Solving the Helmholtz equation for membranes of arbitrary shape
SSolving the Helmholtz equation for membranes ofarbitrary shape
Paolo Amore ‡ Facultad de Ciencias, Universidad de Colima,Bernal D´ıaz del Castillo 340, Colima, Colima, M´exicoand Physics Department, University of Texas at El Paso,El Paso, Texas, USA
Abstract.
I calculate the modes of vibration of membranes of arbitrary shape using acollocation approach based on Little Sinc Functions. The matrix representation of thePDE obtained using this method is explicit and it does not require the calculation ofintegrals. To illustrate the virtues of this approach, I have considered a large numberof examples, part of them taken from the literature, and part of them new. Whenpossible, I have tested the accuracy of these results by comparing them with the exactresults (when available) or with results from the literature. In particular, in the caseof the L-shaped membrane, the first example discussed in the paper, I show that itis possible to extrapolate the results obtained with different grid sizes to obtain higlyprecise results. Finally, I also show that the present collocation technique can be easilycombined with conformal mapping to provide numerical approximations to the energieswhich quite rapidly converge to the exact results.PACS numbers: 03.30.+p, 03.65.-w ‡ [email protected] a r X i v : . [ phy s i c s . c o m p - ph ] J a n olving the Helmholtz equation for membranes of arbitrary shape
1. Introduction
This paper considers the problem of solving the Helmholtz equation − ∆ ψ ( x, y ) = Eψ ( x, y ) (1)over a two-dimensional domain, B , of arbitrary shape, assuming Dirichlet boundaryconditions over the border, ∂B . Physically, this equation describes the classical vibrationof a homogenoeous membrane or the behaviour of a particle confined in a region withinfinite walls in quantum mechanics. Unfortunately exact solutions to this equation areavailable only in few cases, such as for a rectangular or a circular membrane, wherethey can be expressed in terms of trigonometric and Bessel functions respectively [1].In the majority of cases, in fact, only numerical approaches can be used: some of theseapproaches are discussed for example in a beautiful paper by Kuttler and Sigillito, [2].The purpose of the present paper is to introduce a different approach to the numericalsolution of the Helmholtz equation (both homogenous and inhomogeneous) and illustrateits strength and flexibility by applying it to a large number of examples.The paper is organized as follows: in Section 2 I describe the method and discussits application to the classical problem of a L-shaped membrane; in Section 3 I consideran homogenous membrane, with the shape of Africa and calculate few states; in Section4 I consider two inequivalent membranes, which are known to be isospectral, obtaininga numerical indication of isospectrality; in Section 5 I study an example of irregulardrum; in Section 6 the method is applied to study the emergence of bound states ina configuration of wires of neglegible transverse dimension, in presence of crossings;in Section 7 I show that even more precise results can be achieved by combining thecollocation method with a conformal mapping of the boundary. Finally, in Section 8 Idraw my conclusions.
2. The method
The method that I propose in this paper uses a particular set of functions, the
LittleSinc functions (LSF) of [13, 14], to obtain a discretization of a finite region of thetwo-dimensional plane. These functions have been used with success in the numericalsolution of the Schr¨odinger equation in one dimension, both for problems restrictedto finite intervals and for problems on the real line. In particular it has been provedthat exponential convergence to the exact solution can be reached when variationalconsiderations are made (see [13, 14]).Although Ref.[13] contains a detailed discussion of the LSF, I will briefly reviewhere the main properties, which will be useful in the paper. Throughout the paper Iwill follow the notation of [13].A Little Sinc Function is obtained as an approximate representation of the Diracdelta function in terms of the wave functions of a particle in a box (being 2 L the size olving the Helmholtz equation for membranes of arbitrary shape s k ( h, N, x ) ≡ N (cid:26) sin ((2 N + 1) χ − ( x ))sin χ − ( x ) − cos ((2 N + 1) χ + ( x ))cos χ + ( x ) (cid:27) . (2)where χ ± ( x ) ≡ π Nh ( x ± kh ). The index k takes the integer values between − N/ N/ − N being an even integer). The LSF corresponding to a specific value of k is peaked at x k = 2 Lk/N = kh , h being the grid spacing and 2 L the total extensionof the interval where the function is defined. By direct inspection of eq. (2) it is foundthat s k ( h, N, x j ) = δ kj , showing that the LSF takes its maximum value at the k th gridpoint and vanishes on the remaining points of the grid.It can be easily proved that the different LSF corresponding to the same set areorthogonal [13]: (cid:90) L − L s k ( h, N, x ) s j ( h, N, x ) dx = h δ kj (3)and that a function defined on x ∈ ( − L, L ) may be approximated as f ( x ) ≈ N/ − (cid:88) k = − N/ f ( x k ) s k ( h, N, x ) . (4)This formula can be applied to obtain a representation of the derivative of a LSFin terms of the set of LSF as: ds k ( h, N, x ) dx ≈ (cid:88) j ds k ( h, N, x ) dx (cid:12)(cid:12)(cid:12)(cid:12) x = x j s j ( h, N, x ) ≡ (cid:88) j c (1) kj s j ( h, N, x ) d s k ( h, N, x ) dx ≈ (cid:88) j d s k ( h, N, x ) dx (cid:12)(cid:12)(cid:12)(cid:12) x = x j s j ( h, N, x ) ≡ (cid:88) j c (2) kj s j ( h, N, x ) , (5)where the expressions for the coefficients c ( r ) kj can be found in [13]. Although eqs.(4) isapproximate and the LSF strictly speaking do not form a basis, the error made withthis approximation decreases with N and tends to zero as N tends to infinity, as shownin [13]. For this reason, the effect of this approximation is essentially to replace thecontinuum of a interval of size 2 L on the real line with a discrete set of N − x k , uniformly spaced on this interval.Clearly these relations are easily generalized to functions of two or more variables.Since the focus of this paper is on two dimensional membranes, I will briefly discuss howthe LSF are used to discretize a region of the plane; the extension to higher dimensionalspaces is straightforward. A function of two variables can be approximated in terms of( N x − × ( N y −
