique if and only if the corresponding homogeneous equation (3.13) has only the trivial solution. The next result now follows from Theorem 1.

Choosing as $Z$ a two-dimensional uniqueness set and employing Theorem 3, we obtain that the nonlinear coefficient inverse problems $[Y, Z]$ and $[Z, Y]$ are uniquely solvable in the spatially non-overdetermined setting, where the dimension of the spatial data support $Y\times Z$ coincides with that of the support $D$ of the original function. It is worth pointing out that in the preceding papers [26]–[29] uniqueness theorems were put forward for problems with spatial data supports of the form $\{ (y,z)\colon y\in \Sigma,\, z\in O_{\varepsilon}(y) \}$, $\{ (y,z)\colon y,\, z\in \Sigma \}$ (of dimension at least 4). Here $\Sigma$ is a closed surface surrounding $\overline{D}$. In [26], a uniqueness theorem for non-overdetermined phaseless inverse problem was established if (4.1) involves a unique spatially distributed sounding source. This result was obtained under quite stringent assumptions about the form and smoothness of the sought-for function $n(x)$.

§ 5. Axial symmetry of inhomogeneities

Note that the expressions in the denominator of the integrand in equations (3.10), (4.10) are invariant under orthogonal transformations of the variables. With a suitable arrangement of sets of sources and detectors, this property in combination with Theorems 2, 3 can be used for establishing symmetry properties of the inhomogeneity given analogous symmetry in the input data of the inverse problems $\{ Y, Z\}$, $[Y, Z]$. Let us illustrate this for the axial symmetry property.

We first consider the problem $\{ Y, Z\}$ equivalent to the integral equation (3.10). For definiteness, we will assume that

Let $R_{\varphi}$ be the matrix of rotation of $\mathbb{R}^3$ by an angle $\varphi\in [-\pi,\pi)$ about the $x_3$-axis. Suppose that

Since, for a fixed $\varphi$, the range of all points $\widetilde{z}=R_{-\varphi}z$ is the whole set $Z$ as $z$ varies over $Z$, from (5.4) we get

As a well-know analogue of Theorem 4, we mention S. N. Karp’s theorem [24], § 5.1, § 7.1, and its generalizations [31]. However, these results are concerned with spherical symmetry of the scattering amplitude $A(\eta,\eta')$ qua a function of two vectors $\eta$, $\eta'$ from the unit sphere $S$. Hence, testing the corresponding symmetry condition $A(U\eta, U\eta')=A(\eta,\eta')$ for all $\eta,\eta' \in S$ and all orthogonal $(3\times 3)$-matrices $U$ with $\operatorname{det}U=1$ calls for a solution of the overdetermined scattering problem with data in the four-dimensional manifold $S \times S$. The hypotheses of Theorem 4 can be tested in solving the inverse problems $\{ Y,Z \}$, $\{ Z, Y \}$ with three-dimensional data support $Y \times Z$. At the same time, in Theorem 4 and Remark 1, one deals with narrower transformation groups.

Let us now consider the inverse problem $[Y,Z]$ in which $Y$, $Z$, $D$ are chosen according to (5.1). It is easily checked that, under the condition

The author is grateful to the referee for calling his attention to the last remark.

§ 1. Statement of the problem. The main theorem

§ 2. Proof of Theorem 1

§ 3. The Lavrent’ev equation

§ 4. The inverse sounding problem in the phaseless setting

§ 5. Axial symmetry of inhomogeneities

Bibliography