Estimates of complex eigenvalues and an inverse spectral problem for the transmission eigenvalue problem

This work deals with the interior transmission eigenvalue problem: $y'' + {k^2}\eta \left( r \right)y = 0$ with boundary conditions ${y\left( 0 \right) = 0 = y'\left( 1 \right)\frac{{\sin k}}{k} - y\left( 1 \right)\cos k},$ where the function $\eta(r)$ is positive. We obtain the asymptotic distribution of non-real transmission eigenvalues under the suitable assumption for the square of the index of refraction $\eta(r)$. Moreover, we provide a uniqueness theorem for the case $\int_0^1\sqrt{\eta(r)}dr>1$, by using all transmission eigenvalues (including their multiplicities) along with a partial information of $\eta(r)$ on the subinterval.


Introduction and main results
Consider the interior transmission problem y + k 2 η (r) y = 0, 0 < r < 1, y (0) = 0 = y (1) sin k k − y (1) cos k, (1) We introduce two key quantities. Denote which is explained physically as the time needed for the wave to travel from r = 0 to r = 1. Introduce the characteristic function where y (r, k) is the solution of y + k 2 η (r) y = 0 with the initial conditions y (0, k) = 0 and y (0, k) = 1. Obviously, the transmission eigenvalues coincide with the squares of zeros of d (k). The problem (1) was first studied by McLaughlin and Polyakov [13], they showed that if a = 1 then there are infinitely many real eigenvalues {(k n ) 2 } n≥n 0 , which have the asymptotics where q(x) is defined in (11). Some aspects of the asymptotics of large (real and non-real) transmission eigenvalues for the case a = 1 were discussed in [19]. In 2015, Colton and co-authors [6] studied the existence and distribution of the non-real transmission eigenvalues. They showed that if a = 1 and η (1) = 0 (this assumption can be weakened [7]), then there exists infinitely many real and non-real transmission eigenvalues, moreover, the imaginary parts of the non-real eigenvalues go to infinite. In particular, they give an example to show the distribution of the transmission eigenvalues, which is η(r) = 16 (r + 1) 2 (r − 3) 2 . It is easy to calculate η(1) = 1, η (1) = 0 and η (1) = 1 = 0. For this η(r), the distribution of the eigenvalues is shown numerically in the Figure 1 (see [6]).
From Figure 1, we see that the locations of the non-real zeros {x n + iy n } of d(k) in the right half-plane seem to satisfy asymptotically a logarithmic curve y n = log(cx n ), where c may be some complex number. We will prove in theory that this is indeed true in the more general case (see Theorem 1.1).
For the inverse spectral problem, many scholars contribute a lot of works (see [1-5, 13, 18, 21] and the references therein). However, for the case a > 1 there are only a few results. It is known [5,13] that the determination of η(r) on [0, 1] with η(1) = 1 and η (1) = 0 is equivalent to the determination of q(x) on [0, a] defined in (11). McLaughlin and Polyakov [13] first showed that if a > 1 and η(r) is known a priori on a subinterval [ε 1 , 1] with ε 1 satisfying In 2013, Wei and Xu [18] suggested to specify all transmission eigenvalues (including their multiplicities) and the norming constants, corresponding to the real eigenvalues, to obtain the unique determination of η(r) on [0, 1].
In this paper, we will prove a new uniqueness theorem for the inverse spectral problem in the case a > 1 (see Theorem 1.2), by using the less known information on η(r) and all eigenvalues (including real and non-real). Moreover, with the help of some ideas in [8,10,16], we give a relationship between the proportion of the needed eigenvalues and the length of the subinterval on the given η(r) (see Theorem 1.3).
The main results in this article are as follows.

