Spectral invariants of Dirichlet-to-Neumann operators on surfaces

We obtain a complete asymptotic expansion for the eigenvalues of the Dirichlet-to-Neumann maps associated with Schr\"odinger operators on Riemannian surfaces with boundary. For the zero potential, we recover the well-known spectral asymptotics for the Steklov problem. For nonzero porentials, we obtain new geometric invariants determined by the spectrum. In particular, for constant potentials, which give rise to the parameter-dependent Steklov problem, the total geodesic curvature on each connected component of the boundary is a spectral invariant. Under the constant curvature assumption, this allows us to obtain some interior information from the spectrum of these boundary operators.

Given Ω and τ , the map DN λ is well-defined for all λ ∈ V ⊂ C, where C \ V is a discrete set consisting in the Dirichlet eigenvalues of the linear operator pencil ∆ g + λτ . For fixed λ ∈ V, the Dirichlet-to-Neumann map is an elliptic pseudodifferential operator of order one. When λ is real, the Dirichlet-to-Neumann map is self-adjoint. Its spectrum is discrete and accumulating only at infinity, i.e. σ 1 (Ω; τ ; λ) ≤ σ 2 (Ω; τ ; λ) ≤ · · · ∞.
A survey of the general properties of the Steklov problem, i.e. the problem for λ = 0, is found in [9].
1.2. Spectral asymptotics. Since DN λ is an elliptic, self-adjoint pseudodifferential operator of order one, it follows from Weyl's law with sharp remainder (see [11]) that for any λ, τ the eigenvalues satisfy σ j (Ω; τ ; λ) = πj per(Σ) where per(Σ) denotes the length of Σ. When λ = 0 and Ω is simply connected Rozenblum [19] and Guillemin-Melrose (see [5]) obtained independently the precise asymptotics where for any sequence the notation a j = O (j −∞ ) means that a j ≤ C N j −N for any N ∈ N. Our first result is an extension of this result to the Dirichletto-Neumann operators associated to Schrödinger operators. Given K ∈ Z, we say that a sequence a j has complete asymptotic expansion Theorem 1.1. Let (Ω, g) be a simply connected Riemannian surface with smooth boundary Σ. For λ ∈ R ∩ V, the eigenvalues of DN λ (Ω; τ ) are asymptotically double and admit a complete asymptotic expansion given by where L = per(Σ) 2π . The coefficients s n are polynomials in λ of degree at most n with vanishing constant coefficients. They depend on both τ and the metric in a neighborhood of Σ. If τ ≡ 1, the first two terms are given by where k g is the geodesic curvature on Σ.
When Ω is an arbitrary surface, we obtain a generalisation of Theorem 1.1. Note that Theorem 1.2 obviously implies Theorem 1.1. However, the statement for simply connected surfaces is cleaner and obtained as an intermediate step in proving Theorem 1.2. Hence we state them separately. When λ = 0, Girouard, Parnovski, Polterovich and Sher proved this result in [8], whereas Arias-Marco, Dryden, Gordon, Hassannezhad, Ray and Stanhope proved in [2] the equivalent statement for the eigenvalues of DN 0 on orbisurfaces.  depend only on λ, τ and the metric in a neighborhood of Σ m in the same way as in (2) (including the case when τ ≡ 1). Let Ξ = ξ (1) , . . . , ξ ( ) . For λ ∈ R ∩ V, the eigenvalues of DN λ (Ω; τ ) are given by (5) σ j = S(Ξ) j + O j −∞ .
1.3. Inverse spectral geometry. Inverse problems consist in recovering data of some PDE -the domain of definition Ω, the metric, the potential, etc. -from properties of the operator alone, and inverse spectral geometry consists in recovering that data from the spectrum only. One of the seminal questions in that field was asked for the Dirichlet Laplacian by Mark Kac in [14] and answered negatively by Gordon, Webb and Wolpert in [10]: "Can one hear the shape of a drum?" For this reason, we often say that any geometric data that one can recover from the spectrum of an operator can be "heard". It is long known and follows from Weyl's law that the total boundary length can be heard from DN λ . It also follows from the standard theory of the wave trace asymptotics as developped by Duistermaat and Guillemin [4] that the length spectrum -that is the length of the closed geodesics -of the boundary Σ can be heard as well. For DN 0 , it is shown in [8] that we can recover the number of connected components, as well as their lengths. It is also shown that from polynomial eigenvalue asymptotics alone in dimension two nothing more can be recovered. This can be seen as a consequence of Theorems 1.1 and 1.2 since the coefficients s n and s (m) n are all polynomials in λ that vanish when λ = 0.