1) functions, corresponding to the direct product of the N x − N y − x and y axis: each term in this set corresponds to a specific pointon a rectangular grid with spacings h x and h y (in this paper I use a square grid with N x = N y = N and L x = L y = L ). olving the Helmholtz equation for membranes of arbitrary shape k, k (cid:48) ) identifies a unique point on the grid, I can select this point using asingle index K ≡ k (cid:48) + N N − (cid:18) k + N − (cid:19) (6)which can take the values 1 ≤ K ≤ ( N − . I can also invert this relation and write k = 1 − N/ (cid:20) KN − ε (cid:21) (7) k (cid:48) = K − N/ − ( N − (cid:20) KN − ε (cid:21) , (8)where [ a ] is the integer part of a real number a and ε → Hψ n ( x, y ) ≡ [ − ∆ + V ( x, y )] ψ n ( x, y ) = E n ψ n ( x, y ) (9)using the convention of assuming a particle of mass m = 1 / (cid:126) = 1. TheHelmholtz equation, which describes the vibration of a membrane, is a special case of(9), corresponding to having V ( x, y ) = 0 inside the region B where the membrane liesand V ( x, y ) = ∞ on the border ∂ B and outside the membrane.The discretization of eq. (9) proceeds in a simple way using the properties discussedin eqs. (4) and (5): H kk (cid:48) ,jj (cid:48) = − (cid:104) c (2) kj δ k (cid:48) j (cid:48) + δ kj c (2) k (cid:48) j (cid:48) (cid:105) + δ kj δ k (cid:48) j (cid:48) V ( x k , y k (cid:48) ) (10)where ( k, j, k (cid:48) , j (cid:48) ) = − N/ , . . . , N/ −
1. Notice that the potential part of theHamiltonian is obtained by simply ”collocating” the potential V ( x, y ) on the grid, anoperation with a limited computational price. The result shown in (10) corresponds tothe matrix element of the Hamiltonian operator ˆ H between two grid points, ( k, k (cid:48) ) and( j, j (cid:48) ), which can be selected using two integer values K and J , as shown in (6).Following this procedure the solution of the Schr¨odinger (Helmholtz) equationon the uniform grid generated by the LSF corresponds to the diagonalization of a( N − × ( N − square matrix, whose elements are given by eq. (10).I will now use a specific problem, the vibration of a L-shaped membrane, representedin Fig.1, to illustrate the method, and discuss different implementations of the methoditself. This problem has been widely used in the past to test the performance of thedifferent numerical methods (see for example refs. [3, 4, 5, 2, 7, 8, 9, 10, 11]) and it istherefore a useful tool to assess the strength of the present approach. Because of thereentrant corner, corresponding to the angle θ = 3 π/ , ψ ( x, y ) in the radial direction are unbounded (see [3]).Reid and Walsh in [3] obtained a numerical approximation for the two lowest modesof this membrane using finite differences and a confomal map which eliminates thereentrant corner (see fig.5 of [3]); a more precise result was later obtained by Fox, Henriciand Moler who used the Method of Particular Solutions (MPS) in [4] exploiting thesymmetries of the problem (the reader may find a detailed discussion of the symmetries olving the Helmholtz equation for membranes of arbitrary shape Figure 1.
L shaped membrane. The dots are the collocation points corresponding to N = 14. for this problem in [2]): the first eight digits of the lowest eigenvalue reported by theauthors are correct. Mason has obtained numerical estimates for the first few modes ofthe L-shaped membrane in terms of a two dimensional Chebyshev series [5]. Milstedand Hutchinson [6] have obtained finite element solutions to this problem. Sideridis in[7] used a conformal mapping of the L-shaped region onto a square and then solved theresulting equation on a uniform rectangular mesh, obtaining the first four digits of thelowest mode. Schiff, ref. [8], has calculated the first 15 lowest modes of this membraneusing finite elements, with a refined grid covering the region surrounding the reentrantcorner.More recently Platte and Driscoll have solved the boundary value problem on theL-shape membrane using radial basis functions [9]. Finally Betcke and Trefethen haverevisited the MPS in [10]; in that paper they have observed that the MPS reachesa minimal error for a certain value of N (the number of collocation points on eachof the sides non adjacent to the corner where the expansion is performed) but thenit starts to grow as N increases. The modified version of the method discussed in[10], which samples the Fourier-Bessel functions also in the interior points, corrects thisproblem and provides a convergent behaviour for the error. In this way Betcke andTrefethen were able to obtain the first 14 digits of the lowest eigenvalue of the L-shapedmembrane, E ≈ . N there is a total of 3 / N − N + 1 points; the gridrepresented in the figure corresponds to N = 14 and therefore to a total of 120 internalpoints. In this case the collocation of the Hamiltonian on the uniform grid generated by olving the Helmholtz equation for membranes of arbitrary shape n E ( − ) n E (+) n n E ( − ) n E (+) n n E ( − ) n E (+) n Table 1.
First 108 eigenvalues of the L-shaped membrane calculated with a grid with N = 60. olving the Helmholtz equation for membranes of arbitrary shape ×
120 matrix, which can then be diagonalized. The eigenvaluesof this matrix provide the lowest 120 modes of the membrane, while the eigenvectorsprovide the lowest 120 wave functions. Alternatively I can pick all the points of the gridinternal to the membrane, including those falling on the border: in such case a total of3 / N − N points is found, corresponding to a total of 133 points in the case of theFigure.Table 2 contains the first 108 eigenvalues of the L-shaped membrane calculatedusing a grid with N = 60 and selecting the grid points according to the prescriptionsjust explained. I have used the notation E ( ± ) n for the energy of the n th state when thecollocation points on the border are either rejected ( E (+) n ) or kept ( E ( − ) n ). The notation( ± ) is used since the two sets approach the exact results either from above (+) or frombelow ( − ), as one can see comparing these numbers with the precise results containedin [10, 11]. The reader will certainly notice that the results of Table 2 contain ratherlarge errors: in the case of the fundamental state, for example, one has an error of about1% from E (+) n and a much larger error of almost 5% for E ( − ) n .The left panel of Fig.2 shows the eigenvalues E (+) n (solid line) and E ( − ) n (dashed line)for the L-shaped membrane corresponding to a grid with N = 60. The reader may noticethat the higher end of the spectrum displays a curvature, contrary to the behaviourpredicted by Weyl’s law, i.e. (cid:104) N (cid:105) ∝ E for large energies. It is easy to show that sucheffect is artificial: consider for example the case of a particle confined in a unit square,whose energies are given by E n x ,n y = ( n x + n y ) π . The diagonalization of the Hamiltonian(10) for this problem would provide the energies corresponding to the ( N − statesobtained taking the first N − n x and n y . This means that for energieshigher than E N = [ N − N + 2] π the method will provide only the eigenvaluescontained inside a square of side N − n x , n y ) plane), up to a maximal energy E MAX = 2 [ N − N + 2] π . For this reason, the states above E N are incomplete andshould not be taken into account for inferring the asymptotic behavior of (cid:104) N (cid:105) . The rightpanel of Fig.2 displays the asymmetry defined as A n = 2( E (+) n − E ( − ) n ) / ( E (+) n + E ( − ) n )for the same grid: this quantity provides an upper estimate for the error.Fig. 3 displays the ground state energy of the L-shaped membrane as a functionof the number of grid points and compares it with the precise result of [10]: as alreadypointed out the two sets approach the exact value from above and below.Much more precise results can be obtained by performing an extrapolation of theresults corresponding to finite grids: this is a common procedure used in the literature(see for example [2]). I have considered four different extrapolation sets using thenumerical results obtained working with grids with N ranging from N = 10 to N = 60(only even values). Calling h = 2 L/N the grid spacing the sets are: f ( h ) = ¯ N (cid:88) n =0 c n h n (11) f ( h ) = (cid:80) ¯ N/ n =0 c n h n (cid:80) ¯ N/ n =1 c n h n (12) olving the Helmholtz equation for membranes of arbitrary shape n Set 1 Set 2 Set 3 Set 41 ( − ) 9.63959383529194 9.63970774930113 9.63972385784876 9.63972384404696 ∗ ∗ − ) 15.1972518419212 15.1974702475024 15.1972519362081 15.1972519266011 ∗ ∗ − ) 19.7392087861784 19.7392088017282 19.7392073765870 19.7392088020095 ∗ ∗ − ) 29.5178267971821 29.5214811097206 − ∗ ∗ − ) 31.9159767579531 31.9125745966885 − ∗ ∗ − ) 41.474267306813 41.4744740922213 41.4761914432832 41.4745099148779 ∗ ∗
20 ( − ) 101.776561675314 ∗ ∗
50 ( − ) - 246.740564791939 - 246.602432808866 ∗
50 (+) 250.784799377301 250.785244396338 - 250.785494606618 ∗
104 ( − ) - 410.08260648211 - -104 (+) 493.480067984180 ∗ Table 2.
Extrapolation of the nine eigenvalues of the L-shaped membrane using thefour different sets. The first 6 states correspond to extrapolating the results for gridsgoing from N = 10 to N = 60, with 25 unknown coefficients; the last two statescorrespond to extrapolating the results for grids going from N = 18 to N = 60, andwith 21 unknown coefficients. For a given state, the set with the asterisk correspondsto the minimal value taken by the least squares. The results which do not converge tothe exact value have been omitted. f ( h ) = c + ¯ N (cid:88) n =1 c n h n/ / (13) f ( h ) = c + (cid:80) ¯ N/ n =1 c n h n/ / (cid:80) ¯ N/ n =1 c n h n/ / (14)where ¯ N is an even integer which determines the number of coefficients used in the fits.The continuum limit is reached taking h → ∞ , where only the coefficient c survives. The unknown coefficients in the expressions (11), (12), (13) and (14) areobtained using a Least Square approach: I show the results of this procedure in Table2. In general, the last set provides the best results and indeed it reproduces the first 11digits of E correctly, using either the values of E ( − )1 or those of E (+)1 . In the case of E ,for which the exact value is known ( E = 2 π ), I obtain the first 14 digits correct using E (+)3 and the first 11 digits correct using E ( − )3 . olving the Helmholtz equation for membranes of arbitrary shape Figure 2.
Left panel:Energy of the ground state of the L shaped membrane as afunction of the number of grid points N . The horizontal line is the precise result of[10]. The set approaching the exact result from above (below) corresponds to E (+)1 ( E ( − )1 ). Right panel: The asymmetry A n = 2( E (+) n − E ( − ) n ) / ( E (+) n + E ( − ) n ) calculatedwith a grid with N = 60. Figure 3.
Energy of the ground state of the L shaped membrane as a function ofthe number of grid points N . The horizontal line is the precise result of [10]. The setapproaching the exact result from above (below) corresponds to E (+)1 ( E ( − )1 ). In [15] Berry has devised an algorithm for obtaning successive approximations tothe geometric properties K j of a closed boundary B given the lowest N eigenvalues E n . The partition function Φ( t ) ≡ (cid:80) ∞ n =1 e − E n t obeys an asymptotic expansion for smallvalues of t Φ( t ) ≈ t ∞ (cid:88) j =0 K j t j/ , (15)where the coefficients K j are related to the geometric properties of B . For example K = A/ π and K = − γL/ √ π . Using this asymptotic expansion Berry has obtainedaccelerated expressions for the geometrical constants of B . In particular for the area of olving the Helmholtz equation for membranes of arbitrary shape B he has found the approximant (eq.(20) of [15]) A m ( t ) = 2 πtm ! ∞ (cid:88) n =1 e − ξ n ξ m − n H m +1 ( ξ n ) (16)where ξ n ≡ √ E n t . Figure 4.
Left Panel: The area approximant A ( t ) obtained using the expression ofBerry. The thin solid and dashed lines are obtained with the first 1000 eigenvaluescorresponding to the sets E (+) n and E ( − ) n respectively. The bold solid and dashed linescorrespond to the sets obtained through an extrapolation from the original sets. RightPanel: The perimeter approximant L ( t ) obtained with the improved expression ofBerry. The same sets of eigenvalues have been considered. In the left panel of Fig.4 I show the area approximant A ( t ), obtained using theexpression of Berry. The thin lines correspond to using the the sets E (+) n and E ( − ) n (solid and dashed lines respectively); the thick lines correspond to using the eigenvaluesobtained from the extrapolation of the sets E (+) n and E ( − ) n (solid and dashed linesrespectively). I call ¯ E ( ± ) n the eigenvalues obtained extrapolating the eigenvalues E ( ± ) n ;the extrapolation is carried out using the results obtained with grids with N going from N = 48 to N = 60 and assuming E n ( N ) ≈ ¯ E n + (cid:15) n N . The approximants obtained withthe extrapolated eigenvalues provide excellent approximations to the area and perimeterof the membrane, as seen in Fig.4.Fig. 5 shows the first two eigenfunctions of the L shaped membrane obtainedwith a grid corresponding to N = 30. The solid lines appearing in the ”forbiddenregion” correspond to the level ψ ( x, y ) = 0: the effect observed in the figure is due tothe approximation of working with a finite number of grid points. In fact, although aparticular LSF vanishes on the points defining the grid, except on a particular point,where it reaches its maximum, it is non-zero elsewhere. This means that the numericalsolution can take small values even in the region where the exact solution must vanish;however, the size of this effect decreases as the number of grid points is increased (takinginto account that the computational load roughly increases as N ). In the Appendix wepropose an alternative procedure which does not involve the diagonalization of largermatrices and which can be used to improved the results obtained with a given grid. olving the Helmholtz equation for membranes of arbitrary shape Figure 5.
