DEGENERACY CURVES, GAPS, and DIABOLICAL POINTS in the SPECTRA of NEUMANN PARALLELOGRAMS P Overfelt

DEGENERACY CURVES, GAPS, AND DIABOLICAL POINTS IN THE SPECTRA OF NEUMANN PARALLELOGRAMS P Overfelt

To cite this version:

P Overfelt. DEGENERACY CURVES, GAPS, AND DIABOLICAL POINTS IN THE SPECTRA OF NEUMANN PARALLELOGRAMS. 2020. hal-03017250

HAL Id: hal-03017250 https://hal.archives-ouvertes.fr/hal-03017250 Preprint submitted on 20 Nov 2020

HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. DEGENERACY CURVES, GAPS, AND DIABOLICAL POINTS IN THE SPECTRA OF NEUMANN PARALLELOGRAMS

P. L. OVERFELT

Abstract. In this paper we consider the problem of solving the Helmholtz equation over the space of all parallelograms subject to Neumann boundary conditions and determining the degeneracies occurring in their spectra upon changing the two parameters, angle and side ratio. This problem is solved numerically using the finite element method (FEM). Specifically for the lowest eleven normalized eigenvalue levels of the family of Neumann parallelograms, the intersection of two (or more) adjacent eigenvalue level surfaces occurs in one of three ways: either as an isolated point associated with the special geometries, i.e., the rectangle, the square, or the rhombus, as part of a degeneracy curve which appears to contain an infinite number of points, or as a diabolical point in the Neumann parallelogram spectrum. The degeneracies associated with the special geometries may occur as isolated points but are often part of a degeneracy curve. Each of these degeneracy curves comprises an extended seam between adjacent eigenvalue levels of slightly differing geometries where the normalized eigenvalues of the paired levels vary in value but preserve the level degeneracy. For the Neumann parallelogram case, some of the degeneracy curves are separated by degeneracy gaps, i.e., values of angle and side ratio for which degeneracies do not occur. Also the structure of these degeneracy curves is very different from those obtained in the Dirichlet parallelogram case [6]. For odd/even eigenvalue levels, the degeneracy curves have the appearance of bifurcation curves [14, 15, 16]. The even/odd normalized eigenvalue levels contain isolated point degeneracies of general Neumann parallelograms that appear unrelated to any of the degeneracy curves of those same levels. These are the equivalent of the diabolical triangles as in [2] and are thus called diabolical Neumann parallelograms. The lowest normalized eigenvalue of these diabolical parallelograms occurs for the 6, 7 levels at α = 46.4088 deg, l2 = 0.7624 with Λ = 3.1558. l1 There are two notable exceptions to the rule of degeneracy curves: the 2, 3 eigenvalue levels and the 4, 5 eigenvalue levels. Only isolated degeneracies associated with the rectangle, the rhombus, or the square have been determined for these levels. In solving the degeneracies of the spectrum of Neumann parallelograms, an unexpected benefit occurred. There are some eigenvalues and eigenvalue degeneracies that can be determined exactly for parallelograms with opening angle of α = 60 deg where the side ratio is rational. Although this is well- known for the 60 deg rhombus, as far as we are aware this has not been known for parallelograms with other rational side ratios.

1. Introduction Based on an old argument of Von Neumann and Wigner [1], and revisited by Berry [2, 3], it is known that for real operators (such as the Helmholtz operator)