For DN 0 , to extract more information different authors have turned to spectral quantities that have a more global nature. In [17], Polterovich and Sher obtain an asymptotic expansion as t → 0 for the heat trace of DN 0 . From the coefficients, they obtain that the total mean curvature is a spectral invariant for d ≥ 3. See also [16] for further works. In the case of DN λ (Ω; τ ), heat trace asymptotics as well as invariants deduced from them have also been obtained by Wang and Wang in [20], again in dimension d ≥ 3. We also refer to the works of Jollivet and Sharafutdinov [12,13] where they find invariants for simply connected domains from the zeta function associated with DN 0 .
Our main theorem shows that for non-zero potential, one can hear more information from polynomial eigenvalue asymptotics.
The spectral inverse problem for the Dirichlet-to-Neumann map consists in extracting information about Ω, g, τ and λ (or a subset of these parameters) from the eigenvalues {σ j : j ∈ N}. As an application of our methods, we will find spectral invariants when τ ≡ 1, and show that we can recover λ as well as geometric data on Ω. For λ = 0, the problem has been studied already and is referred to as the Steklov problem. Lee and Uhlmann have shown in [15] that the map DN 0 (but not necessarily its spectrum) determines the Taylor series for g close to the boundary. Girouard, Parnovski, Polterovich and Sher show in [8] that from polynomial order spectral asymptotics, one can determine the number of boundary components and each of their lengths, but nothing more. Our goal is to obtain more information from the spectrum when λ = 0. Theorem 1.3. For any λ ∈ (R ∩ V) \ {0}, the spectrum of DN λ determines the following quantities: • the number of connected components of the boundary, and their respective perimeters; (4), and in particular if τ ≡ 1: the parameter λ; the total geodesic curvature on each boundary component.
The previous theorem along with the Gauss-Bonnet theorem also yield. Theorem 1.4. Let Ω be a smooth Riemannian surface with smooth boundary and genus γ. Suppose further that the Gaussian curvature K of Ω is constant. Then, the quantity is a spectral invariant of DN λ (Ω; 1).
Here the genus γ of Ω corresponds to the minimal genus of a closed surface in which Ω can be topologically embedded. Equivalently, it is the genus of the closed surface obtained from Ω by gluing topological disk onto each boundary component. By restricting the choice of Ω, we can gain more interior geometric information from the spectrum. Note that while the Steklov spectrum is not known to determine interior information in general, for planar domains it is already known from the work of Edward [6,Theorem 4] that we can get lower bounds for the area. If Ω is a domain of the standard sphere S 2 , its area is a spectral invariant.
If Ω is a domain in a flat space form, its genus is a spectral invariant.
Proof. If Ω is a domain in a flat space form, then K(Ω) = 0 and only 4πγ remains in (6).
The inverse problem for DN λ (Ω; τ ) has a concrete interpretation in terms of the inverse scattering problem. In this context, Ω ⊂ R 2 has anisotropic refraction index τ . Non-destructive testing is the process of using the farfield data to measure the scattering of an incoming wave at frequency √ λ by the obstacle Ω. The inverse scattering problem consists in recovering then the refraction index τ , as well as the geometry of Ω. In [3], it is shown that the far-field data determines the spectrum of DN λ (Ω; τ ), so that any spectral invariant of DN λ can be obtained from the far-field data. We have explicit expressions for geometric quantities related to the boundary of Ω when the refraction index is isotropic, i.e. constant. When it is not, we do not give an explicit value of the coefficients s n , however the algorithmic procedure to compute them in Sections 3 and 6 applies. Similarly, Theorem 4.2 is also valid in that context, giving an exact expression for the first few invariant quantities. Note that the coefficients s n are polynomials of order at most n in λ with vanishing constant coefficient. This means that it is possible to decouple the coefficients of this polynomial by knowing the asymptotics for λ 1 , . . . , λ n . Physically, this simply means measuring the scattered far-field data for incoming waves at n different frequencies.
Our first step will be to show that we can reduce Theorems 1.1 and 1.2 for Problem (7) to proving them for In other words, by introducing this extra parameter ρ they only need to be proved in the case where Ω is a disk, and g is the flat metric g 0 . This reduction will be done by following the strategy set out in [8], where they glue a disk to a tubular neighborhood of every boundary component, and discard the rest of the surface. Since the symbol of DN λ depends solely on data obtained from a neighborhood of the boundary, this doesn't change the symbol of the Dirichlet-to-Neumann map. Mapping these topological disks conformally to the unit disk in R 2 will multiply the factors τ and ρ by a conformal factor, in other words it doesn't change the structure of the problem.
