THE NEUMANN ISOSPECTRAL PROBLEM FOR TRAPEZOIDS

Transcription

1 THE NEUMANN ISOSPECTAL POBLEM FO TAPEZOIDS HAMID HEZAI, ZHIQIN LU, AND JULIE OWLETT Abstract. We show that non-obtuse trapezoids with identical Neumann spectra are congruent up to rigid motions of the plane. The proof is based on heat trace invariants and some new wave trace invariants associated to certain diffractive billiard trajectories. We use the method of reflections to express the Dirichlet and Neumann wave kernels in terms of the wave kernel of the double polygon. Using Hillairet s trace formulas for isolated diffractive geodesics and one-parameter families of regular geodesics with geometrically diffractive boundaries for Euclidean surfaces with conical singularities [Hil05], we obtain the new wave trace invariants for trapezoids. To handle the reflected term, we use another result of [Hil05], which gives a Fourier Integral Operator representation for the Keller-Friedlander parametrix [Kel58, Fri81] of the wave propagator near regular diffractive geodesics. The reason we can only treat the Neumann case is that the wave trace is more singular for the Neumann case compared to the Dirichlet case. This is a new observation which is of independent interest. 1. Introduction Two-dimensional plane billiards, although they may be geometrically rather simple, provide a rich setting for the study of their spectrum and dynamics. For example, their dynamical properties can range from integrable to ergodic and mixing, hence reflecting such properties in their Laplace spectra in the light of Berry-Tabor [BT77] and Bohigas-Giannoni-Schmit [BGS84] conjectures. A natural approach to relate the spectrum and the billiard flow is via semiclassical trace formulas which is the main interest of this article. More precisely, we study the inverse spectral problem for polygonal billiards using semi-classical trace formulas near diffractive periodic billiard orbits (see the works of Bogomolny-Pavloff-Schmit [BPS00] and Hillairet [Hil05]). Although the existence of non-convex non-isomorphic isospectral polygonal tables is well known [GWW92a], many questions remain unanswered in spite of significant interest. These questions often have a deceptively simple statement which conceal the underlying technical difficulties. For example, it is an open problem to determine whether there exist isospectral convex plane polygonal billiards. Certain classes of planar domains are known to be spectrally determined. Disks are determined among all planar domains [Kac66]. Certain oval domains [Wat00] are Key words and phrases. isospectral; trapezoid; polygons; conical singularities; diffraction; heat invariants; wave invariants; inverse spectral problems. MSC primary 58C40, secondary 35P99. 1

2 2 HAMID HEZAI, ZHIQIN LU, AND JULIE OWLETT Figure 1. The top edge orbit, γ b, which bounces between two diffractive conical singularities, contributes a singularity at t = 2b of order ( 1 2 )+ to the wave trace of the double of the trapezoid (as an ESCS) if there are no other orbits of the same length. Hence 2b is a spectral invariant if both Dirichlet and Neumann spectra are known. In fact, as we prove, 2b is a spectral invariant for the Neumann spectrum. determined among smooth domains. See also [Zel09, HZ12, DSKW17, PT16] where inverse results are obtained for Z 2 -symmetric analytic domains, ellipses, nearly circular Z 2 -symmetric smooth convex domains, and certain completely integrable manifolds with boundary. The simplest planar polygonal billiards are triangles, and these are spectrally determined in the moduli space of triangles [Dur88, GM13]. See the surveys [Mel96, Zel14, DH13, L15] for more background on positive inverse results. Our main result is the following: Theorem 1. Let T 1 and T 2 be two non-obtuse trapezoidal domains in 2. Then if the spectra of the Euclidean Laplacian with Neumann boundary conditions coincide for T 1 and T 2, the trapezoids are congruent, that is equivalent up to rigid motions of the plane. Our proof 1 relies on heat trace invariants and also some new wave trace invariants associated to some diffractive billiard trajectories. We first use the method of reflections to express the Dirichlet and Neumann wave kernels in terms of the wave kernel of the double of the trapezoid, which can be realized as a Euclidean surface with conical singularities (ESCS). We then obtain new wave trace invariants using two results of Hillairet [Hil05] for ESCSs (see also [BPS00] for results in the physics literature). The first is a parametrix construction of the wave propagator near diffractive geodesics as a Fourier Integral Operator (FIO). We use this parametrix construction for the reflected term. Such parametrices were found by Keller [Kel58], Friedlander [Fri86, Fri81], and Cheeger-Taylor [CT82a, CT82b], however expressing them in the language of FIOs was first done by Hillairet in [Hil05]. For the non-reflected term (i.e. the first term in Proposition 11) we use trace formulas of [Hil05] associated to isolated diffractive geodesics and to one-parameter families of regular geodesics with geometrically diffractive boundaries. More precisely we apply the trace formulas of [Hil05] to the diffractive bouncing ball orbit associated to the top edge of the trapezoid (Figure 1), and 1 For the benefit of readers who are less familiar with the language and the methods used here, we recall the needed tools from microlocal analysis in the appendix.

3 NEUMANN ISOSPECTAL TAPEZOIDS 3 Figure 2. The vertical lines show the interior of a one-parameter family of non-diffractive bouncing ball orbits of length 2h. This family sweeps a rectangle of area hb twice. Here h and b are defined in Figure 3. The two boarder line orbits are both geometrically diffractive. The family, which we call γ h, contributes a singularity at t = 2h of order 1 + to both Dirichlet and Neumann wave traces if there are no other periodic orbits of length 2h with the same order of singularity. to the one-parameter family of bouncing ball orbits associated to the altitudes of the trapezoid (Figure 2). The lengths of these orbits and the principal terms of the singularity expansions of the Neumann wave trace at these lengths provide new spectral invariants for the trapezoid. Together with the well known heat trace invariants, these can be used to prove spectral uniqueness of a trapezoid amongst all trapezoids. The reason we treat the Neumann case is that in some sense the wave trace is more singular for the Neumann case when compared to the Dirichlet case, and this more singular behavior facilitates our proof. The crux of the matter lies in orbits which have some part contained in the boundary of the trapezoid. This is a new feature which is of independent interest: the Neumann wave trace has a larger singularity at 2b (see Figure 1) than the Dirichlet wave trace. It would be interesting to study the singularity of the Dirichlet wave trace at 2b (if singular at all), but since we do not require this for our inverse result for the Neumann boundary condition, we refrain from exploring this question here. In the sequel to this work we shall study the isospectral problem in the Dirichlet case. 2. Background The isospectral problem is: if two iemannian manifolds are isospectral, then are they isometric? For a iemannian manifold (M, g) the spectrum in question is for the Laplace operator n 1 ( = i g ij ) det(g) j. det(g) i,j=1 The answer in this generality is no, and was proven by Milnor in 1964 [Mil64]. He used a construction of Witt [Wit41] of two self-dual lattices L 1 and L 2 in 16 such that no rotation of 16 maps one to the other, but such that the spectra of the iemannian manifolds 16 /L i are identical for i = 1, 2. Around the same time, M. Kac wrote a popular article