Preliminaries
In this section, we provide some known auxiliary results. Using the Liouville transformation, we can write the equation y + k 2 η (r) y = 0 with y (0, k) = 0 and y (0, k) = 1 as where Using the transformation operator theory (see, e.g. [14]), we have where K(x, t) satisfies the following integral equation (see, e.g. [2]) where 0 ≤ t ≤ x ≤ a. In particular, 2K(x, x) = x 0 q(s)ds and K(x, 0) = 0. On the other hand, from Eq.(1.2.9) in [14], we know that where By virtue of (9) and η(1) = 1 and η (1) = 0, we have ϕ(a, k) = y(1, k) and ϕ (a, k) = y (1, k). Thus, and Using Eq.(13), by tedious calculation, we have and To get Theorem 1.1, we introduce the following transcendental equation where λ is a constant. It is known (see, e.g. [9]) that Eq. (20) has a unique solution for sufficiently large w. We will transform the equation d(k) = 0 to the equation with the form of (20), and then use (21) to obtain the asymptotics of non-real transmission eigenvalues. We also mention that this method, which can be used to obtain the asymptotics of non-real eigenvalues, was applied by some authors [17,20]. For the inverse spectral problem, we shall use the following three lemmas.
Let G(k) be analytic in C + and continuous in Then, for k ∈ C + , there holds where ϕ(x, k) andφ(x, k) are defined by ( 10) corresponding to q andq, respectively. Lemma 2.3 (See Chapter IV of [12]). For any entire function g(k) ≡ 0 of exponential type, the following inequality holds, where N g (r) is the number of zeros of g(k) in the disk |k| ≤ r (r > 0) and h g (θ) := lim r→∞ log |g(re iθ )| r with k = re iθ .
Case (i): by virtue of (33a), we know that d(k) = 0 for |k| → ∞ in C ± is equivalent to that Taking logarithm on both sides of the above equation, we get that for sufficiently large n ∈ Z, It follows from (20) and (21) and z = ik that Clearly, the above sequences belong to the domain C ± for all large n. Substituting (38) into (26) and (27), we get that ε j (k ± n )e −a|Imk ± n | ∈ l 2 for j = 1, 4, which implies ε 5 (k ± n ) ∈ l 2 . It follows from (35) and (36) that ε 8 (k ± n ) ∈ l 2 . Taking (20) and (21) into account, we can obtain α ± n ∈ l 2 . Case (v): by virtue of (34a), we know that d(k) = 0 for |k| → ∞ in C ± is equivalent to that which implies that for sufficiently large n ∈ Z It follows from (20) and (21) and z = ik that for n → ∞, Using a similar argument, one gets γ ± n ∈ l 2 . Through similar arguments, one obtains asymptotics of other cases. The proof is finished.
Proof of Theorem 1.2. Since the function d(k) is an entire function of k of order 1 and even with respect to k, by Hadamard's factorization theorem, where s is the multiplicity of the zero eigenvalue. Using (2), (9) and (11), it can be verified that η(r) is known a priori on [ε, 1] with ε satisfying (6) is equivalent to that q(x) is known a priori for x ∈ [ a+1 2 , a]. Let us prove that q(x) on [0, a] is uniquely determined by E(k) and the known q(x) on [ a+1 2 , a]. If it is true, then η(r) on [0, 1] with η(1) = 1 and η (1) = 0 is uniquely determined by E(k) and the known η(r) on [ε, 1]. (See [13]).
If we can prove |G(k)| ≤ C for k ∈ R (see ( * ) below), then it follows from Lemma 2.1 that for all k ∈ C + |G(k)| ≤ C.
Note that G(k) is an entire function of k from the above argument, thus, zeros of sin k can not be poles of G(k). Thus, it follows from (48) that a 0 K(a, t) −K(a, t) sin(nπt)dt = 0, n = 0, ±1, ±2 · · · .
Proof of Theorem 1.3. By a similar argument to the proof of Theorem 1.2, we know that it is enough to show the function g(k) ≡ 0, where g(k) is defined in (40) with (a + 1)/2 replacing by a − b (because now q(x) =q(x) on [a − b, a] from (8)). From (42) and (43), together with the boundary condition in (1), we get g(k) = 0 for k ∈ D ∪ {k n } n≥n 0 .