We then follow the general theory set out by Rozenblum in [18] to obtain a complete asymptotic expansion of the eigenvalues of a pseudodifferential operator on a circle in terms of integrals of its symbol. Note that in [18], an abstract algorithm is given to do so, but as is often the case with pseudodifferential symbolic calculus the expressions become unwieldy very quickly, and the difficulty resides in extracting actual geometric information out of it. The symbol is easy to compute for ρ = 1, λ = 0, where it is simply |ξ|, with no lower order terms. However, when λ = 0, this is no longer the case, and it will lead to the full asymptotic expansion that we obtain.
We obtain the following theorem for the disk.
Theorem 1.7. The eigenvalues of Problem (8) satisfy the asymptotic expansion where the coefficients b n depend only on ρ, λ and the values of τ in a neighborhood of S 1 , as well as their derivatives.
We will then specialize the previous theorem to the values of τ and ρ coming from the conformal mapping between the disk and Ω. We obtain explicit values of the coefficients b n in that situation.
1.5. Plan of the paper. In Section 2, we make clear our reduction to the disk and compute the full symbol of the Dirichlet-to-Neumann map. In Section 3 and Section 4, using the method laid out in [18], we transform the symbol of a general Dirichlet-to-Neumann map on a circle to extract the asymptotic expansion of its eigenvalues. In Section 5, we specify our results to the case of the parametric Steklov problem in order to show Theorem 1.1. Finally, in Section 6, we prove Theorem 1.3. There, we use Diophantine approximation to decouple the sequences obtained in Theorem 1.2 recursively. metric g in a neighborhood of the boundary Σ. This will allow us to show that we can reduce the problem at hand to the situation where Ω is the unit disk D. In the second part of this section, we explicitly compute the value of the symbol for the disk.
For each x ∈ Σ, let γ x be the unit speed geodesic starting at x, normal to Σ. Since ε < inj(Ω), for every x ∈ Υ, there is a unique x ∈ Σ such that x ∈ γ x , and set x = (x, t) where t is the parameter along γ x . The boundary Σ is characterised by {t = 0}, and the outward normal derivative is given by ∂ ν = −∂ t . In these coordinates, the metric has a much simpler form as for some positive function g. The Laplacian reads Proposition 2.1. There is a family A(x, t, D x ) of pseudodifferential operators depending smoothly on t such that Proof. The proof follows that of [15, Proposition 1.1] in computing the symbol of A recursively. Their construction only relies on ellipticity of H, and the fact that the only derivatives in t are in ∆ g .
Remark 2.2. In subsection 2.2, we make this recursive computation of the symbol explicit for the disk, as we need to obtain concrete values of the coefficients in that case. The reader interested in a more detailed proof of Proposition 2.1 can see that this recursive computation also works for a general Ω.
Proposition 2.1 admits the same corollary as in [15].
In other words, In particular, the symbol of 1 ρ DN λ (Ω; τ ) depends only on λ, ρ and the boundary values of g, τ and of their derivatives.

Proof. By Corollary 2.3, the operators
It follows from [8, Lemma 2.1] that their eigenvalues are close to infinite order.

Lemma 2.5.
Let Ω be a smooth simply connected surface with boundary Σ. Let ϕ : D → Ω be conformal. Then, the Steklov problem (7) on Ω is isospectral to the problem Proof. It follows directly from the observation, see [12], that the Laplacian and normal derivatives transform under a conformal mapping ϕ : (D, g 0 ) → (Ω; g) as respectively.
This leads us to the main theorem of this subsection, reducing the problem to the one on the unit disk.
Theorem 2.6. Let (Ω, g) be a Riemannian surface whose smooth boundary Σ has connected components, and let Ω be a union of identical unit disks D 1 , . . . , D with boundary Σ the union of circles S 1 m . There exist τ 0 : Ω → C and ρ 0 : Σ → C.
such that Proof. The proof follows that of [8,Theorem 1.4]. For 1 ≤ m ≤ , let Ω m be a topological disk with a Riemannian metric that is isometric to a collar neighborhood Υ m of Σ m , and denote by Ω the union of the disks Ω m . We abuse notation and denote also by τ any smooth function on Ω whose value on Υ m coincides with τ on Ω. This is justified since only its value in a neighborhood of the boundary affects eigenvalue asymptotics. It follows from Lemma 2.4 that For every m the Riemann mapping theorem implies the existence of a conformal diffeomorphism ϕ m : (D m , g 0 ) → (Ω m , g m ). Given that ϕ * m g m = e 2fm g 0 , define τ 0 and ρ 0 for x ∈ D m and S 1 m respectively as

It follows from Lemma 2.5 that
. The conclusion then follows from the fact that the spectrum of the Dirichletto-Neumann map defined on a disjoint union of domains is the union of their respective spectra.