Date: November 15, 2020. 1 2 P. L. OVERFELT at least two changing parameters are necessary to produce degeneracies in a family of geometrical shapes. This need for two changing parameters to produce degeneracies was considered theoretically by Arnol’d [4], Appendix 10, and demonstrated specifically by Berry and Wilkinson [2] who solved the Helmholtz equation on the space of all triangles subject to Dirichlet boundary conditions. For this family of planar shapes a number of accidental degeneracies or diabolical points were found for certain scalene triangles. These degeneracy points or degenerate triangles were relatively rare; they were isolated points in the space of all triangles and the number of them for each set of adjacent eigenvalue levels was finite. Also there were degeneracies that occurred for certain isosceles triangles and the equilateral triangle. These degeneracies were isolated points as well in the space of all triangles and only a finite number of them was determined for each pair of adjacent eigenvalue levels. These same general characteristics were noted for the problem of solving the Helmholtz equation over the space of all triangles subject to Neumann boundary conditions [5]. In this paper we consider the problem of solving the Helmholtz equation over the space of all parallelograms subject to Neumann boundary conditions and determining the degeneracies occurring in their spectra upon changing the two parameters, angle and side ratio, using the finite element method (FEM) [13]. As previously in the Dirichlet boundary condition case [6], the addition of the fourth boundary, and the fact that the parallelogram has an extra 180 degree rotation or Z2 symmetry as opposed to the family of all triangles introduces quite a difference in the degeneracies found for the two families of shapes. Specifically for the lowest eleven normalized eigenvalue levels of the family of Neumann parallelograms, the intersection of two (or more) adjacent eigenvalue level surfaces occurs in one of three ways: either as an isolated point associated with the special geometries, i.e., the rectangle, the square, or the rhombus, as part of a degeneracy curve which appears to contain an infinite number of points, or as a diabolical point in the Neumann parallelogram spectrum. The degeneracies associated with the special geometries may occur as isolated points but are often part of a degeneracy curve. Each of these degeneracy curves comprises an extended seam between adjacent eigenvalue levels of slightly differing geometries where the normalized eigenvalues of the paired levels vary in value but preserve the level degeneracy. For the Neumann parallelogram case, some of the degeneracy curves are separated by degeneracy gaps, i.e., values of angle and side ratio for which degeneracies do not occur. These degeneracy gaps appear for the 5, 6, the 7, 8, and the 9, 10 normalized eigenvalue levels. Also the structure of these degeneracy curves is very different from those obtained in the Dirichlet parallelogram case. Particu- larly for the above levels, the Neumann degeneracy curves have the appearance of bifurcation curves [14, 15, 16]. Additionally for the 6, 7, the 8, 9, and the 10, 11 normalized eigenvalue levels, isolated point degeneracies of general Neumann parallelograms have been determined that appear unrelated to any of the degeneracy curves of those same levels. These are the equivalent of the diabolical triangles as in [2] and are thus called diabolical Neumann parallelograms. The lowest normalized eigenvalue of these diabolical parallelograms occurs for the 6, 7 levels at α = 46.4088 deg, l2 = 0.7624 l1 with Λ = 3.1558 (see Table 3). There are again two notable exceptions to the rule of degeneracy curves: the 2, 3 eigenvalue levels and the 4, 5 eigenvalue levels. Only DEGENERACY CURVES, GAPS, AND DIABOLICAL POINTS 3 isolated degeneracies associated with the rectangle, the rhombus, or the square have been determined for these levels. No other general parallelogram degeneracies have been found for these levels. But all other levels (within the lowest eleven eigenvalue levels) contain at least one degeneracy curve and except for the 3, 4 levels, the odd/even curves for the Neumann case contain degeneracy gaps. The differences between the odd/even eigenvalue levels and the even/odd levels are interesting. The odd/even levels have degeneracy curves that extend through most of the (α, l2 ) parameter space, and such curves are organized by the degen- l1 eracies of the rectangle, square, and rhombus. They are separated by degeneracy gaps, and perhaps most importantly, we have been unable to determine diabolical parallelograms for these eigenvalue levels. The even/odd levels are not organized by the special geometry degeneracies. So far these levels contain only one degeneracy curve which is generally localized in the parameter space for the larger values of both l2 and α. They also contain the diabolical parallelograms. This dependence l1 on the parity of adjacent levels of parallelograms was commented on by Korsch [12]. In solving the degeneracies of the spectrum of Neumann parallelograms, an unexpected benefit occurred. There are some eigenvalues and eigenvalue degeneracies that can be determined exactly for parallelograms with opening angle of α = 60 deg where the side ratio is rational. Although this is well-known for the 60 deg rhombus, as far as we are aware this has not been known for parallelograms with other rational side ratios. Some of these lower level exact degeneracies are given in Table 4 of the Appendix. In the following the eigenvalues, λi, and the eigenfunctions, ψi(x, y) are com- puted numerically by solving the Helmholtz equation

(1) ∆ψi(x, y) + λiψi(x, y) = 0; (x, y) ∈ Ω with ∂ψ (x, y) (2) i = 0; (x, y) ∈ ∂Ω ∂n where (2) refers to the Neumann boundary condition and n denotes the outer normal to the boundary. Ω is the interior of any parallelogram while ∂Ω is its boundary. The normalized eigenvalues are given by λ A (3) Λ = n n 4π where A is the parallelogram area. Throughout the sequel unless otherwise spe- ciﬁed, the term “eigenvalues” refers to normalized eigenvalues.

