Communicated by Notices Associate Editor Reza Malek-Madani
Introduction
The study of eigenvalue problems for partial differential equations has a long history during which a variety of themes has emerged. Although historically such efforts have focused on eigenvalue problems defined on bounded domains, the importance of scattering theory in modern mathematical physics has led to an intensive study of eigenvalue problems in unbounded domains connected with the Schrödinger equation and the wave equation for propagation in an inhomogeneous medium. A particularly noteworthy development in this latter direction has been the theory of scattering resonances which now play a central role in mathematical scattering theory. For a magisterial presentation of this theory we refer the reader to the monograph by Dyatlov and Zworski DZ19. More recently, a new eigenvalue problem in scattering theory has attracted increased attention both inside and outside the scattering community. This new problem is called the transmission eigenvalue problem and in a certain sense exhibits a duality relation to the theory of scattering resonances. The purpose of this short survey is to introduce this class of nonselfadjoint eigenvalue problems to the wider mathematical community. For further information on transmission eigenvalues, including applications to inverse scattering theory, we refer the reader to the monograph CCH16 and Chapter 10 of CK19.
The transmission eigenvalue problem arises in the study of wave propagation in an inhomogeneous medium. Hence, for the benefit of the reader who is not an expert in scattering theory, we begin by describing the basic elements of acoustic scattering theory (see CK19). Broadly speaking, acoustic scattering theory is concerned with the effect an inhomogeneous medium has on an incident wave. In particular, if the total field $u$ is viewed as the sum of an incident field $u^i$ and a scattered field $u^s$, then the scattering problem is to determine $u^s$ from a knowledge of $u^i$ and the differential equation governing the wave motion. More specifically, assume that the incident field is given by the time-harmonic acoustic plane wave
where $x\in {\mathbb{R}}^3$,$t$ denotes time, $k=\omega /c_0$ is the wave number, $\omega$ is the frequency, $c_0$ is the speed of sound, and the unit vector $\hat{y}$ is the direction of propagation. Then factoring out the term $e^{-i\omega t}$, the simplest acoustic scattering problem for the case of an inhomogeneous medium is to find the total field $u$ such that
where $u^i(x)=e^{ikx\cdot \hat{y}}$,$r=|x|$,$n=c_0^2/c^2$ is the refractive index where $c=c(x)$ is the speed of sound in the inhomogeneous medium, and 3 is the Sommerfeld radiation condition which holds uniformly with respect to $\hat{x}\coloneq x/|x|$ and guarantees that the scattered field is outgoing. It is assumed that $1-n$ has compact support in a bounded region $D$ having piecewise smooth boundary $\partial D$ such that ${\mathbb{R}}^3\setminus D$ is connected and that $n\in L^\infty (D)$ is such that $n(x)>0$ for $x\in D$. It can be shown that there is a unique solution $u^s\in H^2_{loc}({\mathbb{R}}^3)$ to the scattering problem 1–3 and that $u^s(x)\coloneq u^s(x,\hat{y}, k)$ (for fixed $\hat{y}$ and $k$) has the asymptotic behavior
where the function $u^\infty (\hat{x})\coloneq u^\infty (\hat{x},\hat{y},k)$ is called the far field pattern and is an infinitely differentiable function of $\hat{x}$ on the unit sphere $S^2$.Rellich’s lemma says that $u^\infty ( \hat{x})$ for $\hat{x} \in S^2$ determines $u^s(x)$ for $x\in {\mathbb{R}}^3\setminus \overline{D}$. Note that if we vary the incident direction $\hat{y}\in S^2$,$u^\infty (\hat{x},\hat{y},k)$ is also infinitely differentiable with respect to $\hat{y}$, and in fact it can be shown that $u^\infty (\hat{x},\hat{y},k)=u^\infty (-\hat{y}, -\hat{x} ,k)$.
Figure 1.
Sketch of the scattering problem.
Associated with the scattering problem 1–3 are two basic nonselfadjoint eigenvalue problems. The first of these, and by far the most studied, is the theory of scattering resonances which seeks values of the wave number $k$ such that for $u^i=0$ there exists a nontrivial solution $u$ of 1–2 with an appropriate modification of the radiation condition 3 accounting for $\Im (k) <0$. Such values of $k$ are called scattering resonances and can be shown to form a discrete set lying in the lower half of the complex $k$-plane. The second class of eigenvalue problems associated with the scattering problem 1–3, and one of more recent origin, is the theory of transmission eigenvalues and this is the topic of our survey article. Now, instead of asking for a nontrivial solution $u$ of 1–3 for which $u^i=0$, we ask if there is a nontrivial solution $u$ of 1–3 for which $u^s=0$. In other words, we ask the question of whether we are able to construct an incident field which does not scatter. Values of $k$ for which this is possible will lead to the theory of transmission eigenvalues. Of particular interest to us in the sequel will be incident fields $u^i=v_g$, where $v_g$ is defined by
and $g\in L^2(S^2)$ is referred to as the kernel of $v_g$. Solutions of the Helmholtz equation of the form 5 are called Herglotz wave functions and are extensively discussed in CK19.