2.2.
The symbol of the Dirichlet-to-Neumann map on the disk. We now compute the full symbol of Λ := 1 ρ DN λ (D; τ ) on S 1 = ∂D from the factorisation obtained in Proposition 2.1. Let us introduce boundary normal coordinates (x, t) for the collar neighborhood S 1 × [0, δ), for some small but fixed δ. The flat metric in these coordinates reads and the Laplacian reads as We are therefore looking for a factorisation of the form Rearranging, this implies finding A such that which at the level of symbols is tantamount to finding a(x, t, ξ) such that is the symbol of A and the coefficients a m are positively homogeneous of degree m in ξ.
By gathering the terms of degree two, we obtain while gathering the terms of degree one yields One can observe that neither a 1 nor a 0 depend on λτ . However, by gathering the terms of order 0, we get For m ≤ −1, a m−1 is found recursively by gathering the terms of order m and is given by Note that this is the same recurrence relation as the one appearing in [15] as soon as m < −1. For the sequel, we will require explicit knowledge of the term of order −2. From the previous equation we deduce that As indicated by corollary (2.3), the symbol of Λ is given by where the sign is chosen so that Λ is a positive operator. Note that ∂ t is the interior normal derivative hence ∂ t = −∂ ν . Writing f (x) := f (x, 0) for the restriction of any function to the boundary, the first few terms of the symbol of Λ read as

2.3.
Symmetries of the symbol. When λ and τ are real, we see from these first expressions, that the real part of the symbol is an even function of ξ, while its imaginary part is an odd function of ξ. This is equivalent to the following definition.
We now show recursively that the symbol of Λ is hermitian.
The proposition follows from (9) and the following lemma whose proof is straightforward.
Lemma 2.9. Let a and b be two hermitian symbols corresponding to operators A and B. Then (1) ∂ x a and D ξ a are hermitian; (2) a + b and ab are hermitian; (3) The symbol of AB is hermitian.
Proof. The first two claims are a trivial computation. The third claim follows from the fact that the symbol of AB is obtained from a and b using the operations described by the first two claims.

Transformation of the symbol
In this section, we follow and make explicit the strategy laid out in [18], [1, Section 2] and [7] in the specific case of the parametric Dirichlet-to-Neumann map.
Specifically, we want to find a sequence P N ∈ Ψ 1 such that • The symbol of P N depends only on the cotangent variable ξ up to order 1 − N .
Such a procedure (making the symbol dependent solely on ξ) will be referred to as a diagonalisation of the symbol. It is motivated by the following proposition resulting from [18, Theorem 9].
Proposition 3.1. Let A be an elliptic, self-adjoint pseudodifferential operator of order 1 and let P be the operator with symbol where p 1−k depends only on ξ and is positively homogeneous of order 1 − k.
Suppose that AU − U P ∈ Ψ −N for some bounded operator U . Then the eigenvalues of A are given by the sequences Diagonalisation of the principal symbol. We start by diagonalising the principal symbol of Λ = 1 ρ DN λ (D; τ ). Let The function S is a generating function for the canonical transformation (y, ξ) = T (x, η) given by the relations We define the Fourier integral operator Φ with phase function S as where u is the Fourier transform of u. We use Φ to diagonalise the principal symbol of Λ in the following proposition. Proof. We are looking for the symbol of B in the form with b j (x, ξ) positively homogeneous of order j in ξ. Let us first study the operator ΛΦ. It acts on smooth functions as where k(x, ξ) = r(x, η)e i(x−y)η e i(S(y,ξ)−S(x,ξ)) dy dη.