2. Degeneracies of the Neumann Rectangle The rectangle shape is one of the few planar domains for which the eigenvalues and eigenfunctions of the Laplacian are known exactly. As a result degeneracies of the adjacent eigenvalue levels for the Neumann rectangle can be found exactly b also. Assuming a standard rectangle with side ratio, a , as in [6] (Figure 1) shows the lowest eleven eigenvalue levels of the Neumann rectangle vs. side ratio. The points of degeneracy for the rectangle occur near the close approaches of adjacent eigenvalue levels in Figure 1 and can be found exactly using the eigenvalue formula 4 P. L. OVERFELT

! πa b 2 (4) Λ (a, b) = m2 + n2 m,n 4b a

with m, n ∈ Z (m = 0, n = 0 are now allowed values of (4)). For a = 1,

π (5) Λ (1, b) = m2b2 + n2 m,n 4b Figure 1 shows that the lowest Neumann rectangle degeneracy occurs for ei- b genvalue levels 2, 3 at a = 1 (the square). The 3, 4 eigenvalue levels have one b 1 degeneracy which occurs at a = 2 . The rectangle degeneracies for the lowest eleven eigenvalue levels are given in Table 1. There is one degeneracy for the 2, 3 levels, one for the 3, 4 levels, two for the 4, 5 levels, three for the 5, 6 levels, four for the 6, 7 levels, ﬁve for the 7, 8 levels, six for the 8, 9 levels, seven for the 9, 10 levels, and nine for the 10, 11 levels. There is a total of 38 rectangle degeneracies b 2 for the lowest eleven eigenvalue levels. For all degeneracies ( a ) is rational [7, 8].

3. Degeneracies of the Neumann Rhombus Assuming a rhombus with side ratio, l2 = 1, and an opening angle of α (where l1 0 < α ≤ 90 deg), by plotting the adjacent eigenvalue levels versus the opening angle, degeneracies (if they are present) are found at those points where close approaches of adjacent eigenvalue levels occur (see Figure 2). A root ﬁnding technique was used to compute the rhombus degeneracies accurately (see Table 2). For the lowest eleven eigenvalue levels of the rhombus shape, there are 21 total degeneracies, not including those of the square. It is important to know the rhombus and rectangle degeneracies as accurately as possible for each set of adjacent eigenvalue levels because they are often the limiting cases for the degeneracies of the family of all parallelograms. In many instances (mainly for the odd/even eigenvalue levels) these special geometries organize the parameter space of the family of general parallelograms. Within the lowest eleven eigenvalue levels of the rhombus (and omitting the degeneracies of the square), the lowest rhombus degeneracy occurs for the 3, 4 eigenvalue levels at α = 60 deg. There is one rhombus degeneracy for the 3, 4 levels, one for the 4, 5 levels, two for the 5, 6 levels, three for the 6, 7 levels, two for the 7, 8 levels, ﬁve for the 8, 9 levels, four for the 9, 10 levels, and three for the 10, 11 levels (see Table 2). In Table 2, the square degeneracies have been included for completeness. It is worth noting that there are two degeneracies that occur for the α = 60 deg rhombus for the 3, 4 and the 8, 9 levels. These are degeneracies that can be known exactly, and any of the various formulas for the eigenvalues of the equilateral triangle [9, 10, 11] can be used to obtain a formula for some of the eigenvalues of the 60 deg rhombus given by

2π 2 2 (6) Λm,n = √ 3m + n 3 3 with m, n ∈ Z. In (6) using m = 0, n = 1 gives the 3, 4 level 60 deg rhombus degeneracy, and using m = 1, n = 1 gives the 8, 9 level 60 deg rhombus degeneracy. DEGENERACY CURVES, GAPS, AND DIABOLICAL POINTS 5