4 4 HAMID HEZAI, ZHIQIN LU, AND JULIE OWLETT [Kac66], Can one hear the shape of a drum? He popularized the isospectral problem for planar domains. Although this may seem like an easier setting, it turned out to be quite difficult to prove that the answer is in general negative. For a bounded domain Ω in 2, we consider the Euclidean Laplacian with Dirichlet (D) or Neumann (N) boundary conditions, (2.1) u(x, y) := 2 u x 2 2 u = λu, Bu = 0, y2 where Bu = u Ω when B = D, and Bu = ν u Ω when B = N. For both boundary conditions, the eigenvalues, which depend on B, form a discrete subset of of the form 0 λ 1 < λ 2 λ The Dirichlet case physically corresponds to a fixed, stationary boundary. In this case the spectrum is in bijection with the resonant frequencies a drum would produce if Ω were its drumhead. With a perfect ear one could hear all these frequencies and therefore know the spectrum. This is the origin of the title of Kac s paper [Kac66]. On the other hand, the Neumann boundary condition physically corresponds to an insulated boundary from which heat neither leaves nor enters. In both cases the eigenfunctions and eigenvalues determine the fundamental solution and thereby all solutions for the heat equation on Ω with given initial conditions. Gordon, Webb, and Wolpert answered Kac s question in the negative [GWW92a, GWW92b], based on Sunada s method [Sun85] and Buser s work [Bus86]. The isospectral problem for surfaces was previously demonstrated to have a negative answer by [Vig78]. Buser s method relied on a pasting procedure for pairs of surfaces. In [GWW92b], they determined how to suitably fold two such curved surfaces to create isospectral non-isometric planar domains. This general idea of folding paper was later presented in an accessible style by Chapman [Cha95]. On the other hand, in some cases the isospectral problem has a positive answer. If one considers triangular domains in the plane, then if two such domains are isospectral, the triangles are congruent. The first proof of this fact is contained in the doctoral thesis of C. Durso [Dur88]. She used the fact that the heat trace implies that the area and perimeter are spectral invariants, so any two triangles which are isospectral must have the same area and perimeter. To complete the proof, she used the wave trace and demonstrated that the length of the shortest closed geodesic in a triangular domain is also a spectral invariant. More recently Grieser and Maronna [GM13] realized that if one used an additional spectral invariant from the heat trace, then this together with the area and perimeter uniquely determine the triangle. That is a much simpler proof. Other types of domains which are known to be spectrally determined are analytic planar domains with reflective symmetries; see the works of Colin de Verdière [CdV84, CdV73] and Zelditch [Zel00, Zel09]. 3 2 In fact, for the Dirichlet case, we have 0 < λ1. 3 Note that Zelditch s results require a further genericity hypothesis.

5 NEUMANN ISOSPECTAL TAPEZOIDS 5 b l π-α h π-β l' α B β Figure 3. Parameters of a trapezoid (from [L15] Figure 7). After triangles, one is naturally interested to know whether the same result may hold for quadrilaterals. For rectangles, this is a straightforward exercise to prove that if two rectangles are isospectral, then they are congruent. For parallelograms, it is also a straightforward argument using the first three heat trace invariants as in [L15]. Of course the next natural generalization is to trapezoids. In this case, one can rather easily prove that the geometric information which can be extracted from the heat trace is, currently, insufficient to prove that isospectral trapezoids are congruent. To fix a trapezoid requires four parameters, whereas the heat trace invariants provide only three smooth functions. It is therefore necessary to use the wave trace in the spirit of [Dur88], which is a much more delicate matter. In 3, we review heat trace invariants associated to polygons and we also discover some new wave trace invariants for trapezoids. In 4, we prove the main propositions on wave trace invariants using [Hil05]. Finally, in 5 we prove our inverse spectral result. 3. Spectral invariants of trapezoids Definition 2. A trapezoid is a convex quadrilateral which has two parallel sides of lengths b and B with B b. The side of length B is called the base. The two angles α, β adjacent to the base are called base angles. The base angles satisfy 0 < β α π 2. The other two sides of the trapezoid are known as legs of lengths l and l, respectively. If l = l, then we say the trapezoid is isosceles. The distance between two parallel sides is called the height. More precisely, the trapezoids considered here are non-obtuse trapezoids, due to the fact that the base angles are assumed to be non-obtuse. Any quantity which is uniquely determined by the spectrum is known as a spectral invariant. Notation 3. If we are considering different boundary conditions on a domain Ω, we shall use the notation B Ω to indicate the boundary condition B. If we are considering

6 6 HAMID HEZAI, ZHIQIN LU, AND JULIE OWLETT a compact iemannian manifold M without boundary we shall use M. We shall use these notations when Ω is a polygon, and when M is the double of a polygon as a compact Euclidean surface with conical singularities (ESCS) The heat trace invariants. The heat trace Tr e t B Ω = e tλ k( B Ω ), k 1 is a spectral invariant. It is an analytic function for t > 0 and has a singularity at t = 0. It is well known in this setting (see [Kac66, MS67, vdbs88, L15]) that the heat trace on a polygonal domain P admits an asymptotic expansion 4 as t 0, P P (3.1) Tr e t B P + ( 1)s(B) 4πt 8 πt + n k=1 π 2 θ 2 k 24πθ k + O(e c t ), t 0, where c > 0, and s(b) = 1 when B = D, and s(b) = 0 if B = N. Above, P and P denote respectively the area and perimeter of the domain P, and θ k are the interior angles. Since the angles of a trapezoid are α, π α, β, and π β, we therefore have the following: Proposition 4. For a trapezoidal domain, the area A = P, perimeter L = P, and the angle invariant 1 q = q α,β = α(π α) + 1 β(π β), are spectral invariants. emark 1. Note that by the definition of a trapezoid, q 8 π 2, and equality holds if and only if the trapezoid is actually a rectangle. One can see that these quantities A, L, and q are insufficient to determine a trapezoid because one needs four independent variables to determine a trapezoid. In other words, considering any trapezoid T, up to congruence via rigid motions of the plane, there are (in general) infinitely many different trapezoids which have the same A, L, and q. Although the heat trace asymptotic expansion is insufficient to uniquely determine a trapezoid, it was shown in [L16] that the heat trace encodes the presence, or lack, of corners in a planar domain. The remainder term in the heat trace decays like e c/t for a constant c > 0 as t 0; this was shown by [vdbs88, Kok11]. It is therefore not feasible to extract further geometric information from the heat trace so we shall turn to a more subtle spectral invariant: the wave trace. 4 In fact in [vdbs88], this is proved only for the Dirichlet Laplacian. That a similar asymptotic is valid for the Neumann case follows easily from the Dirichlet case and the works of [Kok11, Fur94] on heat trace asymptotics on ESCS; see also emark 4.