We now look for the asymptotic expansion of k as a symbol on S 1 , up to symbols of order −∞. Note that the expressions here have sense in terms of distributions, see [7, Section 2.2.2]. By following the method of proof in [7, Theorem 6.5], we can localize the symbol by finding smooth cut-off functions h 1 (x, y) and h 2 (ξ, η) supported in suitable neighborhoods of x = y and ξ = η such that if then Op(k − k ) ∈ Ψ −∞ . By Taylor's theorem, we can write We can rewrite k as Changing variables as η = η − R(x, y, ξ)(y − x) and ξ = ∂S(x,ξ) ∂x = ξρ L , we obtain that k is of the form K(x, y, ξ, η) = r (x, η + R(x, y, ξ)) h 1 (x, y)h 2 (ξ, η + R(x, y, ξ)).
From [7, Lemma 2.13], we know that k (x, ξ) is a symbol given by By the choice of cut-off functions, when x is close to y and η is close to ξ, we have that h 1 and h 2 are constant and equal to one. Hence, they don't intervene in the symbol's calculation and We now make the following observation : if r(x, η + R(x, y, ξ)(y − x)) is a symbol of order m, then applying ∂ α η D α y results in a symbol of order m − α. In fact, for α = 1, and denoting by ∂ 2 the derivative with respect to the second argument, we have It is clear from this last equation that it is a symbol of order m−1. Induction on α is then straightforward. This yields the asymptotic symbolic expansion We can compute the first few terms of the symbolic expansion, using the fact that in R \ {0} the second derivative of a 1 in the second variable vanishes identically. This gives Let us now compute the symbol of ΦB. We have where f (x, ξ) = b(y, ξ)e i(S(x,η)−S(x,ξ)) e iy(ξ−η) dy dη.
As above, this integral only converges in the sense of distributions. As in [7], we can find a smooth cut-off function h(ξ, η) supported in a neighborhood of ξ = η such that the symbol Let us observe that and that F (x) = ∂S ∂ξ (x, ξ). After the change of variable y = y + x − F (x), the equation for f becomes Once again from [7, Lemma 2.13], we have that f is a symbol in S 1 and Since Q is constant in η close to ξ, the derivatives in η always vanish. Hence, the symbol of B N is given by To have the terms of the same order of homogeneity cancel out, we need to choose where s(x) is the number s such that This concludes the proof.

3.2.
Diagonalisation of the full symbol. Let us denote by P 1 the operator with symbol The diagonalisation of the full symbol is based on the following lemma inspired by the methods laid out by Rozenblum [18] and Agranovich [1]. We include it for completeness. Lemma 3.3. Let N ≥ 0 and suppose that there exists a bounded operator U N such that ΛU N −U N P N ∈ Ψ −∞ where P N is a pseudodifferential operator whose symbol is given by Then if and K is the pseudodifferential operator with symbol there exists an operator P N +1 with symbol Proof. Starting off with the pseudodifferential operator P N , we would like to find a bounded operator K and a pseudodifferential operator P N +1 whose symbol p (N +1) satisfies with k −N positively homogeneous of order −N in ξ. The symbol of P N K − KP N +1 is then given by comes from the diagonalisation of the principal symbol and is given by p Hence, we see that the terms of order −N cancel if the symbol of K is given by (16) and since 0 = k −N (0, ξ) = k −N (2π, ξ), we must take p (N +1) −N as in (15). In order to get that P N K − KP N +1 ∈ Ψ −∞ knowing that the symbol of P is given by we need to take P N +1 with symbol which is calculated inductively as (17) The previous lemma gives us a family of operators P N that diagonalise Λ down to any desired order. By applying it N − 1 times starting from P 1 , we get that there exists P N with symbol such that ΛU N − U N P N is smoothing for some bounded operator U N . We summarize the properties of the operators P N that were proved along the discussion above in the following proposition.   Proof. We denote by deg(p) the degree of a function p(x, ξ) as a polynomial in λ. We proceed by induction on both N and m.
It is easily seen from (9)  m 0 −1 . Its expression is given by (17) and we can see that the term of highest degree in λ in the sum is obtained when α = 0. Hence, −N ) ≤ N by the induction hypothesis. Since m 0 −1+N ≥ m 0 , the induction hypothesis yields (20) deg(p Therefore, by combining (18), (19) and (20), deg(p m vanishes, it suffices to show that it is the case for a m . Proceeding inductively, since a 0 = 0, notice from (9) that the only term in a m−1 that could be constant in λ is Remark 3.6. That p (N ) m = 0 whenever λ = 0 is not surprising. Indeed, this corresponds to the classic Dirichlet-to-Neumann operator whose symbol is precisely |ξ|.
If one is interested in computing the symbols explicitly in a given example the calculations quickly become very involved. The following lemma allows us to reduce the number of computations to obtain the k-th term in the diagonalised symbol.
