## Abstract

We consider the inverse spectral problem for the Laplace operator on triangles with Dirichlet boundary conditions, providing numerical evidence to the effect that the eigenvalue triplet (*λ*_{1},*λ*_{2},*λ*_{3}) is sufficient to determine a triangle uniquely. On the other hand, we show that other combinations such as (*λ*_{1},*λ*_{2},*λ*_{4}) will not be enough, and that there will exist at least two triangles with the same values on these triplets.

## 1. Introduction

Ever since Marc Kac (1966) posed the question of whether or not it would be possible to identify a domain by its spectrum, much effort has been put into the study of such problems. The problem had actually been around for some years by the time Kac’s paper appeared, and for the case of compact manifolds it had already been shown by Milnor (1964) that there existed sixteenth-dimensional tori which were isospectral but not isometric. Although it has been known since the work of Gordon *et al.* (1992) that in fact the spectrum does not uniquely determine the domain even in the case of planar domains, there are not many results in the way of defining broad classes of domains within which the spectrum does determine one element in a unique way. Even such a restrictive condition as convexity by itself is not sufficient in general, as may be seen from the result by Gordon & Webb (1994) showing that there exist convex domains in four-dimensional Euclidean space which are isospectral but not isometric.

On the other hand, if one restricts the universe of possible domains to those having an analytical boundary and some symmetry, it has been shown in Zelditch (2009) that the full spectrum will then be sufficient to determine the domain in a unique way. A similar result for the class of Euclidean triangles was obtained by Durso (1990) where it was proved that the knowledge of the entire spectrum also allows for the determination of a triangle uniquely up to congruence. The study of Chang & DeTurck (1989) stated that the spectral identification of a triangle is possible from a finite number of eigenvalues, this number being determined by the first two eigenvalues. Related to this, and in a more recent paper, Laugesen & Siudeja (2009) raised the question of knowing whether the first three Dirichlet eigenvalues *λ*_{1}, *λ*_{2} and *λ*_{3} would uniquely determine (or not) a triangle.

The purpose of the current paper is to address these and related questions from a computational point of view. More precisely, and in a similar spirit to our previous work related to spectral problems (Antunes & Freitas 2006, 2008), we present a numerical study of the inverse spectral problem for Euclidean triangles and give evidence to the fact that these are indeed determined by their first three eigenvalues. On the other hand, we shall see that not all triplets of eigenvalues are enough to determine a triangle, one such case being the triplet (*λ*_{1},*λ*_{2},*λ*_{4}) for which there exist combinations corresponding to at least two triangles. In fact, the first three eigenvalues seem to be the exception rather than the rule, in that all other triplets which were analysed in our study did not determine a triangle in a unique way. We stress the fact that no data on the triangle are assumed to be known from the start, and that in particular the area is not known *a priori*.

As a by-product of our work, we determine the ranges for certain quotients of eigenvalues and also the minimizers for the first four eigenvalues of triangles with fixed area.

We would like to point out that although it might be argued that the class of triangles is very specific and simple, it has recently become apparent that triangles do play an important role in a spectral context. On the one hand, they appear as solutions to some shape optimization problems such as those studied in Antunes & Freitas (2008) and Harrell & Henrot (2010). On the other hand, they are in a class of their own when it comes to spectral properties related to isoperimetric inequalities. One such example is illustrated by the following upper bound proved by Pólya (1960)
which is valid for general planar doubly connected domains—here *A* and *L* denote the area and the perimeter of *Ω*, respectively. Although the constant *π*^{2}/4 cannot be improved by restricting ourselves to quadrilaterals (even if we take them to be convex), for triangles it was conjectured in Antunes & Freitas (2006) and proved the study of Siudeja (2007) that

In order to carry out a study of this type, it is necessary to process a large number of domains, for which one must have an efficient algorithm to compute the eigenvalues. In this case, the computation of the eigenvalues was done using a mesh-free numerical method known as method of fundamental solutions (MFSs), as studied in Alves & Antunes (2005). In some stiffer cases such as for thin triangles, we used an enriched version of the MFS as described in Antunes & Valtchev (2010) instead. The optimization problems were solved by the gradient method, using the MFS as forward solver.