Levels (m,n) Λ(exact) Λ(FEM) b/a(exact) b/a(num) 2, 3 (0, 1), (1, 0) π/4 0.7854 1 1.0000 3, 4 (0, 1), (2, 0) π/√2 1.5708 1/√2 0.5000 4, 5 (1, 1), (2, 0) π/ 3 1.8138 1/ 3 0.5774 4, 5 (0, 1), (3, 0) 3π/√4 2.3562 1/√3 0.3333 5, 6 (1, 1), (3, 0) 9π/(8 2) 2.4991 1/ 8 0.3535 5, 6 (0, 1), (4, 0) π 3.1416 1/4 0.2500 5, 6 (0, 2), (2, 0) π√ 3.1416 1√ 1.0000 6, 7 (2, 1), (3, 0) 9π/(4√ 5) 3.1612 1/√ 5 0.4472 6, 7 (1, 1), (4, 0) 4π/ √15 3.2446 1√/ 15 0.2582 6, 7 (0, 2), (2, 1) 2π/ 3 3.6276 3/2 0.8660 6, 7 (0, 1), (5, 0) 5π/√4 3.9270 1/√5 0.2000 7, 8 (2, 1), (4, 0) 2π/ 3 3.6276 1/(2 3) 0.2880 7, 8 (1, 2), (2, 1) 5π/4√ 3.9270 1√ 1.0000 7, 8 (1, 1), (5, 0) 25π/(8 6) 4.0080 1/(2 6) 0.2041 7, 8 (0, 1), (6, 0) 3π/2 4.7124 1/6 0.1667 7, 8 (0, 2), (3, 0) 3π/√2 4.7124 2√/3 0.6667 8, 9 (2, 1), (5, 0) 25π/(4√ 21) 4.2847 1/ √21 0.2182 8, 9 (3, 1), (4, 0) 4π/√ 7 4.7496 1/√ 7 0.3780 8, 9 (1, 1), (6, 0) 9π/ √35 4.7792 1/ √35 0.1690 8, 9 (1, 2), (3, 0) 9π/(4√ 2) 4.9982 1/√2 0.7071 8, 9 (0, 2), (3, 1) π 3 5.4414 1/ 3 0.5774 8, 9 (0, 1), (7, 0) 7π/4 5.4978 1/7 0.1429 9, 10 (3, 1), (5, 0) 25π/√16 4.9087 1√/4 0.2500 9, 10 (2, 1), (6, 0) 9π/(4 √2) 4.9982 1/√32 0.1768 9, 10 (1, 1), (7, 0) 49π/(16 3) 5.5548 1/ 48 0.1443 √ 9, 10 (1, 2), (3, 1) 35π/(8 6) 5.6112 p3/8 0.6124 9, 10 (0, 1), (8, 0) 2π 6.2832 1/8 0.1250 9, 10 (0, 2), (4, 0) 2π 6.2832 1/2 0.5000 √ p 9, 10 (2, 2), (3, 0) 9π/(2√ 5) 6.3223 4√/5 0.8944 10, 11 (3, 1), (6, 0) π 3√ 5.4414 1/(3√3) 0.1924 10, 11 (2, 1), (7, 0) 49π/(12√ 5) 5.7369 1/(3√5) 0.1491 10, 11 (1, 1), (8, 0) 16π/√(3 7) 6.3329 1/(3√ 7) 0.1260 10, 11 (1, 2), (4, 0) 8π/ 15) 6.4892 2/ 15) 0.5164 √ 10, 11 (2, 2), (3, 1) 8π/ 15) 6.4892 p3/5 0.7746 10, 11 (4, 1), (5, 0) 25π/12 6.5450 1/3 0.3333 10, 11 (0, 3), (3, 0) 9π/4 7.0686 1 1.0000 10, 11 (0, 1), (9, 0) 9π/√4 7.0686 √1/9 0.1111 10, 11 (0, 2), (4, 1) 4π/ 3 7.2552 3/4 0.4330 Table 1. Neumann Rectangle Degeneracies 6 P. L. OVERFELT