Therefore, since p The rightmost integral vanishes for all α since k −N is periodic and thus Using that s (x) = L ρ(s(x)) , we get where the terms containing i sgn ξ vanish from the fact that this equality being obtained by integrating by parts. Therefore, by doing a similar calculation for 2π 0 b −1 (x, ξ) dx, we see that the symbol of P 2 is given by (21)

4.
General eigenvalue asymptotics from the symbol 4.1. Self-adjointness. For λ ∈ R ∩ V and τ real-valued, the operator Λ := 1 ρ DN λ (D; τ ) is self-adjoint and therefore has real spectrum. This follows from the fact that DN λ (D; τ ) is self-adjoint and the following lemma applied to P = DN λ (D; τ ).
Lemma 4.1. Let P be a self-adjoint pseudodifferential operator on L 2 (S 1 ; dx) and ρ > 0 be a positive function on S 1 and denote M 1/ρ the operator of multiplication by ρ −1 . For f ∈ Diff(S 1 ), define by K f the composition operator Proof. The operator K g is an invertible isometry from L 2 (S 1 ; dx) to L 2 (S 1 ; ρ(x)/L dx). Indeed, for u, v ∈ L 2 (S 1 ; dx), we have The operator M 1/ρ P is self adjoint on L 2 (S 1 ; ρ(x)/L dx), hence we have proving that Q is self adjoint.

4.2.
General eigenvalue asymptotics. We have shown how to diagonalise the symbol down to any order. We can now deduce the spectral asymptotics of Λ from Proposition 3.1. Eigenvalue asymptotics for an elliptic pseudodifferential operator on a circle are discussed also in [1, Theorem 3.1].
Theorem 4.2. The eigenvalues of Λ are asymptotically double and admit a full asymptotic expansion given by Proof. The fact that the eigenvalues admit a full asymptotic expansion follows from Propositions 3.1 and 3.3. Moreover, (22) follows from equation (21) and Proposition 3.1 . It remains to show that the eigenvalues are asymptotically double. This will follow from Proposition 3.1 if we can show that, for all N ∈ N, there exists a bounded operator U N and a pseudodifferential operator P N with symbol such that p 1−m is an even function of ξ (since then p 1−m (j) = p 1−m (−j)) and such that ΛU N − U N P N is smoothing. To do so, it is sufficient to show that a symbol being hermitian is an invariant property of the diagonalisation procedure, see Definition (2.7). The claim will then follow since Λ is selfadjoint and hence all its eigenvalues must be real. We know from Proposition 2.8 that the symbol of Λ is hermitian. In order to diagonalise the principal symbol, we conjugated by the Fourier integral operator Φ. The resulting symbol is given by where a m is given by (12). It suffices to show that a m is hermitian for all m. This is a consequence of the fact that is hermitian for all α ≥ 0. Indeed, by Leibniz's formula and (11) we have for all β ≥ 0. Hermiticity of (23) then follows from Faà di Bruno's formula since each derivative in the second argument will come with a power of ξ, thus preserving the parity in the real and imaginary parts. Let N ≥ 0 and suppose that ΛU N − U N P N ∈ Ψ −∞ as in the notation of Proposition 3.3 is such that the symbol p (N ) of P N is hermitian. From (16), (17) and Lemma 2.9, we see that the symbol p (N +1) of P N +1 is also hermitian. The fact that the spectrum is asymptotically double then follows from the previous discussion.

Eigenvalue asymptotics
Let (Ω, g) be a simply connected Riemannian surface with smooth boundary Σ. We are now interested in finding the spectral asymptotic for the operator DN λ (Ω; τ ; 1) corresponding to the problem −∆ g u = λτ u in Ω; ∂ ν u = σu on Σ; which we refer as the parametric Steklov problem on Ω. By the Riemann mapping theorem, there exists a conformal diffeomorphism ϕ which maps (D, g 0 ) onto Ω such that ϕ * g = e 2f g 0 for some smooth function f : D → R. Therefore, the parametric Steklov problem on (Ω, g) is isospectral to the problem In the notation of (10), we have 2π 0 e f dx = per g (Σ) 2π .
We are now in a position to prove our main results about eigenvalue asymptotics.