The paper is organized as follows. In §2, we choose a parametrization of triangles which is appropriate for our purposes. In particular, we define a set containing one and only one representant of triangles with angles (*α*,*β*,*γ*). Since we will be dealing with quantities involving the first four eigenvalues, we begin by considering the minimization of the first four Dirichlet eigenvalues for triangles with fixed area. Section 3 then deals with the issues of existence and uniqueness of a triangle given a spectral triplet (*λ*_{1},*λ*_{2},*λ*_{3}), and of examples of other spectral triplets for which uniqueness does not hold. We also include some remarks on the Neumann problem for which we show that, given any integer *n*, there are examples of rectangles and circular sectors where *n* eigenvalues alone do not determine the domain. We then consider the case of triangles, but here the situation is not so clear cut. Finally, in §4 we discuss the results obtained.

## 2. Parametrization of triangles

In order to study the existence and uniqueness for the inverse spectral problem, we need to choose an appropriate way of representing triangles. Since three parameters are enough to define a triangle uniquely up to congruence, the question is thus more one of which parameters to choose. We further remark that a key ingredient for the inverse problem is the fact that the eigenvalues of a triangle depend continuously on the triangle itself and thus on any reasonable set of three parameters defining the triangle, such as the lengths of the three sides.

We begin by recalling the definition introduced in Antunes & Freitas (2008) to distinguish between two types of isosceles triangles depending on whether the angle formed by the equal sides is less than or larger than *π*/3. While in that paper we referred to these triangles as being of type I and II, here we shall follow the notation used in Laugesen & Siudeja (2010).

### Definition 2.1

The aperture of an isosceles triangle is the angle between its equal sides. We say a triangle is *subequilateral* if it is isosceles with aperture less than *π*/3, and *superequilateral* if it is isosceles with aperture greater than *π*/3.

### (a) The admissible region

In order to study the inverse problem of determining a triangle from the knowledge of its Dirichlet eigenvalues, it is convenient to define a set of triangles , such that given an arbitrary triangle *T* we can identify one (and only one) triangle in which is similar to *T*. We shall now construct one such set.

Let *R* be the region plotted in figure 1 which is defined by
We have ∂*R*=*Γ*_{0}∪*Γ*_{1}∪*Γ*_{2} where
On each of these parts of the boundary of *R* we will have particular types of triangles.

### Definition 2.2

We will say that a triangle is *admissible* if it is defined by the vertices *v*_{1}=(1,0), *v*_{2}=(−1,0) and *v*_{3}=(*x*,*y*)∈*R*. The set of all admissible triangles will be denoted by .

It follows easily that for an arbitrary triangle *T* there is one (and only one) triangle , which is similar to *T*. This fact makes the set suitable for the study of the inverse spectral problem for triangles. For a triangle belonging to the length of its sides is given by *l*_{1}=|*v*_{1}−*v*_{3}|, *l*_{2}=|*v*_{3}−*v*_{2}| and *l*_{3}=|*v*_{1}−*v*_{2}|=2 (figure 1). We have
2.1and
2.2
By definition of the region *R*, then 0<*l*_{1}≤*l*_{2}≤*l*_{3}. Moreover, by equations (2.1) and (2.2) we note that the subequilateral and superequilateral triangles belong (respectively) to the portions of the boundary *Γ*_{1} and *Γ*_{2} (figure 1).

We shall now consider a discretization of the region *R*_{N}:=*R*∩([0,1]×[0.4,1.74]) with a uniform 101×101 grid, and for each point in this grid, corresponding to an admissible triangle , we compute , *i*=1,…,4. The numerical calculation of the eigenvalues was done using the MFS or in some cases with an enriched version of the MFS. The solution of the boundary value problem is approximated by a linear combination of some base functions that satisfy the partial differential equation of the problem,
for some constant *λ*. Thus, the approximations for the eigenvalues are the values of *λ* for which we can determine a linear combination (which is not the zero function) fitting the null boundary conditions of the eigenvalue problem (see Alves & Antunes (2005) and Antunes & Valtchev (2010) for details).