7 NEUMANN ISOSPECTAL TAPEZOIDS The wave trace invariants. The wave trace is the trace of the wave propagator, also known as the trace of the wave group, and is formally w(t) := Tr e it = k 1 e it λ k. This is purely formal, since the wave trace is only well-defined when paired with a Schwartz class test function; it is a tempered distribution by an easy application of the Weyl s law. It is defined in more general settings such as compact iemannian manifolds without boundary as well as with boundary and with various boundary conditions. Chazarain [Cha74] (see also [CdV73]) showed that in the case of compact iemannian manifolds without boundary the singular support of the wave trace is contained in {0} ±L, where L is the set of lengths of closed geodesics. Duistermaat & Guillemin [DG75] found the principal term in the singularity expansion when the orbit is single and non-degenerate or more generally when the fixed points set is a clean manifold of arbitrary dimension. Andersson & Melrose [AM77] studied this problem in the presence of a smooth geodesically convex or concave boundary (see also Theorem of Petkov & Stoyanov [PS92] for smooth domains with no geometric assumptions) and considered the Dirichlet or Neumann boundary conditions. They showed that where SingSupp w(t) {0} ±L, L = {lengths of generalized broken periodic geodesics}. Note that in this case the length spectrum L contains the lengths of all periodic billiard trajectories hitting the boundary transversally, as well as the lengths of tangential orbits traveling completely or partly (ghost orbits) in the boundary. Hence in a smooth convex planar domain only the lengths of transversal billiard trajectories and the boundary (and its multiples) contribute to the length spectrum. Guillemin & Melrose [GM79] found the principal term in the singularity expansion when the orbit is a non-degenerate broken geodesic that hits the boundary transversally. Some experts conjecture that the above containment for SingSupp w(t) is in fact an equality The length spectrum of polygonal domains. A polygonal domain is a planar domain whose boundary is a Euclidean polygon. Propagation of singularities of the wave operator in polygonal tables or in general on manifolds with corners or with conical singularities are more difficult to study because of the diffraction phenomena that take place at the conical singularities. oughly speaking, when a geodesic that carries a singularity of the wave hits a conical singularity, it can reflect in all possible directions. There is a huge literature on the subject of diffraction, and for the sake of brevity we only list the most relevant ones for our purposes: Keller [Kel58], Sommerfeld [Som96], Friedlander [Fri86, Fri81], Cheeger &Taylor [CT82a, CT82b], and Melrose & Wunsch[MW04]. There has also been a lot of research on the contribution of diffractive geodesics to the wave trace: Friedlander [Fri86]; Durso [Dur88]; Wunsch [Wun02]; Hillairet [Hil02, Hil05]; Ford & Wunsch [FW17]; and more recently Ford, Hassell & Hillairet [FHH15]. In the physics literature, we also note the works of Bogomolny, Pavloff & Schmit [BPS00] and Pavloff & Schmit [PS95].

8 8 HAMID HEZAI, ZHIQIN LU, AND JULIE OWLETT Figure 4. Some diffractive periodic orbits of a trapezoid. A standard technique used in studying the wave trace on polygonal tables is to double the polygon along its edges to obtain a compact Euclidean surface with conical singularities, or ESCS, as commonly abbreviated in the literature. A compact n-dimensional ESCS is a compact manifold with finitely many conical singularities which is locally isometric to n away from the conical points, and near conical points it is isometric to a neighborhood of the vertex of a Euclidean cone, C α. Let P be a polygon, and let P be a copy of P, disjoint from P, and : P P be the identity map. We use for (P P )/ where we have identified the points of P and P under the map. There is a canonical extension of to an involution :. The surface is smooth everywhere except at the vertices which are isolated conical singularities. We note that the cone angles are doubled under this procedure, meaning that the cone angle in the surface is twice the interior angle at the corresponding vertex in the polygon. We consider the Friedrichs extension of the Euclidean Laplace operator on C0 (( ) 0), where ( ) 0 is with conical singularities removed. We will denote this extension simply by. There is a ellich theorem for (see [CT82a, CT82b]) which shows that the spectrum of is discrete, and the usual Weyl s asymptotic holds. In fact since the involution commutes with, there is an orthonormal basis (ONB) consisting of eigenfunctions of both operators, and. The eigenvalues of are ±1, and hence the joint eigenfunctions of and are even and odd eigenfunctions of with respect to. The even eigenfunctions of correspond to the eigenfunctions of the Neumann Laplacian on P, N P, and the odd eigenfunctions correspond to the ones of Dirichlet Laplacian, D P. It is now clear that counting multiplicities we have Spec = Spec D P Spec N P. It was shown by Hillairet [Hil02] (and in a more general setting by Wunsch [Wun02]) that SingSupp Tr e it {0} ±L To describe L precisely, we first need to describe the geodesics on an ESCS and to do so we need some definitions. The conical points are separated into two groups. A conical point is called non-diffractive if its angle is equal to 2π N for some positive integer N, otherwise it is called diffractive. We also use the same terminology for polygons except that non-diffractive angles are of the form π N. For example, if a trapezoid T is not a rectangle, then the top vertex with angle π β > π/2, is diffractive, while the bottom two vertices could be either diffractive or non-diffractive. The top vertex with angle π α is non-diffractive if and only if α = π 2.