Proof of Theorem 1.1. The theorem follows directly from Theorem 4.2 for the existence of the complete asymptotic expansion. The fact that s n is a polynomial in λ of degree at most n follows directly from Lemma 3.5. For the explicit values of s −1 and s −2 when τ ≡ 1, we replace in (22) the values of τ and ρ by the conformal factor. The second term in (22) is given by Finally, the third term is given by λL By Green's theorem, we have Recall that the Gaussian curvature of (D, ϕ * g) is given by Hence, since ϕ * K g = K ϕ * g and ϕ * dA g = e 2f dA, Combining everything and using the Gauss-Bonnet theorem yields since Ω is simply connected, and hence its Euler characteristic is 1.
Proof of Theorem 1.2. Let (Ω, g) now be any Riemannian surface with smooth boundary Σ. Suppose that Σ has connected components Σ 1 , . . . , Σ and let Ω m be a smooth topological disk with a Riemannian metric that is isometric to Ω in a neighborhood of Σ m . Denote by Ω the union of the disks Ω m . From Lemma (2.4), we know that Since Ω is a union of disks, its spectrum is given by the union of each disk's spectrum. Applying Theorem (1.1) to each Ω m , and using that the parametric Steklov spectrum of a disjoint union of surfaces is the union of their spectra we see that the spectrum of Ω is the union of different sequences taking the form of equation (2). This is the statement of Theorem 1.2, as claimed.

Spectral invariants
When the surface Ω is simply connected, the search for spectral invariants is easier. From the first two terms of the eigenvalue asymptotic expansion, we can deduce uniquely the values of both L and λ. Hence, from the third term, we can deduce uniquely the value of Σ k g ds and it is a spectral invariant. By restricting ourselves to surfaces of constant curvature, we get the following. Corollary 6.1. Let (Ω, g) be a simply connected Riemannian surface with smooth boundary Σ. Suppose further that the Gaussian curvature K of Ω is constant, then the quantity is a spectral invariant of the parametric Steklov problem on Ω.
In the multiply connected case, we need to introduce some definitions to talk about functions between two multisets. To determine the number of boundary components and the lengths of them, we will use methods from Diophantine approximation. This is in the spirit of [8], where they obtained those quantities as invariants of the Steklov problem with λ = 0. There, they had an asymptotic expansion of the form (4)- (5), where all the coefficients s n were 0. However in order to obtain the number of boundary components and their lengths as spectral invariants, they need only that the second term is o (1), which we do have.
Recovering λ as well as the total geodesic curvature of the boundary is more complicated and requires an algorithmic procedure to recover subsequences (which can be explicitly constructed) once we know the number of boundary components and the length of the largest one. We start by introducing terminology found in [8, Section 2.3] Definition 6.2. Let A, B be two multiset of positive real numbers. We say that F : A → B is close if it has the property that for every ε > 0, there are only finitely many x ∈ A with |F (x) − x| ≥ ε. We say that F is an almost-bijection if for all but finitely many y ∈ B, the pre-image F −1 (y) consists in a single point.
where 0 is repeated times and the union is understood in the sense of multisets, i.e. multiplicity is conserved. : j ∈ N : 1 ≤ m ≤ be a set of asymptotically double sequences with complete asymptotic expansion Then, M and the quantities s Let us first describe heuristically how the proof goes. In the first step, we simply show that [8, Lemmas 2.6 and 2.8] apply to this situation. This will allow us to recover M from S(Ξ), and we assume from then on that M , and therefore R(M ), are already known to be spectral invariants.
In the second step, we show that for any α m ∈ M which is not an integer multiple of another strictly smaller element of M , we can identify a subsequence along which S(Ξ) j = ξ (m) k(j) where k : N → N is a function that can be computed explicitly. For this, we use Dirichlet's simultaneous approximation theorem.
In the third step, we obtain the coefficients of those sequences α m that we decoupled in the previous step. Obviously, if α m appears only once in M this is trivial, the difficulty comes when α m has multiplicity.
In the fourth step, we proceed inductively and show that if α m is an integer multiple of some other α n ∈ M , but we already know the coefficients of the relevant sequences for α n , then we can apply the same procedures as in steps 2 and 3 to recover the coefficients of the asymptotic expansion for the sequence ξ (m) .

Proof.
Step 1: We obtain M from S(Ξ). From [8][Lemmas 2.6 and 2.8], it suffices to show that there is a close almost-bijection from R(M ) to S(Ξ). Now, it is not hard to see that the map F : R(M ) → S(Ξ) that maps R(M ) j to S(Ξ) j is a close almost-bijection. Indeed, it follows from the definition of the sequences ξ (m) that which implies that F is a close almost-bijection.