First two eigenfunctions of the L-shaped membrane obtained using N = 30.The black lines correspond to the level ψ ( x, y ) = 0.
3. The Africa drum
I will now examine the case of a membrane with an irregular shape. The application ofthe method proceeds exactly as in the case of the L-shaped membrane: once a grid ischosen, the points of the grid which are internal to the membrane are used to build amatrix representation of the Hamiltonian which, once diagonalized, provides the energiesand wave functions of the problem.As a paradigm of this class of membranes I have studied the vibrations of a drumwith the shape of Africa. Unlike in the previous example the border does not crossthe grid points, a feature which affects the precision of the results. The plots in Fig.6display the energies of the first two states of the Africa drum for grids with different N (the dots in the plots) and compare them with the best fit obtained assuming that E ( N ) = a + b/N , where a and b are constants independent of N . The irregularity ofthe border is reflected in the behavior of the eigenvalues which decay with N but at thesame time oscillate.In Fig.7 I show the density plot of four different states of the Africa drum, obtainedusing a grid with N = 60. In Fig.8 I show the wave function of the ground state of theAfrica drum, obtained using a grid with N = 60.
4. Isospectral membranes
In a classic paper dated 1966, [16], Kac formulated an interesting question: whetherit is possible to hear the shape of a drum, meaning if the spectrum of frequencies ofa given drum is unique to that drum or drums with different shapes can have thesame spectrum. The question was finally answered in 1992, when Gordon, Webb and olving the Helmholtz equation for membranes of arbitrary shape Figure 6.
Left: Energy of the fundamental mode of the Africa shaped membrane as afunction of the number of grid points. The continous line is the fit E = a + b/N , with a = 20 . E = a + b/N ,with a = 32 . Figure 7.
Density plot for the fundamental state (upper left), first excited state(upper right), 200 th excited state (lower left) and 300 th excited state (lower right) ofthe Africa shaped membrane. In all plots the absolute value of the wave function isshown and a grid with N = 60 is used. olving the Helmholtz equation for membranes of arbitrary shape Figure 8.
Ground state of the Africa shaped membrane obtained using N = 60. Wolpert found a first example of inequivalent drums having the same spectrum [17].An experiment made by Sridhar and Kudrolli reported in [18] used microwave cavitieswith the shape of the drums of [17] to verify the equality of the spectrum for the lowest54 states. More recently the same experiments have been carried out on isospectralcavities where the classical dynamics changes from pseudointegrable to chaotic [19].Numerical calculations of the first few modes of the isospectral drums found in [17] havebeen performed with different techniques: Wu, Sprung and Martorell [20] have used amode matching method to calculate the first 25 states of these drums and compared theresults with those obtained with finite difference; using a different approach Driscoll [21]has also calculated the first 25 states obtaining results which are accurate to 12 digits;Betcke and Trefethen [10] have used their modified version of the method of particularsolutions to obtain the first three eigenvalues of these drums, reporting results whichare slightly more precise than those of Driscoll.I will now discuss the application of the present method to the calculation of thespectrum of these isospectral membranes: whereas in the case of the L-shaped membranethe border of the membrane was sampled by the grid, regardless of the grid size (keeping N even), in the case of the isospectral membranes this happens only for grids where N = 6 k , with k integer. It is important to restrict the calculation to this class of grids toavoid the oscillations observed in the case of the Africa membrane. I have thus appliedthe method with grids ranging from N = 6 to N = 120 § . § The numerical results presented in the case of the L-shaped membrane were obtained with a 40-digitprecision in the eigenvalues, using the command N[,40] of Mathematica: in this case, since I need toresort to larger grids I have worked with less digits precision using the command N [] in Mathematica. olving the Helmholtz equation for membranes of arbitrary shape Figure 9.
Energy of the ground state of the first isospectral membrane as a functionof the number of grid points N . The horizontal line is the precise result of [10]. Theset approaching the exact result from above (below) corresponds to E (1+)1 ( E (1 − )1 ). The plot in Fig.9 displays the ground state energy of the first isospectral membranecalculated at different grid sizes. The horizontal line is the precise value given in [10].The set approaching this value from above (below) corresponds to the application of themethod rejecting (accepting) the grid points falling on the border. The correspondingplot for the second isospectral membrane is almost identical and therefore it is notpresented here.In Table 4 I report the energies of the first 30 states obtained using Richardsonextrapolation [22] on the results for grids going from N = 66 to N = 120. The secondand third columns are the energy of the first isospectral membranes obtained with thesets which reject ( E (+) n ) or accept ( E ( − ) n ) the grid points falling on the border, whichas seen in the case of the L-shape membrane provide a sequence of numerical valuesapproaching the exact eigenvalue from above and from below respectively. The last twocolumns report the analogous results for the second isospectral membrane. Notice thatsome of the energies in the third column are clearly incorrect.A further empirical verification of the isospectrality of the two membranes ispresented in Fig.10, where I have plotted the asymmetry A n ≡ ( E (1+) n − E (2+) n ) / ( E (1+) n + E (2+) n ) for the first 2000 states of the isospectral membranes. In this case E (1+) n ( E (2+) n )is the energy of the n th state of the first (second) membrane obtained using Richardsonextrapolation of the grids with N = 114 and N = 120.
5. An unusual drum
I will now consider a further example by looking at a particular membrane originallystudied by Trott in [23]: this drum is shown in Fig.12 and consists of a total of308 units squares which are joined into a rather irregular form. Theoretical andexperimental studies carried out on drums with fractal or irregular boundaries haveshown that the wave excitations for these drums are drastically altered [24, 25, 26]: inparticular, the Weyl law for these membranes is modified in a way which depends on olving the Helmholtz equation for membranes of arbitrary shape n E (1+) n E (1 − ) n E (2+) n E (2 − ) n Table 3.
First 30 eigenvalues of the isospectral membranes obtained with Richardsonextrapolation of the results obtained with grids from N = 66 to N = 120. olving the Helmholtz equation for membranes of arbitrary shape Figure 10.