9 NEUMANN ISOSPECTAL TAPEZOIDS 9 When a geodesic in a ESCS hits a non-diffractive conical singularity with cone angle 2π N, it continues on a straight line in the cone C 2π (which is isomorphic to 2 ), as an N-fold covering space of C 2π/N. Hence if the incoming angle of a geodesic is θ in, its outgoing angle θ out is Π N (π + θ in ) where Π N : C 2π C 2π/N is the natural covering map. In contrast, when a geodesic hits a diffractive conical point, it reflects according to Keller s democratic law of diffraction, meaning that it reflects in all possible directions, and we call θ out θ in the angle of diffraction. A geodesic is called diffractive if it goes through at least one diffractive singularity (see Figure 4). A geodesic geometrically diffracts at a diffractive conical point with angle α if it is locally the limit of a family of non-diffractive geodesics (see Figure 2), which happens when θ out θ in = ±π mod αz. All the above are defined similarly on polygons except that geodesics reflect on the edges according to the law of reflection. More precisely, geodesics on P are the projections of geodesics on under the natural folding projection map π : P. Then the following wavefront relations hold ([CT82a, CT82b]) for the integral kernels of the propagators e it and e it B P. (3.2) WF e it { (t, τ, x, ξ, y, η) T ( ) τ = ξ, Φ t (x, ξ) = (y, η) }, (3.3) WF e it B P { (t, τ, x, ξ, y, η) T ( P P ) τ = ξ, Φ t P (x, ξ) = (y, η) }, where Φ and Φ P are geodesics flows 5 on and P, respectively, and geodesics are defined above. Consequently, L and L P are defined to be the lengths of closed geodesics, where geodesics follow the above rules of diffractions. We point out that in fact (3.3) follows from (3.2) as we will discuss in the appendix. emark 2. We note that L L P but they are not necessarily equal. For example when P is a tall trapezoid, an easy observation shows that the length of the orthic triangle (see Figure 5) is in L P but it is not in L Singularities of wave trace on polygons. As we discussed before, with counting multiplicities we have Spec = Spec D P Spec N P, and therefore (3.4) Tr e it = Tr e it D P + Tr e it N P. This in particular shows that SingSupp Tr e it SingSupp Tr e it D P SingSupp Tr e it N P. emark 3. Again, this inclusion may not be an equality because for example the length of the orthic triangle in a tall trapezoid belongs to the singular support of both traces on the right hand side but it does not necessarily belong to the singular support of Tr e it. It turns out that the singularities of Dirichlet and Neumann wave traces at the length of orthic triangle cancel each other on the right hand side of (3.4). 5 Note that these flows are multi-valued maps (more precisely they are relations) when the geodesic hit the diffractive corners.

10 10 HAMID HEZAI, ZHIQIN LU, AND JULIE OWLETT Figure 5. The vertices of the orthic triangle are the feet of the heights of the triangle that extends a trapezoid. It does not exists if the trapezoid is too short or if it is acute, i.e., α + β π 2. Its linearized Poincare map is I, hence it is a non-degenerate orbit, and by a result of Guillemin- Melrose [GM79] it contributes a singularity of order to both Dirichlet and Neumann wave traces if there are no other periodic orbits of the same length with the same order of singularity. emark 4. We point out that a similar relationship holds between the Dirichlet and Neumann heat traces and the heat trace of as a ESCS. More precisely, Tr e t = Tr e t D P + Tr e t N P. The asymptotic expansion (3.1) was proved in [vdbs88] for the Dirichlet heat trace. We have not found a reference in literature stating the asymptotic expansion (3.1) for the Neumann heat trace, however it follows immediately from the above identity and the asymptotic expansion Tr e t = P 2πt + n k=1 π 2 θ 2 k 12πθ k + O(e c t ), t 0, proved by Kokotov (see Theorem 1 of [Kok11]) and Fursaev [Fur94]. A common way to measure the singularity of a tempered distribution is to study the decay and growth properties of its local Fourier transform (smoothed resolvent). The following propositions are crucial to prove our inverse problems for trapezoids. Proposition 5. Let T be a trapezoid that is not a rectangle. Suppose there are no other closed geodesics in T of length 2h or arbitrarily close to 2h, other than the one-parameter family in Figure 2. Let ˆρ(t) C0 () be a cutoff function supported near t = 2h whose support does not contain any lengths in {0} ±L T other than 2h. Then as k + ˆρ(t)e ikt Tr e it B e iπ/4 e 2ihk T dt = ˆρ(2h)A()k o(k 2 ), 4πh where B = D or N, and A() is the area of the inner rectangle of T.

11 NEUMANN ISOSPECTAL TAPEZOIDS 11 Corollary 6. If the conditions of Proposition 5 are satisfied then 2h and A() = bh, the area of the inner rectangle of T, are spectral invariants. The above proposition, with A() replaced by 2A(), was proved by Hillairet [Hil05] for the trace of the wave group of 2T. In fact it was proved in a more general context, namely for ESCSs and for any one-parameter family (a cylinder) of regular periodic geodesics whose boundary components consist of geodesics with only one diffraction (necessarily a geometric diffraction). Hence it immediately applies to the double of the trapezoid in Figure 2. However, recalling (3.4), it does not immediately imply anything about the asymptotics of the traces of the wave groups associated to D T nor N T. We emphasize that Hillairet s theorem implies that if both Dirichlet and Neumann spectra are known, then 2h and A() are spectral invariants. We do not wish to make this strong assumption, but we will nonetheless show using the method of reflections and a wavefront calculation that the Dirichlet and Neumann wave traces have an identical singularity at t = 2h, showing that indeed Proposition 5 follows from Hillairet s result. Note that this is special for the orbits in Figure 2 and does not necessarily hold for other orbits. For example as we will see in Proposition 7, the orbit in Figure 1 contributes a singularity at t = 2b to the trace of the Neumann wave group which is larger than the singularity at t = 2b of the trace of the Dirichlet wave group. emark 5. We note that as k we have ˆρ(t)e ikt Tr e it B P dt = O( k ). This is because by the Fourier inversion formula ˆρ(t)e ikt Tr e it ( ) B P dt = 2πTr ρ B P k. However since ρ is rapidly decaying near infinity, and since by the Weyl s law the eigenvalues grow linearly in dimension 2, the trace Tr ρ( B P k) decays rapidly as k. This, together with Proposition 5, shows that for trapezoids (i.e. P = T ) locally near t = 2h, the wave trace Tr e it B T belongs to the Sobolev spaces H s () for all s > 1 but does not belong to H 1 (). For this reason we say that t = 2h is a singularity of order 1 +. In general (for any Laplacian ) if Tr e it has an isolated singularity at t = t 0, and if for some ˆρ supported near t 0 we have as k + ˆρ(t)e ikt Tr e it dt = ce ikt 0 k a + o(k a ), for some a and nonzero constant c, then t 0 is a singularity of order (a )+. Near t = t 0 the wave trace belongs to H s () for all s > a but does not belong to H s () for s = a + 1 2, which motivates this definition of the order of the singularity. emark 6. The discussion in emark 5 also shows that as k + ˆρ(t)e ikt Tr cos(t )dt = 1 ˆρ(t)e ikt Tr e it dt + O(k ). 2 This shows a relationship between Tr cos(t ) and Tr e it.