As a by-product of our work, we numerically determined the triangles minimizing *λ*_{i} (*i*=1,…,4) among all triangles with unit area, as well as the triangles for which the second and third eigenvalues have a multiplicity two. This task was done using a classical gradient method (or steepest descent) for which the optimization problem is numerically solved by determining, in an iterative way, the greatest decrease in the objective function along the direction defined by its gradient. The results obtained indicate that the minimizers are all isosceles triangles, which are shown in table 1 as a function of the aperture of the optimizer. The equilateral triangle minimizes the first and third eigenvalues. The result for the first eigenvalue was already proved by Pólya & Szegő (1951), while the result for the third eigenvalue is not surprising, taking into account that among general planar domains, the ball is conjectured to be the minimizer (cf. Bucur & Henrot 2000; Oudet 2004). The minimizer for the second eigenvalue is a subequilateral triangle that may be related to the minimizer among general convex domains, which is known to be a stadium-like domain, as studied by Henrot and Oudet (see §4*b* of Henrot’s book (Henrot 2006) or Oudet (2004). Moreover, the numerical results gathered indicate that *λ*_{2}=*λ*_{3} only for the equilateral triangle, while *λ*_{3}=*λ*_{4} only for a subequilateral triangle which is also the minimizer of *λ*_{4}.

## 3. The inverse eigenvalue problem

We are now finally ready to address the inverse spectral problem, which we will formulate as follows

### Problem 3.1

Given , i = 1,2,3 with ℓ_{1}<ℓ_{2}≤ℓ_{3},

Is there a triangle

*T*, for which*λ*_{i}(*T*)=ℓ_{i}, i = 1,2,3?If yes, is

*T*unique up to congruence?

These two points will be the object of discussion of the next two sections.

### (a) Existence of solution for the inverse eigenvalue problem for triangles

In this section, we consider the first part of problem 3.1. From previous results it is known that in order to obtain (at least) one solution of problem 3.1 there are some restrictions on the values ℓ_{i} to be taken into consideration. For example Ashbaugh & Benguria (1991) proved that for a bounded planar domain *Ω* (in particular for a triangle), we have
3.1
where *j*_{n,k} is the *k*th positive zero of the Bessel function *J*_{n}. This yields the restriction
which although not sharp for triangles shows that it is important to know what the admissible values for ℓ_{i}, *i*=1,2,3 are, i.e. to study the range of the first three Dirichlet eigenvalues of triangles. A similar study for general domains was performed in Levitin & Yagudin (2003). We begin with the range of the first Dirichlet eigenvalue. Define
and
and similar sets for isosceles triangles
and
We will use the notation
where denotes any of the domains defined above. The Faber–Krahn inequality applied to planar domains of unit area (Faber 1923; Krahn 1924) yields
Moreover, if we denote by a rectangle with unit area whose largest side length is equal to *L*, we know that
and thus *λ*_{1}(*Ω*) is not bounded from above within and thus

Some similar results exist for triangles. Pólya & Szegő (1951) proved that the equilateral triangle minimizes *λ*_{1} over all triangles with the same area. This can be written as
Again it is easy to check that *λ*_{1} is not bounded from above within each of the relevant sets of triangles with fixed area (see, for instance, Freitas 2007) and then
3.2

Now we consider the quotient of Dirichlet eigenvalues,
and for any set of planar domains we will use the notation
The Ashbaugh–Benguria bound (3.1) allows us to conclude that
but this is clearly not optimal for triangles. Some recent advances in this direction were obtained by Siudeja (in press), who proved that, among all acute triangles, *ξ*_{2,1} is maximized by the equilateral triangle. It had been conjectured in Antunes & Freitas (2008) that this result is actually valid for any triangle, and more generally that the quantity *ξ*_{2,1} over each class of polygons is maximized by the regular polygon. Still regarding results for *ξ*_{2,1} within the class of triangles, we recall that several asymptotic expansions were obtained in Freitas (2007), and, in particular, if *T*_{S} is a superequilateral triangle with aperture *π*−*β*, then
where *a*_{1}≈−2.33811 and *a*_{1}′≈−1.01879 are, respectively, the first negative zero of the Airy function of the first kind and of its first derivative.