Levels α(deg) Λ(Exact) Λ(FEM) 2, 3 90 π/4√ 0.7854 3, 4 60 2π/(3 3) 1.2092 4, 5 40.2714 1.7147 5, 6 31.1283 2.1252 5, 6 90 π 3.1416 5, 6 73.2123 3.1819 6, 7 24.7088 2.5734 6, 7 75.9545 3.3065 6, 7 51.3594 3.3981 7, 8 20.7292 2.9788 7, 8 36.2267 3.7812 7, 8 90 5π/4 3.9270 8, 9 17.6483 3.4098 8, 9 28.8459 √ 4.1367 8, 9 60 8π/(3 3) 4.8368 8, 9 51.1805 4.9940 8, 9 48.2867 5.0470 9, 10 15.4670 3.8130 9, 10 23.4978 4.5423 9, 10 48.9601 5.1170 9, 10 37.6231 5.2971 10, 11 13.6764 4.2356 10, 11 19.9984 4.9279 10, 11 29.1706 5.6418 10, 11 90 9π/4 7.0686 Table 2. Neumann Rhombus Degeneracies

4. Degeneracies of General Neumann Parallelograms As with the Dirichlet parallelogram case [6], the intersection of two (or more) eigenvalue levels occurs (if degeneracy curves are present) as an extended seam between adjacent eigenvalue levels of slightly diﬀering geometries where the eigenvalues of the adjacent levels vary in value but preserve the level degeneracy. Each pair of parallelogram adjacent eigenvalue levels appears to be composed of at least one eigenvalue degeneracy curve (with the 2, 3 levels and the 4, 5 levels as notable exceptions) within the eleven lowest eigenvalue levels, and for most levels there may be an uncountably inﬁnite number of values of angle, α, vs. side ratio, l2 , that pro- l1 duce degeneracies. Additionally for the even/odd eigenvalue levels, isolated point degeneracies of general Neumann parallelograms have been determined that appear unrelated to any of the degeneracy curves of those same levels. These are the equivalent of the diabolical triangles as in [2] and are thus called diabolical Neumann parallelograms. The lowest eigenvalue of these diabolical parallelograms occurs for the 6, 7 levels at α = 46.4088 deg, l2 = 0.7624 with Λ = 3.1558 (see Table 3). Also l1 the Neumann parallelogram case exhibits degeneracy gaps in a number of instances for the odd/even eigenvalue levels. DEGENERACY CURVES, GAPS, AND DIABOLICAL POINTS 7

Because the pairs of adjacent levels behave somewhat diﬀerently from one an- other and because in general the degeneracy values must be determined numerically subject to the mesh size used, we will use conjectures to show our conclusions, and then oﬀer the numerical evidence leading to these conjectures. The following conjectures refer to the space of all general parallelograms governed by Neumann boundary conditions within the lowest eleven eigenvalue levels. In these conjectures “lowest” refers to the eigenvalue level as well as the eigenvalue magnitude.

Conjecture 1 (2, 3 Eigenvalue Levels). The lowest eigenvalue degeneracy occurs at the 2, 3 eigenvalue levels for the square. There are no other 2, 3 level eigenvalue degeneracies.

Conjecture 2 (3, 4 Eigenvalue Levels). b 1 (a) There is one 3, 4 level eigenvalue degeneracy for the rectangle at a = 2 . (b) There is one 3, 4 level eigenvalue degeneracy for the rhombus at l2 = 1, l1 α = 60 deg. (c) If 0 < l2 < 1 , then there are no 3, 4 level eigenvalue degeneracies. l1 2 (d) If 1 ≤ l2 ≤ 1, then for every value of l2 in this region, there is one corres- 2 l1 l1 ponding value of α that gives a 3, 4 level eigenvalue degeneracy.

Conjecture 3 (4, 5 Eigenvalue Levels). b 1 (a) There are two 4, 5 level eigenvalue degeneracies for the rectangle at a = 3 q 1 and 3 . (b) There is one 4, 5 level eigenvalue degeneracy for the rhombus at l2 = 1, α = l1 40.2714 deg. (c) There are no other 4, 5 level eigenvalue degeneracies.