12 12 HAMID HEZAI, ZHIQIN LU, AND JULIE OWLETT The next proposition concerns the diffractive orbit γ b in Figure 1. Proposition 7. Let T be a trapezoid with α π 2 and β π 2. Suppose there are no closed geodesics in T of length 2b other than γ b in Figure 1. Let ˆρ(t) C0 () be a cutoff function supported near t = 2b whose support does not contain any lengths in {0} ±L T other than 2b. Then as k + ˆρ(t)e ikt Tr e it B T dt = 4πiˆρ(2b)e 2bki C α,β k 1 + O(k 2 ), for B = Neumann, and ˆρ(t)e ikt Tr e it B T dt = O(k 2 ), for B = Dirichlet. Here the constant C α,β is given by (3.5) C α,β = π2 cot( 2π 2α ) cot( π 2 (π α)(π β) 2π 2β ). When α = π 2 and β π 2, as k + we have ˆρ(t)e ikt Tr e it B 1 T dt = (4πb) 2 ˆρ(2b)e πi 4 e 2bki C β k O(k 3 2 ), for B = Neumann, and for B = Dirichlet. Here ˆρ(t)e ikt Tr e it B T dt = O(k 3 2 ), (3.6) C β = π2 cot( 2π 2β ). π β As a quick corollary we obtain a new angle invariant. Corollary 8. Let T be a trapezoid such that there no orbits of length 2b other than the bouncing ball orbit γ b corresponding to the top edge. If β α < π 2, then 2b and C α,β defined by (3.5) are spectral invariants for the Neumann spectrum. If β < α = π 2, then 2b and C β defined by (3.6) are spectral invariants for the Neumann spectrum. emark 7. Again, this proposition follows from Hillairet [Hil05] with required modifications to separate the Dirichlet and Neumann wave traces, which we will discuss in the proof. emark 8. Note that the principal coefficient C α,β as α π 2. This is consistent with the order of singularity being stronger in the case α = π 2 as the above proposition proves. A striking fact about diffraction in general is that each diffraction reduces the order of singularity, so orbits with many diffractions contribute milder singularities to the wave trace. This is proved for ESCSs in [Hil05].

13 NEUMANN ISOSPECTAL TAPEZOIDS 13 emark 9. The following proposition might be useful when one studies the isospectral problem on trapezoids for the Dirichlet Laplacian. It concerns the wave trace contribution of the orthic orbit in Figure 5. Up to the principal part, it is a direct consequence of Guillemin-Melrose trace formula [GM79] for simple and non-degenerate periodic orbits. We state the proposition without the proof because we do not use it in this paper. In the following we use l F for the length of the orthic (also called Fagnano) triangle which by the notations of Figure 3 equals 2B sin α sin β. Proposition 9. Let T be a trapezoid with α and β π 2, and α + β > π 2. Suppose T is tall enough that the orthic triangle lies in T and is non-diffractive as in Figure 5. Suppose there are no other closed geodesics in T of length l F, other than the orthic triangle. Let ˆρ(t) C0 () be a cutoff function supported near t = l F whose support does not contain any lengths in {0} ±L T other than l F. Then as k + ˆρ(t)e ikt Tr e it B T dt ( 1) s B l F e iklf ˆρ(l F ) c j k j, where s B = 0 if B = D and s B = 1 if B = N. The constant c 0 is nonzero. Moreover, the constants {c j } j=0 depend only on l F and are independent of B. Hence, the invariants {c j } j=0 do not introduce any spectral invariants other than l F. Corollary 10. Under the conditions of Proposition 9, l F = 2B sin α sin β is a spectral invariant for both Dirichlet and Neumann spectra. j=0 4. Proofs of Propositions 5 and 7 Let P be a polygonal domain and define and the involution map : as in the previous section. We denote and we use U (t) = e it, U D P (t) = e it D P, U N P (t) = e it N P, U (t, x, y), U D P (t, x, y), U N P (t, x, y) for their integral kernels. The following proposition expresses the Dirichlet and Neumann wave kernels in terms of U (t, x, y). Proposition 11. For all t and all x, y P : U D P (t, x, y) = 1 2 (U (t, x, y) U (t, x, y)), U N P (t, x, y) = 1 2 (U (t, x, y) + U (t, x, y)). The proof is obvious from the expansion of U (t, x, y) in terms of an ONB of eigenfunctions of consisting of even and odd eigenfunctions with respect to. As an immediate corollary we obtain:

14 14 HAMID HEZAI, ZHIQIN LU, AND JULIE OWLETT Corollary 12. We have the following equalities, as tempered distributions: Tr UP D (t) = 1 ( ) Tr U (t) U (t, x, x)dx, 2 Tr UP N (t) = 1 ( ) Tr U (t) + U (t, x, x)dx. 2 To prove Propositions 5 and 7, we use Corollary 12 to reduce the problem to studying the asymptotics of the tempered distributions Tr U (t) and U (t, x, x)dx. Theorem 2 of Hillairet [Hil05] gives the asymptotics of the trace Tr U (t), but the term U (t, x, x)dx is a new ingredient which is relatively easy to study. Before going over the proofs let us introduce our coordinates: 4.1. Local coordinates on. We first write = int P int P ( P \C) C, where C is the set of conical singularities. For points in intp or intp, we will use the Cartesian coordinate induced by P as an open set in 2. Note that in these coordinates the map : T ( ) T ( ) becomes (x, ξ) (x, ξ) for x int(p ) int(p ). In this way we identify T x and T x for x int(p ) int(p ). For a point x in P \C we first consider the edge, say of length l, that contains x and choose our coordinates near x in such a way that this edge is identified with the open line segment in 2 connecting the points (0, 0) and (l, 0), P is below the x 1 -axis and P is above the x 1 axis. In these coordinates the map : T ( ) T ( ) becomes (x, ξ) (x, ξ ) for x P \C, where ξ is the natural reflection about the x 1 axis defined by ξ = (ξ 1, ξ 2 ) given ξ = (ξ 1, ξ 2 ). Note that since is an involution x = 1 x. If x C, i.e. x is a conical singularity, we will consider the Euclidean cone C α as a neighborhood of x. Here α is the cone angle at x as a conical point on the double polygon A parametrix for the wave kernel near a regular diffractive orbit. To prove propositions 7 and 5 we will need the following theorem of [Hil05] which gives a parametrix for U (t, x, y) microlocalized near a regular (i.e. non-geometric) diffractive geodesic connecting a point x 0 to a point y 0. Theorem 13 (Hillairet). Let P be a polygon and γ be a diffractive geodesic on of length t 0, with initial and terminal points x 0 and y 0 in, going through n diffractions at conical points p 1, p 2,..., p n of angles α 1, α 2,..., α n, with angles of diffractions β 1, β 2,..., β n all different from ±π. Let (r, θ) and (, Θ) be polar coordinates centered at p 1 and p n, chosen in such a way that the line segments x 0 p 1 and p n y 0 correspond to θ = 0 and Θ = 0 respectively. Then microlocally 6 near γ, U is a FIO, and near (t 0, x 0, y 0 ) and away from the conical points, has a parametrix of the form Ũ,γ (t, x, y) = 6 See Section 2.3 of [Hil05]. ξ>0 e iξ(t r(x) (y) n 1 j=1 L j) aγ (t, x, y, ξ)dξ,

15 NEUMANN ISOSPECTAL TAPEZOIDS 15 where L j = d(p j, p j+1 ) and as ξ + the amplitude a γ is a classical symbol (see the Appendix) of the form a γ (t, x, y, ξ) a m (t, x, y)ξ n 1 2 m with leading term Here and where m=0 a 0 (t, x, y) = (2π) (n 3)/2 e (n 1)iπ/4 S γ(x, y) Lγ (x, y). S γ (x, y) = S α1 (β 1 θ(x))s α2 (β 2 ) S αn 1 (β n 1 )S αn (Θ(y) β n ), which at η = 0 simplifies to S δ (0) = 1 δ L γ (x, y) = r(x)l 1 L 2 L n 1 (y), sin( 2π2 δ S δ (η) = ) 2δ sin( π δ (π + η)) sin( π δ (π η)), cot( π2 δ ). Proof of Proposition 7. Theorem 2 of [Hil05] gives the asymptotics for Tr U (t) near t = 2b, which are exactly those given in Proposition 7. Hence, by Corollary 12, to prove this proposition it suffices to show that ˆρ(t)e ikt U (t, x, x)dxdt = ˆρ(t)e ikt U (t, x, x)dxdt + O(k n 2 1 ) (4.1) = ˆρ(t)e ikt Tr U (t) dt + O(k n 2 1 ), where n = 2 if β α π 2, and n = 1 when β < α = π 2. We note that n corresponds to the number of diffractions because there is no diffraction at the top left vertex when α = π 2. We now apply Theorem 13 to γ b in Figure 1. First we choose the Cartesian coordinates so that the top left corner of T is at C 1 := (0, 0), and the top right corner is at C 2 := (b, 0), hence γ b lies on the x 1 axis. We then reflect T about the x 1 axis. In particular, this would give a natural neighborhood of the interior of γ, and the involution map becomes (x 1, x 2 ) = (x 1, x 2 ). We also choose three cutoff functions, χ C1, χ C2, and χ on 2T, all invariant under. These are chosen to satisfy: χ C1 + χ C2 + χ = 1 near γ b ; χ C1 and χ C2 are supported in small Euclidean balls B ɛ (C 1 ) and B ɛ (C 2 ) of C 1 and C 2 with respect to the Euclidean metrics on the cones with vertices C 1 and C 2, respectively; and χ is supported away from C 1 and C 2. Since microlocally near the orbit γ we have U (t, x, y) = Ũ,γ(t, x, y), by a wavefront calculation we can see that ˆρ(t)e ikt Tr(U (t) χ) dt = ˆρ(t)e ikt Tr(Ũ,γ(t) χ) dt + O(k ). Next, we substitute the parametrix given for Ũ,γ(t, x, y) in Theorem 13. An immediate observation shows that (x) = (x), and Θ(x) = Θ(x). Since β 2 = 0 and S α2 is an even function, this implies that a 0 (t, x, x) = a 0 (t, x, x).

16 16 HAMID HEZAI, ZHIQIN LU, AND JULIE OWLETT Hence the phase functions and the leading terms of the amplitudes of the oscillatory integrals Ũ,γ(t, x, x) and Ũ,γ(t, x, x) agree on Supp χ. By the stationary phase lemma, as performed in the proof of Theorem 5 of [Hil05], we get ˆρ(t)e ikt Tr(Ũ,γ(t) χ) dt = ˆρ(t)e ikt Tr(Ũ,γ(t) χ) dt + O(k n 2 1 ). This implies (4.1) with the cutoff χ inserted. Near the conical points C 1 and C 2 we can use the cyclicity of the trace, as used by [Dur88, Hil05, FHH15, FW17] to move the support of the integrands away from the conical points and reduce to the setting above. We will only perform this procedure for the conical point C 1 since the argument is identical for C 2. We first use the group property of U (t) and the cyclicity of the trace to write Tr(U (t) χ C1 ) = Tr(U (t 0 )U (t t 0 ) χ C1 ) = Tr(U (t t 0 ) χ C1 U (t 0 )), where t 0 > 0 is fixed and is sufficiently small. We now parametrize the closed geodesic γ by g(t), t [0, 2b], such that g(0) = C 1. Then we choose an appropriate cuttoff function ψ near g( t 0 ), invariant under, so that near t = 2b Tr(U (t t 0 ) χ C1 U (t 0 )) Tr(U (t t 0 ) χ C1 U (t 0 )ψ), is smooth. This is possible by the finite speed of propagation of singularities and a wavefront computation because the operator χ C1 U (t 0 )(1 ψ), is smooth. Note also that the cutoff function ψ, and the values ɛ and t 0 are chosen so that ψ is supported away from the conical point C 1. Next, we write χ C1 = 1 (1 χ C1 ) and obtain (up to a smooth function near t = 2b) Tr(U (t) χ C1 ) = Tr(U (t t 0 ) U (t 0 )ψ) Tr(U (t t 0 ) (1 χ C1 ) U (t 0 )ψ). Since commutes with U (t) and χ C1, we get Tr(U (t) χ C1 ) = Tr(U (t) ψ) Tr(U (t t 0 )(1 χ C1 ) U (t 0 ) ψ), up to a smooth function near t = 2b. Since both traces in the above expression are microlocalized near the regular diffractive orbit γ, and since ψ and 1 χ C1 are supported away from the conical points, we can replace U (t), U (t 0 ), and U (t t 0 ) with their parametrices provided by Theorem 13. We may then follow the same argument as we did for the cutoff χ to show that the leading terms of the singularity expansions at t = 2b of Tr(U (t) χ C1 ) and Tr(U (t)χ C1 ) agree. This reduces our problem to studying the principal term of Tr(U (t)χ C1 ) at t = 2b which was done in [Hil05]. Proof of Proposition 5. By Corollary 12, we know that as distributions Tr UP D (t) = 1 ( ) Tr U (t) U (t, x, x)dx, 2 Tr UP N (t) = 1 ( ) Tr U (t) + U (t, x, x)dx. 2