We shall now proceed to present the results obtained numerically for this quantity over triangles. In figure 2, we plot *ξ*_{2,1}(*T*) as a function of *λ*_{1}(*T*). Points corresponding to subequilateral and superequilateral triangles are marked in blue and red, respectively, and are illustrated by isosceles triangles in the figure. By equation (3.2), we know that for each there exists subequilateral and superequilateral triangles, *τ*^{ℓ1}_{s} and *τ*^{ℓ1}_{S}, respectively, such that . These results together with figure 2 suggest

### Conjecture 3.2

Let *T* be an arbitrary triangle for which *λ*_{1}(*T*)=ℓ_{1}. Then we have
Moreover
with equality if and only if *T* is equilateral.

Under conjecture 3.2, we have
3.3
The last equality in equation (3.3) is actually a theorem proved by Siudeja (in press). In figure 3, we plot *ξ*_{3,1}(*T*) as a function of *ξ*_{2,1}(*T*). Again the blue (and red) points correspond to subequilateral (and superequilateral) triangles. In a similar fashion, if equation (3.3) holds, for each *q*_{2,1}∈[1,7/3] there exist subequilateral and superequilateral triangles *σ*^{q2,1}_{s} and *σ*^{q2,1}_{s}, such that . Moreover, the triangle maximizing the quotient *ξ*_{3,1} is the same which minimizes *λ*_{4}, and for which we have *λ*_{4}=*λ*_{3}. This numerical optimizer and the corresponding value of *ξ*_{3,1} are shown in table 2. We note that the optimal value is attained for the triangle , being thus an improvement upon the corresponding value of 2.827 presented in Levitin & Yagudin (2003).

These numerical results suggest that the following result holds.

### Conjecture 3.3

Let *T* be an arbitrary triangle with *ξ*_{2,1}(*T*)=*q*_{2,1}. Then we have

Moreover
with equality if and only if *T* is the subequilateral triangle with aperture approximately equal to 0.155923*π*.

If conjectures 3.2 and 3.3 are true, then the range of the first eigenvalues of triangles with unit area is the set 3.4 which is plotted in figure 4. The larger point corresponds to the equilateral triangle.

### (b) Uniqueness of solution of the inverse eigenvalue problem for triangles

In this section, we are interested in finding two non-similar triangles *T*_{1}, *T*_{2} (or in proving that they do not exist) for which
3.5
If *T*_{1} and *T*_{2} have the same eigenvalue ratios *ξ*_{2,1} and *ξ*_{3,1}, and if one of the triangles is then re-scaled so that the two triangles have the same first eigenvalue *λ*_{1}, then equation (3.5) will hold. Thus, the problem of finding non-similar triangles satisfying equation (3.5) can be replaced by the problem of finding two admissible triangles and for which
3.6
In figure 5*a*, we plot the contours of *ξ*_{2,1} (with a dashed black line) and *ξ*_{3,1} (with different colours). One fundamental issue here is that *ξ*_{2,1} is increasing with *y* for fixed *x*. Thus, different dashed-line contours in figure 5 correspond to different values of *ξ*_{2,1}. This means that it is possible to find two admissible triangles satisfying equation (3.6) if and only if there is at least one contour of *ξ*_{2,1}, which has at least two intersections with one of the level curves of *ξ*_{3,1}. From figure 5*a*, we see that this does not happen, except possibly in a neighbourhood of the equilateral triangle (upper left corner), where the different level curves become very close and it is not possible to have a reasonable degree of certainty just by analysing these graphs.

This is basically a consequence of the fact that the two portions *Γ*_{1} and *Γ*_{2} of the boundary of the admissible region consist of isosceles triangles. Since any neighbourhood of these points includes at least two congruent triangles, one inside and the other outside the admissible region, any two level curves for *ξ*_{2,1} and *ξ*_{3,1} passing through one such boundary point must have the same tangent at that point. This will cause the whole of the corresponding level lines to become closer and closer as we approach the point corresponding to the equilateral triangle. A zoom for this region close to the equilateral triangle is shown in figure 6*a*. Here, we choose some superequilateral triangles, *T*_{i}, *i*=1,…,*N* and for each of these mark the corresponding contours of *ξ*_{2,1} and *ξ*_{3,1} for the values *ξ*_{2,1}(*T*_{i}), *i*=1,…,*N* (dashed black line) and *ξ*_{3,1}(*T*_{i}), *i*=1,…,*N* (red line), respectively. As we move closer to the equilateral triangle, we indeed see that the level lines of *ξ*_{2,1} and *ξ*_{3,1} are also approaching and it becomes more difficult to tell them apart.