Left panel: log of the asymmetry A n ≡ ( E (1+) n − E (2+) n ) / ( E (1+) n + E (2+) n )for the first 2000 states of the isospectral membranes. E (1+) n ( E (2+) n ) is the energy ofthe n th state of the first (second) membrane obtained using Richardson extrapolationof the grids with N = 114 and N = 120. Right panel: Blow-up of the previous plotfor the first 100 states. the fractal dimension of the perimeter (see for example [27]), the so called Weyl-Berry-Lapidus conjecture. Recently the vibrations of a uniform membrane contained in a Kochsnowflake have been studied in two papers, [28, 29].The paper by Trott is both interesting in its physical and mathematical content andas an example of the excellent capabilities of Mathematica to handle heavy numericalcalculations: as a matter of fact Trott uses a finite difference approximation of theLaplacian on a uniform grid and samples the membrane in 28521 internal points.Explicit numerical values for the first 24 modes are reported.I have therefore considered the same problem using the LSF with grids of differentsize (up to N = 250 which leads to the same grid of [23]). Figure 13 displays theenergy of the fundamental mode of this membrane as a function of the size of N . Thedashed horizontal line in the plot represents the result of [23], E = 6 . N going from 50 to 250, with intervals of50. For these particular values of N the border of the membrane is sampled by thegrid and therefore more accurate results are expected. The grid points on the borderare rejected, which leads to eigenvalues which approach the exact results from above,as seen in the previous examples. The points in the lower part of the plot correspondto grid sizes varying from N = 52 to N = 148, excluding N = 100: in this case thevalues approach the exact result from below, although in doing so they also oscillatereflecting the treatment of the border (a behaviour already observed in the case of theAfrica membrane). As mentioned above the finest grid corresponds to sampling themembrane on 28521 internal points and therefore to working with a 28521 × olving the Helmholtz equation for membranes of arbitrary shape Figure 11.
Upper panel: Wave functions (absolute value) of the first isospectralmembrane (ground state and 100 th excited state); Lower panel: Wave functions(absolute value) of the second isospectral membrane (ground state and 100 th excitedstate). A grid with N = 60 is used. membranes: although this set provides a sequence of values which uniformly approachthe value at the continuum, the number of grid points sampled is quite large because ofthe large perimeter of the membrane. For example, for N = 100, this set samples themembrane on 7029 points, compared with the N = 3801 points used in the other set.The Figure also displays the improved ground state energies obtained using the”mesh refinement” procedure described in the Appendix (the three green points): theeigenvector for a given grid is extrapolated to a finer grid rejecting contributions in the”forbidden region” (i.e. falling outside the border of the membrane). The improvedenergy estimate corresponds to the expectation value of the Hamiltonian in this stateand thus it requires no diagonalization. The results displayed in the figure correspondto extrapolation to a grid which is twice finer. olving the Helmholtz equation for membranes of arbitrary shape Figure 12.
The unusual drum considered by Trott [23]. The black area is the surfaceof the drum; the red points are the collocation points corresponding to N = 50. Figure 13.
Energy of the ground state of the unusual drum as a function of N . Thehorizontal line is the result of [23]; the points approaching the horizontal line fromabove correspond to configurations where the border is sampled by the collocationpoints (and as discussed in the case of the L-shaped membrane are rejected). Thegreen points correspond to the results obtained with the ”mesh refinement” proceduredescribed in the Appendix.
6. Bound states in the continuum
It is well known that the spectrum of the Laplacian with Dirichlet boundary conditionsmay contain bound states even for open geometries, in correspondence of crossings orbendings of the domain. For example, Schult et al.[30] have studied the problem oftwo crossed wires, of infinite length, showing that such geometry supports exactly onebound state, localized at the crossing. Avishai and collaborators have also proved theexistence of a bound state in the broken strip configuration for arbitrarily small angles,see [31] (more recently Levin has proved the existence of one bound state in the brokenstrip for any angle of the strip [32]). Goldstone and Jaffe [33] have given a variational olving the Helmholtz equation for membranes of arbitrary shape Figure 14.
Wave function of the ground state of the configuration for ¯ n = 4 using N = 500. proof of the existence of a bound state for an infinite tube in two and three dimensions,provided that the tube is not straight. Other interesting configurations which supportbound states in the continuum have been studied by Trefethen and Betcke [11].The example which I will consider here is somehow related to the crossed wiresconfiguration studied by Schult at al. I have considered a set of horizontal and verticalwires, of neglegible trasverse dimension, which are contained in a square box of size2. Calling ¯ n the number of wires in each dimension, ¯ n is the number of crossingsbetween these wires (for simplicity the wires are assumed to be equally spaced). Thisconfiguration can be easily studied in the present collocation approach, by samplingthe wires on a grid and by then diagonalizing the Hamiltonian obtained following thisprocedure. The resulting energies calculated in this way will clearly depend on thespacing of the collocation grid, h , and diverge as h is sent to zero. To obtain finiteresults one needs to multiply these eigenvalues by h , which eliminates the divergencecaused by the shrinking of the transverse dimension. Following this procedure I havestudied different configurations, corresponding to choosing different value of ¯ n (goingfrom ¯ n = 1 to ¯ n = 4) and I have found that a given configuration has precisely the samenumber of bound states as the number of crossing. These bound states happen to bealmost exactly degenerate and correspond to wave functions which are localized on thevertices.In Table 4 I report the energy (multiplied by h ) of the bound states and of thefirst unbound state ( E gap ) for the different configurations. These results have beenobtained using a fine grid corresponding to h = 1 /
300 and show that the bound statesare precisely ¯ n as anticipated and they are essentially degenerate; the energy of thebound states and of the gap are also found to be almost insensitive to ¯ n , which can beinterpreted as a sign of confinement of a state to the crossings. I have also checked thedependence of these results upon N (or equivalently upon h ) observing that the energiescan be fitted excellently as E = a + b/N ; for example in the case of the ground stateof the configuration with ¯ n = 4 I have obtained: E = 2 . − . /N . olving the Helmholtz equation for membranes of arbitrary shape n h E h E - 2.59873 2.59869 2.59864 h E - 2.59873 2.59869 2.59864 h E - 2.59876 2.59872 2.59867 h E - - 2.59873 2.59868 h E - - 2.59873 2.59868 h E - - 2.59876 2.59871 h E - - 2.59876 2.59871 h E - - 2.59880 2.59874 h E - - - 2.59874 h E - - - 2.59875 h E - - - 2.59877 h E - - - 2.59877 h E - - - 2.59881 h E - - - 2.59881 h E - - - 2.59887 h E gap Table 4.