17 NEUMANN ISOSPECTAL TAPEZOIDS 17 Now let ɛ > 0 such that there are no lengths other than 2h in the interval (2h ɛ, 2h + ɛ) and assume ˆρ(t) is supported in (2h ɛ, 2h+ɛ). We know by Theorem 2 of Hillairet [Hil05] that ˆρ(t)e ikt Tr U (t)dt = eiπ/4 e 2ihk ˆρ(2h)Ak o(k 2 ), 4πh where A is the area that the family sweeps in. Note that A = 2A(), where A() is the area of the inner rectangle of the trapezoid. Hence to prove Proposition 5 it suffices to prove that ( ) (4.2) ˆρ(t)e ikt U (t, x, x)dx dt = o(k 1 2 ). To do this we first study ( (4.3) WF U (t, x, x)dx) (2h ɛ, 2h + ɛ) We follow the argument as in [DG75] and write U (t, x, x)dx = Π U, where is the pullback by the diagonal map :, and Π is the pushforward by the projection map Π :. The same wavefront calculations as on page 44 of [DG75] (see also the appendix) show that ( ) WF U (t, x, x)dx { (t, τ) τ > 0, (x, ξ) T ( ), Φ t (x, ξ) = (x, ξ) }. Now suppose t 0 (2h ɛ, 2h + ɛ), and Φ t 0 (x, ξ) = (x, ξ) for some (x, ξ) T ( ). We recall again that Φ t is a relation and Φt 0 (x, ξ) = (x, ξ) just means that there is a geodesic (diffractive or non-diffractive) in T ( ) of length t 0 that starts at (x, ξ) and ends at (x, ξ). We claim that in our situation only diffractive orbits lying in the two boundary components of the cylinder corresponding to the 1-parameter family of parallel heights can contribute to (4.3). To see this suppose there is a geodesic {Φ t (x, ξ)} t [0,t 0 ] connecting (x, ξ) to (x, ξ). Then the projection of this geodesic segment onto T P, under the the natural projection π : P, is a closed geodesic in P (as a billiard table) of length t 0. However, by assumption the only periodic orbits in P of length in the interval (2h ɛ, 2h + ɛ) must belong to the one-parameter family in Figure 2. Hence t 0 = 2h, and the projection of {Φ t (x, ξ)} t [0,2h] onto P must be a bouncing ball orbit that is parallel to the altitude of the trapezoid. If this bouncing ball orbit belongs to the interior of the family (hence non-diffractive) then it corresponds to a periodic trajectory on. Hence Φ 2h (x, ξ) = (x, ξ). On the other hand, Φ2h (x, ξ) = (x, ξ). Therefore we must have (x, ξ) = (x, ξ). This is a contradiction because the only points in the phase space satisfying (x, ξ) = (x, ξ) are those with x P and ξ tangential to P at x but the orbit in this problem is transversal to the boundary. On the other hand, if the bouncing ball orbit in P corresponds to one of boundaries of the family then indeed it can be a diffractive geodesic in of length 2h that starts at x and ends at (x). To describe this geodesic, which we call γ, take x P (or P ) on the boundary of the cylinder and ξ parallel to the height. Follow the geodesic starting at (x, ξ) in until it hits the conical singularity. Then the geodesic diffracts with a zero angle of diffraction and backtracks

18 18 HAMID HEZAI, ZHIQIN LU, AND JULIE OWLETT on until it reaches (x, ξ). These are precisely the only geodesics that contribute to (4.3) because if the boundary geodesics diffract geometrically on, then we can draw a contradiction as we did for the non-diffractive ones. Since the described geodesics are regular diffractive geodesics on (note that they are geometric when projected onto P ) we can use the parametrix provided by Theorem (13) to compute the leading singularity of U (t, x, x)dx at t = 2h. Because the computations for these diffractive geodesics are identical we only present the ones corresponding to the diffraction at the top left corner of the trapezoid. In fact the computations are very similar to those in [Hil05] near a periodic geodesic with only one diffraction except that in our case the orbit is not periodic in and travels from x to x. As in the proof of Proposition 7, we first fix coordinates so that the top left corner of T is at C = (0, 0), and the top right corner is at (b, 0). Hence γ lies on the x 2 axis. We then reflect T about its base, i.e. reflect about the line x 2 = h. Note that the involution map becomes (x 1, x 2 ) = (x 1, 2h x 2 ). We also choose cutoff functions, χ C and χ on 2T, all invariant under so that χ C + χ = 1 near γ; χ C is supported in a small Euclidean ball B ɛ (C) centered at C and χ is supported away from C. Since microlocally near the orbit γ we have U (t, x, y) = Ũ,γ(t, x, y), we can write ˆρ(t)e ikt U (t, x, x)χ(x) dxdt = ˆρ(t)e ikt Ũ,γ (t, x, x)χ(x) dxdt+o(k ). Therefore by the parametrix for Ũ,γ(t, x, y) from Theorem 13 we need to study the asymptotic behavior as k of I(k) = ˆρ(t)e ikt+iξ(t r(x) (x)) a γ (t, x, x, ξ)χ(x) dξdxdt. ξ>0 Changing variables ξ kξ, turns this into k ˆρ(t)e ik( t+ξ(t r f(r,θ))) a γ (t, x(r, θ), (x(r, θ)), kξ)χ(x(r, θ)) r dξdrdθdt, where f(r, θ) = (x). We shall perform the stationary phase lemma to this oscillatory integral in the variables (t, θ, ξ) uniformly with respect to r. To find the critical points of the phase function we need to solve the system Since we have f(r, θ) = (x) = Ψ = t + ξ(t r f(r, θ)), 1 + ξ = 0, d dθ f(r, θ) = 0, t r f(r, θ) = 0. x (2h + x 2) 2 = r 2 + 4h 2 4hr cos θ, d f(r, θ) = 0 θ = 0. dθ