In order to obtain more convincing evidence in this region and also as an illustration of how situations like this may be resolved, we shall now consider the values of *ξ*_{3,1} over some level curves of *ξ*_{2,1}. We are mainly interested in studying the region close to the equilateral triangle, for which we have *ξ*_{2,1}=7/3. Then, for some values of *θ*_{i} close to but smaller than 7/3, we determined a set of points on the level curves *ξ*_{2,1}=*θ*_{i}. This was done using a gradient method to compute a number of points (*x*,*y*) on *R* for which the corresponding admissible triangle *T* satisfies
We considered *ϵ*=10^{−9} and analysed the behaviour on several level curves. Figure 6*b* shows the values of *ξ*_{3,1}(*x*) on the level curve defined by *ξ*_{2,1}=2.297, which we consider to be representative. The same behaviour was observed for the other level curves that were studied in this region close to the equilateral triangle. In this way, it becomes apparent that *ξ*_{3,1} is increasing with *x*, while *ξ*_{2,1} remains constant. Thus, it is impossible to have two admissible triangles in this region satisfying equation (3.6).

It is also clear that the points with *y*=0 in the region *R* correspond to degenerate triangles, and any analysis will increase in difficulty as one approaches this part of the boundary. In order to carry out most of our numerical study, we considered the sub-region *R*_{N} excluding the part of *R* satisfying *y*<0.4. Although it is true that it becomes more and more difficult to guarantee sufficient numerical accuracy to go as close as desired to triangles corresponding to points near the *x*-axis, this truncation at *y*=0.4 was arbitrary and we performed several tests in the part of the region *R* being excluded, in the same spirit as above in the region close to the equilateral triangle. In figure 7*a*, we show the contours of *ξ*_{2,1} in *R*_{N} and also a level curve defined by *ξ*_{2,1}=1.4, which falls below the region *R*_{N}. This last level curve was computed with the same technique that we described above to study the region close to the equilateral triangle. In figure 7*b*, we present some values of *ξ*_{3,1}(*x*) over this level curve and we observe that, again, it is impossible to have two admissible triangles in this region satisfying equation (3.6).

In figure 5*b*, we plot contours of *ξ*_{2,1} (dashed black line) and *ξ*_{4,1} (with different colours). In this case, there exist intersections between some contours, such as the two points marked with black dots. The admissible triangles corresponding to this pair of points are plotted in figure 8*a*. These triangles have the same quotients *ξ*_{2,1} and *ξ*_{4,1} and then choosing a suitable scaling, we can obtain a pair of triangles which is congruent to the previous pair but now having the same values of *λ*_{1}, *λ*_{2} and *λ*_{4}. We will denote by and two triangles in those conditions, such as the pair plotted in figure 8*b*. A natural question here is to know if those triangles may or not be distinguished using a fourth eigenvalue *λ*_{i}, *i*>4—we already know that this will be the case if we pick *λ*_{3}. In figure 9, we plot from which it becomes apparent that the triangles can be distinguished using any other eigenvalue *λ*_{i}, *i*∉{1,2,4}.

We remark that the issue behind non-uniqueness observed here is related to the interior saddle point of *ξ*_{4,1} that occurs for an admissible triangle having *v*_{3}≈(0.5705,0.63) and which corresponds to a point interior to the region *R*. Figure 10 shows the contour plot of *ξ*_{4,1} in a neighbourhood of that point, where we represented the level lines of *ξ*_{2,1} with dashed curves. When considering other quotients *ξ*_{i,j}, with 1≤*j*<*i*≤11, this behaviour also appears and for all of those quotients we have interior saddle points. In some cases, such as *ξ*_{7,3}, we also have points of local minima, causing the same effect. Note that although *ξ*_{3,1} also has a saddle point, as can be seen in figure 5, this occurs for an isosceles triangle and it is thus on the boundary of the region *R*. This is then not relevant, as the points which will have the same value correspond to triangles outside the region *R* and correspond thus to triangles which are congruent.