Step 2: Suppose without loss of generality that the smallest element of M is 1. Define on positive real numbers the strict partial order x ≺ y if there is an integer n ≥ 2 such that y = nx, and denote by x y the non-strict version of this partial order, i.e. if n = 1 is allowed. For any multiset U of positive real numbers, we say that x ∈ U is minimal in U if for all y ∈ U , either x y, or x and y are incomparable. Let I ⊂ {1, . . . , } be defined as We claim that there exist δ > 0 and subsets E m ⊂ N of infinite cardinality for each m ∈ I such that for all j ∈ E m , Let Q be the smallest common integer multiple of elements in M 1 . Dirichlet's simulateneous approximation theorem states that there is an infinite subset E ⊂ N such that for all q ∈ E and α m ∈ M 2 there exists p q,m ∈ N such that This means that for all q ∈ E, there is an integer multiple of α m within q −1/ of qQ. Note that for α m ∈ M 1 , the integer multiple is actually exactly qQ.
In that case we put p q,m = Qqα −1 m . Set δ = 1 2 min {|α m − nα k | : m ∈ I, α k = α m , n ∈ N} , and observe that δ > 0 from the assumption that α m is minimal in M for all m ∈ I. Assume that α is the largest element of M and for m ∈ I, set (26) E m := p q,m + 1 : q ∈ E, q −1/ < δ 2α .
We claim that for all j ∈ E m , (25) holds. Indeed, if α k = α m and n ∈ N, we have It follows that no integer multiple of α k = α m is within distance δ of jα m , when j ∈ E m . On the other hand, by definition of R(M ), and assuming without loss of generality that δ < 1, jα m is the only integer multiple of α m in the interval [jα m − δ, jα m + δ], and this happens with multiplicity 2µ(m).
Step 3: For m ∈ I, we recover the quantities s (k) n for all k such that α k = α m . Let j ∈ E m and observe that the indices in the sequence S(Ξ) for the elements in the interval [jα m − δ, jα m + δ] can be uniquely determined from R(M ), which is determined by S(Ξ) as seen in the first step of this proof. It is also easy to see that by (24), for all k such that α m = α k and j ∈ E m large enough, we have .
Consider the set From the definition of E m , we have Consider the accumulation points of X 1 . We claim that those points are exactly the values of s (k) 1 for which α k = α m . In fact, from the previous equation, X 1 is a union of sequences and the claim follows from the fact that Moreover, we can know the number of k such that s Note that from the construction, we cannot directly know which k is associated to each s (k) 1 , but without loss of generality we can label them in any way we choose since we know their multiplicity. For k with α k = α m , we construct the sequences We claim that the accumulation points of X We can also deduce the multiplicity of each s Step 4: We now turn our attention to m ∈ I, and assume that we have already proved the proposition for all k such that α k ≺ α m . Defining this time where µ = 2 α k αm µ(k). We observe that once again, the indices in the sequence S(Ξ) of those elements are uniquely determined by R(M ).
For every k such that α k α m , write r(k) to be the integer such that α m = r(k)α k . Defining X 1 as in step 3, its accumulation points are now given by the values of s (k) 1 r(k) for which α k α m . From the induction hypothesis, we know those values whenever r(k) > 1. Hence, we can disregard them. What is left are the values of s n for any n ∈ N for each k with α k = α m . The set M is finite, hence our inductive procedure necessarily terminates, finishing the proof. , . . . , 2π per(Σ ) .
Therefore, we can recover M from the spectrum of DN λ , or in other words the number of boundary components and their lengths. It follows from Proposition 6.3 that one can recover the coefficients in the sequences (4). In particular, from (3)  Σm k g ds, allowing us to recover the total geodesic curvature on each boundary component.
We can now as well prove Theorem 1.4 Proof of Theorem 1.4. Since the total geodesic curvature on each boundary component is a spectral invariant, the total integral Σ k g ds = m=1 Σm k g ds is a spectral invariant. Applying the Gauss-Bonnet theorem, we get where γ is the genus of Ω. Since the number of boundary components is a spectral invariant, we can deduce that the quantity 4πγ + Ω K g dA g is also a spectral invariant of the parametric Steklov problem.
Remark 6.4. It is impossible to completely decouple the genus and the average of the Gaussian curvature as spectral invariants from the eigenvalue asymptotic expansion since the addition of a handle far from the boundary changes the genus of Ω but leaves the symbol of the Dirichlet-to-Neumann operator unchanged. However, a priori information on Ω, such as being a domain of a specific space form of constant Gaussian curvature can yield additional information, as in Corollaries 1.5 and 1.6.