19 NEUMANN ISOSPECTAL TAPEZOIDS 19 Therefore, as a function of (t, θ, ξ), Ψ has only one critical point at (t, θ, ξ) = (2h, 0, 1). Also, a simple computation reveals that det Hess(Ψ)(2h, 0, 1) = d2 f (r, 0) = 2hr dθ2 2h r, which is uniformly away from zero in the support of χ. Hence (2h, 0, 1) is a (uniform in r) non-degenerate critical point. By the stationary phase lemma (uniform in r) and the asymptotic of a γ from Theorem 13, we get I(k) = O(k 1 2 ). This is in fact much better than what we need in regard to (4.2). To prove that the same estimate, namely ˆρ(t)e ikt U (t, x, x)χ C (x) dxdt = O(k 1 2 ), holds near the conical point C, we use the cyclicity trick used in the proof of Proposition 7 to move the support of the integrand away from the conical point. Since the analysis is very similar, we omit the details. 5. Proof of Theorem 1 Our first simple observation is Proposition 14. The length of any periodic orbit in a trapezoidal table T is strictly larger than 2h or 2b unless the orbit is a bouncing ball corresponding to one of the altitudes or it is the bouncing ball between the top two vertices. Proof. We note that any closed diffractive or non-diffractive geodesic in T that starts from the top edge (including the corners) and is transversal (i.e. not tangent) to the top edge must be of length strictly larger than 2h unless it is parallel to the altitude. Furthermore, any geodesic that touches the left and right edges (including the corners) must be of length larger than 2b unless it is the bouncing ball orbit γ b. If a geodesic touches the bottom edge and the right edge (respectively, left edge) then it must also visit the top edge or the left edge (respectively, right edge) and hence its length is larger than 2h or 2b. The main theorem follows immediately by combining the following four propositions and the heat trace invariants A, L, and q α,β. Proposition 15. Let T 1 and T 2 be two trapezoids with the same Neumann spectra. If T 1 is a rectangle, then T 2 is a rectangle that is congruent to T 1. Proposition 16. Let T 1 and T 2 be two non-rectangular trapezoids with the same Neumann spectra. If h(t 1 ) b(t 1 ), then h(t 1 ) = h(t 2 ) and b(t 1 ) = b(t 2 ).

20 20 HAMID HEZAI, ZHIQIN LU, AND JULIE OWLETT If b(t 1 ) < h(t 1 ) then b(t 1 ) = b(t 2 ). In addition if α(t 1 ) π 2, then α(t 2) π 2, and C α(t1 ),β(t 1 ) = C α(t2 ),β(t 2 ), where C α,β is defined by (3.5). Moreover, if α(t 1 ) = π 2, then α(t 2 ) = π 2 and C β(t 1 ) = C β(t2 ) where C β is given by (3.6). Proposition 17. If two trapezoids have the same area A, perimeter L, height h, and b, then they are congruent up to rigid motions. Proposition 18. If two trapezoids, with α(t 1 ) and α(t 2 ) π 2, have the same area A, angle invariant q α,β, the same b, and the same C α,β, then they are congruent up to rigid motions. Moreover, if two non-rectangular trapezoids, with α(t 1 ) = α(t 2 ) = π 2, have the same area A, angle invariant q α,β, and the same b, then they are congruent up to rigid motions. We now give the proofs of these propositions. Proof of Proposition 15. The angle invariant satisfies q 8 π 2, with equality if and only if the trapezoid is a rectangle. Since T 1 and T 2 are isospectral, they have the same angle invariant. Consequently, since T 1 is a rectangle, T 2 is as well. Furthermore, by isospectrality, T 1 and T 2 have the same area and perimeter, and these uniquely determine a rectangle up to rigid motions. Proof of Proposition 16. First suppose h(t 1 ) < b(t 1 ). Then by Proposition 14, 2h(T 1 ) is the shortest length in L T1 and there are no orbits other than the one-parameter family of altitudes having the same length. Hence by Proposition 5, both Dirichlet and Neumann wave traces of T 1 have a singularity of order 1 + at t = 2h(T 1 ) where up to a constant the leading coefficient equals b(t 1 )h(t 1 ), the area of the inner rectangle. Since T 1 and T 2 are isospectral, the same must hold for the wave trace of T 2. In particular, we must have 2h(T 1 ) = 2h(T 2 ), and b(t 1 )h(t 1 ) = b(t 2 )h(t 2 ), so that b(t 1 ) = b(t 2 ). If b(t 1 ) < h(t 1 ), then again using Proposition 14, 2b(T 1 ) is the shortest length in L T1. By Proposition 7, the Neumann wave trace of T 1 has a singularity at t = 2b(T 1 ). If the order of this singularity is ( 1 2 )+, then we know that α(t 1 ) π 2, and the same type of singularity is found in the Neumann wave trace of T 2, thus α(t 2 ) π 2 as well. Furthermore, 2b(T 1 ) = 2b(T 2 ), and C α(t1 ),β(t 1 ) = C α(t2 ),β(t 2 ). Similarly, if the order of this singularity is 0 +, then we know that there is only one diffraction, meaning that α(t 1 ) = π 2. Since the singularity in the wave trace of T 2 must be the same, we must have α(t 2 ) = π 2, 2b(T 1 ) = 2b(T 2 ), and C β(t1 ) = C β(t2 ). When h(t 1 ) = b(t 1 ), since there are no orbits of length 2h(T 1 ) = 2b(T 1 ) other than γ h and γ b, the singularities of the wave trace at t = 2h(T 1 ) and t = 2b(T 1 ) add. This is because in fact Propositions 5 and 7 are also valid microlocally near their corresponding orbits. In this case since the singularity at t = 2h(T 1 ) is larger, and it contributes to the leading singularity of the wave trace. Hence as in the first case, we have 2h(T 1 ) = 2h(T 2 ), and b(t 1 )h(t 1 ) = b(t 2 )h(t 2 ), thus b(t 1 ) = b(t 2 ).