Next, we explore another way to exhibit the intersections of the contours of *ξ*_{2,1} and *ξ*_{4,1} that we have shown in figure 5. We consider some subsets of *R* obtained for fixed *y*,
In figure 11, we plot curves of type
3.7
for some values of *y*_{i}, which are also marked. It is evident that we have some intersections of the curves and by construction, these intersections correspond to distinct triangles for which we have the same quantities *ξ*_{2,1} and *ξ*_{4,1}.

This means that, in general, we cannot determine uniquely a triangle from the values *λ*_{1}, *λ*_{2} and *λ*_{4} or, in the spirit of Kac (1966), we cannot identify a triangle hearing *λ*_{1}, *λ*_{2} and *λ*_{4}.

### (c) The case of Neumann boundary conditions

We shall now briefly discuss the case of Neumann boundary conditions where the situation changes drastically. In this case, it turns out that for certain domains, given any finite number of eigenvalues, say *m*, it is always possible to identify *m* possible values for these eigenvalues which will not determine the domain uniquely. To understand what happens in this case, we shall analyse the case of rectangles first. For the rectangle [0,*L*]×[0,*l*] (assume that *L*>*l*), we have *μ*_{0}=0 and when *L*>*ml*, we get *μ*_{j}=*π*^{2}(*j*^{2}/*L*^{2}) for *j*=1,…,*m*. Thus, for each *m*, there is a value of *L* sufficiently large for which *ξ*_{j,1} becomes constant and equal to *j*^{2}. If we thus take *ξ*_{j,1}=*j*^{2}, we see that no finite number of eigenvalues can determine the rectangle. A similar result holds for the case of circular sectors, and so we might expect this to be the case also for triangles.

However, from our computational study for triangles, and although it is clear that there is a similar effect at work here, we were not able to obtain decisive evidence showing that it does produce the same *saturation* phenomena as before. In figure 12*a*, we show the quotients *ξ*_{j,1} for rectangles with unit area as a function of *L*, while in figure 12*b* we show *ξ*_{j,1} for subequilateral triangles as a function of the length of the largest side.

A closer look at the numerical values shows that the curves corresponding to triangles in the figure on the right may actually not be straight horizontal lines, but in fact change very slowly. We thus see that a more detailed study is needed before reaching a conclusion.

## 4. Discussion

In the present work, we continue our project of using computational resources to explore the nature of the spectrum of the Laplacian, following the spirit of our previous papers (Antunes & Freitas 2006, 2008). It is our belief that this approach not only provides new insight into these problems in that it raises some conjectures backed by reliable computational evidence, but also that it helps in understanding problems which are, at present, mostly beyond a rigorous analytical treatment.

In the particular case of the present work, we answer a question raised in Laugesen & Siudeja (2009) by providing what we believe to be compelling computational evidence to the fact that, on the one hand, triangles are spectrally determined by their first three Dirichlet eigenvalues and, on the other hand, this is a peculiarity of this spectral triplet in the sense that for other triplets there will exist pairs of triangles having the corresponding eigenvalues in common.

We also showed that the corresponding inverse Neumann problem has quite different properties. Regarding this, we finish by pointing out that the same type of analysis presented in §3*c* implies that a finite number of Dirichlet gaps *λ*_{j}−*λ*_{1}, *j*=2,…,*m* will not, in general, determine a rectangle uniquely.

## Acknowledgements

P.R.S.A. was partially supported by FCT, Portugal, through the scholarship SFRH/BPD/47595/ 2008 and the project PTDC/MAT/105475/2008 and by Fundação Calouste Gulbenkian through programme *Estímulo à Investigação 2009*. Both authors were partially supported by FCT’s project PTDC/MAT/101007/2008.

- Received October 18, 2010.
- Accepted November 9, 2010.

- This journal is © 2010 The Royal Society