I would like to find the spectrum of a triangle (e.g. an equilateral) in using the usual Laplacian. I am not able to find some references on the subject. I'd try to solve the problem myself with a little help. Could someone give me the general idea of the resolution and how I should treat the boundary conditions of this problem?
Help would be appreciated!
With fixed Dirichlet or Neumann boundary conditions, to my knowledge the only triangles with explicitly known Laplace spectra are:
To compute the spectrum of the isosceles right, notice that it is the quotient of the plane by a reflection group, so its eigenfunctions are linear combinations of those of the square which satisfy the given boundary condition on the diagonal.
The equilateral triangle's Laplace spectrum (Dirichlet and Neumann) were worked out by Lame, and in a different fashion by Pinsky. They are related to the action of the equilateral triangle group on the plane. There is a compilation of papers on the equilateral triangle's spectrum by McCartin.
If you sit down and try to work out other examples that aren't the result of a reflection group acting on the plane, the mechanical computation of eigenfunctions/eigenvalues breaks down. Without some extra structure, you're going to run into difficulty in finding finite linear combinations of functions that satisfy the self-adjoint boundary conditions you've chosen.
For example, let's look at a right triangle that is not hemiequilateral or isosceles. Specify the triangle as the domain $$ T = \{(x,y)\ |\ x\geq 0,y\geq 0,2x+y\leq 10\}.$$ Let's choose Dirichlet boundary conditions. The eigenvalue problem is now $$ u_{xx}(x,y) + u_{yy}(x,y) = \lambda u(x,y),\ \ u(0,y) = u(x,0) = u(t,10-2t) = 0 $$ for all $(x,y)\in T$. (I parametrized the line $2x+y=10$ with $t\mapsto (t,10-2t)$.) If we try to separate variables, mechanically following the procedure for finding the spectrum of a rectangle, we get as far as imposing the first two boundary conditions: $$u(x,y) = \sum_c a_c\sin(\sqrt{\lambda}x)\sin((c-\sqrt{\lambda})y)$$ where we have a linear combination of such products of sine functions.
And now we need to find $\lambda$ such that $$ 0 = \sum_c a_c \sin(\sqrt{\lambda}t)\sin((c-\sqrt{\lambda})(10-2t)). $$
Brick wall.
In fact, there are only a few polygonal domains whose eigenfunctions are all trigonometric: c.f. Theorem 2 of this paper, also by McCartin:
The only polygonal domains possessing a complete set of trigonometric eigenfunctions of the form of Equation (2) [i.e., are finite linear combinations of trigonometric functions] are those shown in Figure 1: the rectangle, the square, the isosceles right triangle, the equilateral triangle and the hemiequilateral triangle
Theorem 3 goes on to indicate which domains have some trigonometric eigenfunctions.
For more on triangles in general, this paper by Harmer explores the spectra of Euclidean and spherical triangles from the perspective of finite group actions (but does not more than touch on hyperbolic triangle groups, which are a much subtler topic).
One can still make qualitative statements regarding the spectra of triangles. For example, a paper of Hillairet-Judge proves that the Dirichlet spectrum of a generic Euclidean triangle is simple.
It is also possible to numerically study the spectra of Euclidean triangles. For instance, this paper of Berry studies "diabolical points" in the spectrum, i.e., triangles which have multiplicity. I believe it is conjectured, but not known, that when a triangle has multiplicity at a point in its spectrum, about a small neighborhood of that triangle in modulis space, the graph of the two eigenfunctions has the form of a cone.
One tool for numerically studying the spectrum of triangles is the method of particular solutions, see this paper of Betcke-Trevethen and its description of numerically computing the spectrum of domains. This method has been adopted for other domains and manifolds, particularly hyperbolic triangles and surfaces (e.g. work of Strohmaier-Uski).
References:
Pinsky, Mark A. The eigenvalues of an equilateral triangle. SIAM J. Math. Anal. 11 (1980), no. 5, 819–827. MR0586910
Pinsky, Mark A.(1-NW) Completeness of the eigenfunctions of the equilateral triangle. SIAM J. Math. Anal. 16 (1985), no. 4, 848–851. MR0793926
McCartin, Brian J. Laplacian eigenstructure of the equilateral triangle. Hikari Ltd., Ruse, 2011. x+200 pp. ISBN: 978-954-91999-6-3 MR2918422
McCartin, Brian J. On polygonal domains with trigonometric eigenfunctions of the Laplacian under Dirichlet or Neumann boundary conditions. Appl. Math. Sci. (Ruse) 2 (2008), no. 57-60, 2891–2901. MR2480444
Harmer, Mark The spectra of the spherical and Euclidean triangle groups. J. Aust. Math. Soc. 84 (2008), no. 2, 217–227. MR2437339
Hillairet, Luc; Judge, Chris. Spectral simplicity and asymptotic separation of variables. Comm. Math. Phys. 302 (2011), no. 2, 291–344. MR2770015
Berry, M. V.; Wilkinson, M. Diabolical points in the spectra of triangles. Proc. Roy. Soc. London Ser. A 392 (1984), no. 1802, 15–43. MR0738925
Betcke, Timo; Trefethen, Lloyd N. Reviving the method of particular solutions. SIAM Rev. 47 (2005), no. 3, 469–491 (electronic). MR2178637
Strohmaier, Alexander; Uski, Ville. An algorithm for the computation of eigenvalues, spectral zeta functions and zeta-determinants on hyperbolic surfaces. Comm. Math. Phys. 317 (2013), no. 3, 827–869. MR3009726