Energies of the bound states for configurations with different number ofcrossings, using N = 600, corresponding to a spacing h = 1 / In Fig.14 I have plotted the wave function of the ground state of the configurationcorresponding to ¯ n = 4 using a grid with N = 500. The wave function is clearly localizedat the crossings between the wires. Similar behaviour is observed for the remaining 15bound states.
7. Collocation with conformal mapping
The examples considered in the previous Sections show that it is possible to obtain thespectrum of the negative Laplacian over regions of arbitrary shape by using a collocationscheme, where the boundary conditions need not to be explicitly enforced on the border.Clearly, the precision of this approach should improve if the boundary conditions wouldbe enforced exactly on the border of the membrane. One way of achieving this result isby mapping conformally the shape of the membrane into a square (or a rectangle), onwhose border the LSF obey Dirichlet boundary conditions. I will discuss explicitly twoexamples of how this is done. olving the Helmholtz equation for membranes of arbitrary shape As a first example I consider a circular homogeneous membrane, which is exactly solvable(see for example [1]) and therefore it can be a useful tool to test the precision of thepresent method.The function f ( z ) = e − iπ sn (cid:16) z F (cid:16) sin − (cid:16) e − iπ (cid:17)(cid:12)(cid:12)(cid:12) − (cid:17)(cid:12)(cid:12)(cid:12) − (cid:17) (17)maps the unit square in the w complex plane into the unit circle in the complex z plane,as seen in Fig. 15. Under this mapping the original equation, − ∆ ψ ( w ) = λψ ( w ) (18)with Dirichlet boundary conditions on the unit circle, is mapped to − ∆ χ ( z ) = λσ ( z ) χ ( z ) (19)with Dirichlet boundary conditions on the unit square. Here σ ( z ) ≡ (cid:12)(cid:12) dwdz (cid:12)(cid:12) andeq. (19) describes the vibrations of a non-uniform square membrane. Although in theprevious Sections I have restricted the application of the method to the case of uniformmembranes of arbitrary shapes, the method can be applied also to inhomogenousmembranes straightforwardly. Let me briefly mention how this is done. As a firststep eq. (19) may be written in the equivalent form − σ ( z ) ∆ χ ( z ) = λχ ( z ) . (20)The operator ˆ O ≡ σ ( z ) ∆ is evaluated on a uniform grid in the z -plane using theLittle Sinc Functions (LSF). The action of the operator over a product of sinc functionscan be calculated very easily, as explained in the previous Sections. To make thediscussion simpler, I restrict to the equivalent one dimensional operator and make itact over a single LSF: − σ ( x ) d dx s k ( h, N, x ) = − (cid:88) jl σ ( x j ) c (2) kl s j ( h, N, x ) s l ( h, N, x ) ≈ − (cid:88) j σ ( x j ) c (2) kj s j ( h, N, x ) . (21)The matrix representation of the operator over the grid may now be read explicitlyfrom the expression above. The reader should notice that the matrix will not besymmetric unless the density is constant (cid:107) .Using this approach I have considered grids with N = 10 , , . . . ,
80 and I havehave calculated the first four even-even eigenvalues, which are shown in Table 5. Takinginto account the symmetry of problem I have used symmetrized LSF, which obey mixedboundary conditions (Dirichlet at one end and Neumann at the other hand): in thisway, for a given value of the N a grid of ( N/ points is used. As mentioned before the (cid:107) In general the calculation of the eigenvalues and eigenvectors of non–symmetric matrices iscomputationally more demanding than for symmetric matrices of equal dimension. olving the Helmholtz equation for membranes of arbitrary shape Figure 15.
Unit square in the z plane and the corresponding unit cirle in the w planereached through the trasformation (17). N E E E E
10 5.785633618 26.46056162 30.55061880 57.8818728820 5.783347847 26.37986506 30.47598468 57.6002666930 5.783218252 26.37564237 30.47217988 57.5862620740 5.783196213 26.37493961 30.47155075 57.5839791150 5.783190167 26.37474851 30.47138009 57.5833636360 5.783187992 26.37468004 30.47131902 57.5831440870 5.783187059 26.37465074 30.47129291 57.5830503580 5.783186606 26.37463653 30.47128023 57.58300497LSQ Table 5.
Even-even spectrum of the circular membrane: first four eigenvalues exact eigenvalues for this problem are known (the zeroes of the Bessel functions): theseare reported in the last row.In fig. 17 I have plotted the lowest eigenvalue of the circular membranecorresponding to different N and I have fitted these points using functions like c + c /N r ,with r = 3 , , N (cid:29) /N .Taking into account this behaviour I have considered the quantityΞ Q ≡ (cid:88) k =1 (cid:34) α − Q (cid:88) n =2 α n (10 k ) n +2 (cid:35) , (22)where Q = 8 and I have obtained the coefficients α n by minimizing Ξ Q (notice thatthis expression takes into account the leading 1 /N behaviour just discussed). The rowmarked as LSQ displays the quite precise results obtained following this procedure.I would like to discuss briefly a different issue. In [34] Gottlieb has used the Moebius olving the Helmholtz equation for membranes of arbitrary shape Figure 16.
Density of the inhomogeneous square membrane isospectral to thehomogeneous circular membrane.
Figure 17.
Energy of the ground state of the circular membrane. The dashed, solidand dotted lines correspond to fits using functions like c + c /N r , with r = 3 , , transformation f g ( z ) = ( z − a ) / (1 − az ) (23)to map the unit circle onto itself. This mapping transforms the homogenoeous Helmoltzequation for a circular membrane into the inhomogeneous Helmoltz equation for acircular membrane with density ρ ( x, y ) = (cid:12)(cid:12) f (cid:48) g ( z ) (cid:12)(cid:12) = ρ (1 − a ) [(1 − ax ) + a y ] . (24)Gottlieb uses this result to conclude that membranes corresponding to differentdensities, i.e. different values of a , are isospectral, thus providing a negative answer tothe famous question “Can one hear the shape of a drum?”, posed by Kac in [16]. I wish olving the Helmholtz equation for membranes of arbitrary shape a the mapping of eq. (23)deforms the grid inside the unit circle; as a is changed, the grid points move, as shownin Fig. 18. The case a = 0 is plotted in the right panel of Fig. 15. Clearly, if the densityof the membrane is constant, or symmetric with respect to the center, one expects that a = 0 provide the best grid. In Fig. 19 I have plotted the logarithm of the differencebetween the approximate and exact energy for the ground state of a circular membrane,∆ ≡ Log ( E N − E exact ), using three values of a ( a = 0, 0 . . fora given problem one can improve the numerical accuracy of a calculation by selecting anoptimal grid among those obtained through a conformal map of the region onto itself. The optimization of the parameter a depending on the specific problem considered isin the same spirit of the variational approach used in [35, 13, 14] and could provide auseful computational tool to boost the precision of the results. Figure 18.