We now proceed to outline the basic theory of transmission eigenvalues and in particular their dual relationship to scattering resonances. We begin by noting that a solution $v$ of the Helmholtz equation
where $B_R$ is the ball of radius $R$ centered at the origin. Every $v_g$ in the space of Herglotz wave functions can be uniquely decomposed as $v_g\coloneq u_g-u_g^s$, where the total field $u_g$ and the scattered field $u^s_g$ satisfy 1–3 with $u^i\coloneq v_g$. The scattering operator (matrix) as defined by Lax and Phillips in LP89 maps $v_g\mapsto u_g$ and for $k$ such that $\Im (k)\geq 0$ is an isomorphism in appropriate Banach spaces. A heuristic argument for the latter can be given using the Lipmann-Schwinger equation for the solution of 1–3 with $u^i\coloneq v_g$ in terms of the compact $k$-analytic integral operator $T(k):L^2(B_R)\to L^2(B_R)$,
A fixed point argument implies that for $|k|$ small enough $I-T(k)$ is invertible, and hence by the Analytic Fredholm Theorem we have that $u_g\coloneq (I-T(k))^{-1}v_g$ is meromorphic for $k\in {\mathbb{C}}$. Furthermore, for $k$ such that $\Im (k)\geq 0$, uniqueness of the scattering problem implies that $u_g$ is analytic and thus its poles are in the lower-half complex plane.
Of particular interest to us in the sequel will be the “incoming-to-outgoing” mapping $v_g \mapsto u^s_g\coloneq u_g-v_g$. We shall characterize this in terms of the far field pattern defined in 4. To this end let $u_g^\infty$ denote the far field pattern of the scattered field $u^s_g$ corresponding to the incident field $v_g$. The compact linear operator ${F}(k): L^2(S^2)\to L^2(S^2)$ defined by
where $u^\infty (\hat{x}, \hat{y},k)$ is the far field pattern of the scattered field due to an incident plane wave $e^{ikx\cdot \hat{y}}$4. The scattering operator ${\mathcal{S}}(k): L^2(S^2)\to L^2(S^2)$ can then be expressed as LP89
If $\Im (n)=0$, then ${F}(k)$ is normal and ${\mathcal{S}}(k)$ is unitary for real $k>0$, which is not the case if $\Im (n)>0$ on a subset of $D$ of nonzero measure. Both are analytic operator-valued functions of $k$ in the upper-half complex plane. The scattering poles are the poles of the meromorphic extension of ${\mathcal{S}}(k)$ in the lower-half complex plane.
Now we are ready to formally introduce the transmission eigenvalue problem. An application of Rellich’s lemma implies that the incident field $v_g$ with $g\in \mathrm{Kern}\, {F}(k)$ does not scatter. Straightforward calculation reveals that the kernel of ${F}(k)$ consists of all $g\in L^2(S^2)$ such that, if $v_g$ is the corresponding Herglotz wave function, $v\coloneq v_g|_{D}$ and $u$ satisfy the transmission eigenvalue problem
A value of $k\in \mathbb{C}$ is said to be a transmission eigenvalue if 9 has nontrivial solutions $u\in L^2(D)$,$v\in L^2(D)$, such that $u-v\in H^2_0(D)$. We call the pair $(u,v)$ the corresponding eigenfunction. In general, at a transmission eigenvalue the part $v$ of the corresponding eigenfunction does not take the form of a Herglotz wave function and hence the kernel of the far field operator is in general empty. Thus the set of nonscattering wave numbers$k$ for which $\mathrm{Kern}\ {F}(k)\neq \left\{0\right\}$ is a subset (possibly empty) of the transmission eigenvalues. There is a special configuration for which transmission eigenvalues and nonscattering frequencies coincide, namely the case of spherically stratified media which is discussed in the next section.
Figure 2.
Illustration of a transmission eigenvalue for a dielectric medium formed by two concentric circles with outer radius = 1 and inner radius = 0.3. The refractive index $n=2$ in the annulus and $n=3$ in the inner circle. The wave number is $k=7.223$. Left: Incident field; a Herglotz wave function with kernel $g(\theta ) = exp(-2i\theta )$. Middle: Total field. Right: Scattered field (=$0$ outside the circle of radius 1).
Allowing $u^i$ to be any function $v\in {H}_{inc}(D)$ with
together with the Sommerfeld radiation condition 3. Note that $H_{inc}(D)$ is a Hilbert space which densely contains the Herglotz wave functions $v_g$. Defining the operator ${\mathcal{G}}(k): {H}_{inc}(D)\to L^2(S^2)$ mapping $v\mapsto u^\infty$, where $u^\infty$ is the far field associated with the solution of 10, which is a compact linear operator, then we can equivalently define transmission eigenvalues as the values of $k$ for which $\mathrm{Kern}\ {\mathcal{G}(k)}$ is nontrivial (in fact the part $v$ of the corresponding eigenfunction belongs to $\mathrm{Kern}\ {\mathcal{G}}$). In addition, the relation
and we already observed that $\overline{{\mathcal{H}}(L^2(S^2))}=H_{inc}(D)$. Hence, in general, at a transmission eigenvalue one can construct a Herglotz wave function $v_g$ of unit $L^2(D)$-norm, that produces an arbitrarily small scattered field $u^s_g$ (see Figures 2 and 3).