Grid obtained with the Moebius map corresponding to a = 0 . a = − . Figure 19. ∆ ≡ Log ( E N − E exact ) using three values of a ( a = 0, 0 . . olving the Helmholtz equation for membranes of arbitrary shape The second example of application of conformal mapping to the solution of theHelmholtz equation is taken from the paper of Kuttler and Sigillito [2] (this problemwas also studied earlier by Moler, in ref.[101] of [2]).In Fig.20 two regions of the plane are displayed: the left plot corresponds to a squareof side π centered on the origin in the z = x + iy plane; the right plot corresponds to acircular waveguide with circular ridges in the w = u + iv plane. The function w = tan z maps the first region into the second one.As I have shown for the case of the circular membrane, the homogeneous Helmoholtzequation over the second region may be transformed into an inhomogeneous Helmholtzequation over the square: − ∆ U ( z ) = λσ ( z ) U ( z ) . (25)In the present case σ ( z ) ≡ (cid:12)(cid:12) dwdz (cid:12)(cid:12) = (cos x + cosh y ) and Dirichlet boundary conditionsare assumed on the borders of the two regions.In Tables 1,2 and 3 of their paper, Kuttler and Sigillito report different estimates forthe first 12 even-even eigenvalues, obtained using different approaches. In Table 2 theyalso apply Richardson extrapolation to the eigenvalues obtained with finite difference.In the case of the ground state of this membrane they also mention the precise valueobtained by Moler using the method of point matching λ = 7 . N . The results corresponding to the ground state areplotted in Fig. 21 and fitted using functions like c + c /N r , with r = 3 , , N (cid:29) /N , as for the circular membrane.The results in the Table have also been extrapolated using a least square approachΞ Q ≡ (cid:88) k =1 (cid:34) α − Q (cid:88) n =2 α n (10 k ) n +2 (cid:35) , (27)where Q = 7 , α n are coefficients which are obtained by minimizing Ξ Q . Notice thatthis expression takes into account the leading 1 /N behaviour just discussed. The rowsmarked as LSQ , display the results obtained following this procedure (the comparisonbetween the results for Q = 7 and Q = 8 gives an indication over the precision reached):in particular the energy of the ground state reproduces all the digits of the resultobtained by Moler. It is also remarkable that the energies obtained with the conformal-collocation method decrease monotonically when the number of collocation points isincreased (the only exception is represented by the E for N = 10, probably due to thelimited number of collocation points).As a technical remark, one should notice that the results corresponding to a givenvalue of N are obtained using a set of N/ olving the Helmholtz equation for membranes of arbitrary shape N E E E E
10 7.575738906 29.35369905 44.93667650 68.9953251420 7.569970385 29.12882337 44.84592568 67.9129803030 7.569654735 29.11799633 44.84124500 67.8635706540 7.569601533 29.11623444 44.84047707 67.8559248550 7.569586991 29.11575957 44.84026961 67.8539071060 7.569581767 29.11559019 44.84019553 67.8531950070 7.569579528 29.11551787 44.84016389 67.8528928380 7.569578441 29.11548286 44.84014857 67.85274711LSQ N E E E E
10 76.36327173 105.8649443 127.5818229 147.612811120 74.57343676 104.7105731 123.4501146 137.513674830 74.51254455 104.6448241 123.2916952 137.150875240 74.50340797 104.6345417 123.2690972 137.103303050 74.50101871 104.6318226 123.2633192 137.091479760 74.50017885 104.6308625 123.2613110 137.087423770 74.49982321 104.6304550 123.2604661 137.085729580 74.49965192 104.6302584 123.2600608 137.0849203LSQ N E E E E
10 152.6380731 175.0500571 202.7827432 229.615027820 147.1852075 177.5293898 193.4167694 213.436204830 147.1167888 177.2332164 193.0075863 212.844023040 147.1064916 177.1901085 192.9541314 212.771837450 147.1038082 177.1790645 192.9409198 212.754650760 147.1028673 177.1752263 192.9364009 212.748877470 147.1024696 177.1736121 192.9345164 212.746493880 147.1022783 177.1728379 192.9336173 212.7453635LSQ Table 6.
Even-even eigenvalues of the problem of eq. (25) using collocation with LittleSinc Functions (LSF). olving the Helmholtz equation for membranes of arbitrary shape
Figure 20.
Square in the z plane and corresponding region in the w plane, reachedthrough the conformal map w = tan z . Figure 21.
Energy of the ground state of the circular waveguide. The dashed, solidand dotted lines correspond to fits using functions like c + c /N r , with r = 3 , ,
8. Conclusions
In this paper I have used a collocation method based on LSF to obtain the numericalsolutions of the Helmholtz equation over two-dimensional regions of arbitrary shape. Alarge number of examples has been studied, illustrating the great potentialities of thepresent method. Among the principal virtues of this method I would like to mention itsgenerality (it can be applied to membranes of arbitrary shapes, including inhomogeneousmembranes, and to the Schr¨odinger equation – although I have not done this in thepresent paper), its simplicity (the matrix representation of the Helmholtz operator isobtained directly by collocation, and therefore it does not require the calculation of olving the Helmholtz equation for membranes of arbitrary shape Figure 22.
Upper panel: Even-even wave functions (absolute value): ground state and100 th excited state of the circular waveguide; Lower panel: Even-even wave functions(absolute value): 200 th and 300 th excited states of the circular waveguide. A grid with N = 80 is used. integrals) and the possibility of combining it with a conformal mapping, as done inthe last Section. In this last case, a rapid convergence to the exact eigenvalues isobserved as the number of grid points is increased. In the case where the border isnot treated exactly it has also been observed that the method provides monotonoussequences of approximations to the exact eigenvalue either from above or from below.Readers interested to looking at more examples of application of this method may finduseful to check the gallery of images which can be found at \protect\vrule(cid:32)width0pt\protect\href{http://fejer.ucol.mx/paolo/drum}{http://fejer.ucol.mx/paolo/drum} Appendix A. Mesh refinement
Although the collocation method described in this paper allows one to obtain precisesolutions to the Helmholtz equation over domains of arbitrary shape, in general theDirichlet boundary conditions are not enforced exactly over all the boundary. Asdiscussed in Section 7 the best approach consists of introducing a conformal map, which olving the Helmholtz equation for membranes of arbitrary shape N ). However we can use much simpler procedure, which doesnot require any additional diagonalization. Call N the parameter defining the size ofthe grid: a point in this grid is described by the direct product of the LSF in the x and y directions. In the Dirac notation we write (cid:104) x, y | k, k (cid:48) (cid:105) h ≈ s k ( h, x ) s k (cid:48) ( h, y ) , (A.1)assuming for simplicity that the grid has the same spacing in both directions. Let usnow concentrate on one of the LSF, say the one in the x direction: we take a finer grid,with a spacing h (cid:48) = h/l , where l is a integer. The new grid contains now ( lN −
1) points,including obviously the original grid points. However, it is clear that the original LSFcan be decomposed in the new grid as s k ( h, x ) = lN/ − (cid:88) j = − lN/ s k ( h, ¯ x j ) s j ( h/l, x ) , (A.2)where ¯ x j = 2 Lj/ ( lN ) are the new grid points. Notice that this relation is exact.The wave function of the n th state obtained from the diagonalization of the( N − × ( N −
1) hamiltonian reads ψ n ( x, y ) = 1 h (cid:88) K v ( n ) K s k ( K ) ( h, x ) s k (cid:48) ( K ) ( h, y )= 1 h (cid:88) K v ( n ) K lN/ − (cid:88) j = − lN/ s k ( K ) ( h, ¯ x j ) s j ( h/l, x ) lN/ − (cid:88) j (cid:48) = − lN/ s k (cid:48) ( K ) ( h, ¯ y j (cid:48) ) s j (cid:48) ( h/l, y ) , where v ( n ) is the n th eigenvector. Clearly ψ n ( x, y ) differs from 0 even in points of therefined grid which fall outside the membrane profile. We introduce a new matrix whoseelements are given by η jj (cid:48) = (cid:40) if (¯ x j , ¯ y j (cid:48) ) / ∈ B if (¯ x j , ¯ y j (cid:48) ) ∈ B (A.3) olving the Helmholtz equation for membranes of arbitrary shape ψ n ( x, y ) = N h lN/ − (cid:88) j = − lN/ lN/ − (cid:88) j (cid:48) = − lN/ ˜ V jj (cid:48) s j ( h/l, x ) s j (cid:48) ( h/l, y )where ˜ V jj (cid:48) ≡ η jj (cid:48) (cid:88) K v ( n ) K s k ( K ) ( h, ¯ x j ) s k (cid:48) ( K ) ( h, ¯ y j (cid:48) ) (A.4)and N is a normalization constant that ensures that (cid:90) B ψ n ( x, y ) dx dy = 1 . (A.5)It is easy to show that N = l (cid:113)(cid:80) jj (cid:48) ˜ V jj (cid:48) . (A.6)To simplify the notation I define: V jj (cid:48) ≡ N l ˜ V jj (cid:48) (A.7)and thus write:¯ ψ n ( x, y ) = lh lN/ − (cid:88) j = − lN/ lN/ − (cid:88) j (cid:48) = − lN/ V jj (cid:48) s j ( h/l, x ) s j (cid:48) ( h/l, y )On the other hand we may also calculate the expectation value of the Hamiltonianin this state (cid:104) ˆ H (cid:105) n = − (cid:90) B ¯ ψ n ( x, y ) ∆ ¯ ψ n ( x, y ) dx dy = − (cid:88) jj (cid:48) rr (cid:48) ss (cid:48) V jj (cid:48) V rr (cid:48) h (cid:104) ¯ c (2) rs δ r (cid:48) s (cid:48) + ¯ c (2) r (cid:48) s (cid:48) δ rs (cid:105) (cid:90) B s j ( h/l, x ) s j (cid:48) ( h/l, y ) s s ( h/l, x ) s s (cid:48) ( h/l, y )= − (cid:88) jj (cid:48) rr (cid:48) V jj (cid:48) V rr (cid:48) (cid:104) ¯ c (2) rj δ r (cid:48) j (cid:48) + ¯ c (2) r (cid:48) j (cid:48) δ rj (cid:105) = − (cid:88) jj (cid:48) r ¯ c (2) rj [ V jj (cid:48) V rj (cid:48) + V j (cid:48) j V j (cid:48) r ] (A.8)where ¯ c (2) is the matrix for the second derivative on the refined grid. An example ofapplication of this procedure is shown in Fig.13. References [1] A.L. Fetter and J.D.Walecka, Theoretical mechanics of particles and continua, McGraw Hill, NewYork (1980)[2] J.R.Kuttler and V.G.Sigillito, Eigenvalues of the Laplacian in two dimensions, Siam Review ,163-193 (1984) olving the Helmholtz equation for membranes of arbitrary shape [3] J.K.Reid and J.E.Walsh, An Elliptic Eigenvalue Problem for a Reentrant Region, Journal of theSociety for Industrial and Applied Mathematics , pp. 837-850 (1965)[4] L. Fox, P.Henrici and C.B.Moler, Approximations and bounds for eigenvalues of elliptic operators,Siam J. Numer. Anal. , 89-102 (1967)[5] J.C.Mason, Chebyshev polynomial approximations for the L-membrane eigenvalue problem, SIAMJ Appl. Math. , 172- (1967)[6] M.C.Milsted and J.R.Hutchinson, Use of trigonometric terms in the finite element method withapplication to vibrating membranes, J. of Sound and Vibration , 327-346 (1974)[7] A.B. Sideridis, A numerical solution of the membrane eigenvalue problem, Computing , 167-176(1984)[8] B. Schiff, Finite element eigenvalues for the Laplacian over an L-shaped domain, Journal ofComputational Physics , 233-242 (1988)[9] R.B.Platte and T.A.Driscoll, Computing eigenmodes of elliptic operators using radial basisfunctions, Computer and mathematics with applications , 561-576 (2004)[10] T.Betcke and L.N.Trefethen, Reviving the Method of Particular Solutions, Siam Review , 469-491 (2005)[11] L.N.Trefethen and T.Betcke, Computed eigenmodes of planar regions, AMS ComtemporaryMathematics , 297-314 (2006)[12] D.L.Kaufman, I.Kosztin and K.Schulten, Expansion method for stationary states of quantumbilliards, Am.J. Phys. , 133-141 (1999)[13] P.Amore, M.Cervantes and F.M.Fern´andez, Variational collocation on finite intervals, J.Phys. A
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