Figure 3.
We represent the scattered fields for the same scattering experiment as in Figure 2 with an incident field being a Herglotz wave function with kernel $g(\theta ) = exp(-2i\theta )$ but for different wave numbers. From top to bottom and left to right $k=6$,$k=6.5$,$k=7$,$k=7.223$ (a transmission eigenvalue).
Spherically Stratified Media
As mentioned above, the scattering problem for spherically symmetric media is of great importance since it provides an example where the set of nonscattering frequencies and transmission eigenvalues are the same. More precisely, when $D$ is a ball of radius $a$ centered at the origin and $n\coloneq n(r)$,$r=|x|$, is a radial real-valued function, the part $v$ of a transmission eigenfunction is indeed a Herglotz wave function and hence transmission eigenvalues coincide with the values of $k\in \mathbb{C}$ for which $\mathrm{Kern}\ {F}(k)\neq \left\{0\right\}$. Figure 2 gives an example in two dimensions. To see explicitly what the transmission eigenvalues are in this case, we use as incident field the Herglotz wave function $v=j_\ell (k|x|)Y^m_\ell (\hat{x})$, where $j_\ell$ is a spherical Bessel function and $Y^m_\ell$ is a spherical harmonic of order $\ell \in {\mathbb{N}}_0$,$m=-\ell \cdots \ell$. Straightforward calculations by separation of variables lead to the following expression for the scattered field for $r>a$ and the corresponding far field, respectively,
which as $r\to 0$ behaves like $j_\ell (kr)$, i.e., $\lim _{r\to 0}r^{-\ell }y_\ell (r)=\frac{\sqrt {\pi }k^\ell }{2^{\ell +1}\Gamma (\ell +3/2)}.$ Thus transmission eigenvalues are those values of $k\in {\mathbb{C}}$ such that $C_\ell (k; n)=0$, whereas the scattering poles are $k\in {\mathbb{C}}$ for which $W_\ell (k; n)=0$. The latter set lies in the lower half of the complex plane. If $k$ is a zero of $C_\ell (k; n)$ (i.e., a transmission eigenvalue), then the part $v$ of the corresponding eigenfunction is $v=j_\ell (k|x|)Y^m_\ell (\hat{x})$ which is an entire solution of the Helmholtz equation in ${\mathbb{R}}^3$. All transmission eigenvalues for a spherically stratified medium are obtained in this way by choosing $\ell \in {\mathbb{N}}$. Hence in this case, at any real transmission eigenvalue, there is at least one incident wave $v=j_\ell (k|x|)Y_\ell (\hat{x})$ for some $\ell \in {\mathbb{N}}$ that does not scatter (in fact there are at least $2\ell +1$ linearly independent nonscattering incident waves).
It is possible to give more details on the structure of transmission eigenvalues for spherically stratified media if we focus on the case of spherically symmetric incident fields ($\ell = 0$ in the above discussion), in other words when the transmission eigenfunction is radially symmetric. Hence in the rest of this section we consider only transmission eigenvalues with spherically symmetric eigenfunctions. These are the zeros of $C_0(k; n)$, and at such a zero $v\coloneq j_0(kr)$ and $u\coloneq y_0(r)$ satisfy the transmission eigenvalue problem:
Letting $y_0(r)\coloneq y(r)/r$, where now $y(r)$ satisfies $y^{\prime \prime }+k^2n(r)y=0$, and noting that $j_0(kr)=\sin (kr)/kr$, we obtain that $k$ is a transmission eigenvalue, i.e., $C_0(k;n)=0$, if and only if
We now note that $d(k)$ is an entire function of $k$ that is real for real $k$ and is bounded on the real axis. Hence, by Hadamard’s factorization theorem, if $d(k)$ is not identically zero, then there exists a countable set of transmission eigenvalues (cf. CCH20 for this conclusion in a similar case). It can be shown that if $d(k)$ is identically equal to zero, then $n(r)$ is identically equal to one CCH16, Section 5.1. From now on we assume that
and hence $d(k)$ has an infinite set of real and complex zeros.
This example demonstrates the perplexing structure of transmission eigenvalues and in particular their dependence on the properties of the contrast of the medium. Now let $n\in C^2[0,\,a]$ and assume 12 holds. Then one can prove the following results CK19.
Theorem 1.
Assume that either $1<\sqrt {n(r)}<\delta /a$ or $\delta /a<\sqrt {n(r)}<1$ for $0\leq r\leq a$. Then there exist infinitely many real and infinitely many complex transmission eigenvalues.
Theorem 2.
Assume that $n(a)\neq 1$. Then, if complex eigenvalues exist, they all lie in a strip parallel to the real axis.
The above theorems are quite different if we relax the assumption that $n(a)\neq 1$. In particular if