Conjecture 4 (5, 6 Eigenvalue Levels). b 1 (a) There are three 5, 6 level eigenvalue degeneracies for the rectangle at a = 4 , √1 , and 1. 2 2 (b) There are two 5, 6 level eigenvalue degeneracies for the rhombus at l2 = 1, l1 α = 31.2128 deg and 73.2123 deg. (c) If 0 < l2 < 1 , then there are no 5, 6 level eigenvalue degeneracies. l1 4 (d) If 1 ≤ l2 < √1 , then for every value of l2 in this region there is one 4 l1 2 2 l1 corresponding value of α that gives a 5, 6 level eigenvalue degeneracy. (e) If √1 ≤ l2 ≤ 0.3852, then for every value of l2 in this region there are two (2 2) l1 l1 corresponding values of α that give 5, 6 level eigenvalue degeneracies. (f) If 0.3853 ≤ l2 ≤ 0.4168, then for every value of l2 in this region there are no l1 l1 5, 6 level eigenvalue degeneracies. This is the only 5, 6 level degeneracy gap. (g) If 0.4169 ≤ l2 ≤ 1, then for every value of l2 in this region there are two l1 l1 corresponding values of α that give 5, 6 level eigenvalue degeneracies.

Conjecture 5 (6, 7 Eigenvalue Levels). b 1 (a) There are four 6, 7 level eigenvalue degeneracies for the rectangle at a = 5 , √ q 1 q 1 3 15 , 5 , and 2 . 8 P. L. OVERFELT

(b) There are three 6, 7 level eigenvalue degeneracies for the rhombus at l2 = 1, l1 α = 24.7088 deg, 51.3594 deg, and 75.9545 deg. (c) There are two 6, 7 level diabolical eigenvalue degeneracies for the general parallelogram at l2 = 0.4451, α = 36.6024 deg, and at l2 = 0.7624, α = 46.4088 deg. l1 √ l1 (d) If 0 < l2 < 3 , then there are no 6, 7 level eigenvalue degeneracies (except l1 2 for those in (a) and (c)) √ (e) If 3 ≤ l2 < 1, then for every value of l2 in this region there is one corres- 2 l1 l1 ponding value of α that gives a 6, 7 level eigenvalue degeneracy.

Conjecture 6 (7, 8 Eigenvalue Levels). b 1 (a) There are ﬁve 7, 8 level eigenvalue degeneracies for the rectangle at a = 6 , q 1 q 1 2 24 , 12 , 3 , and 1. (b) There are two 7, 8 level eigenvalue degeneracies for the rhombus at l2 = 1, l1 α = 20.7292 deg and 36.2267 deg. l2 1 (c) If 0 < l < 6 , then there are no 7, 8 level eigenvalue degeneracies. 1 q (d) If 1 ≤ l2 < 1 , then for every value of l2 in this region there is one 6 l1 24 l1 corresponding value of α that gives a 7, 8 level eigenvalue degeneracy. q (e) If 1 ≤ l2 < 0.2154, then for every value of l2 in this region there are two 24 l1 l1 corresponding values of α that give 7, 8 level eigenvalue degeneracies. (f) If 0.2154 ≤ l2 ≤ 0.2347, then for every value of l2 in this region there are no l1 l1 7, 8 level eigenvalue degeneracies. This is the only 7, 8 level degeneracy gap. q (g) If 0.2348 ≤ l2 < 1 , then for every value of l2 in this region there are two l1 12 l1 corresponding values of α that give a 7, 8 level eigenvalue degeneracy. q (h) If 1 ≤ l2 < 1 , then for every value of l2 in this region there are three 12 l1 3 l1 corresponding values of α that give 7, 8 level eigenvalue degeneracies. (i) If l2 = 1 , then there are two corresponding values of α that give 7, 8 level l1 3 eigenvalue degeneracies. (j) If 1 < l2 ≤ 2 , then for every value of l2 in this region there are three 3 l1 3 l1 corresponding values of α that give 7, 8 level eigenvalue degeneracies. (k) If 2 < l2 < 1, then for every value of l2 in this region there are two corres- 3 l1 l1 ponding values of α that give 7, 8 level eigenvalue degeneracies. (l) If l2 = 1 and α = 60 deg, then two of the 7, 8 level degeneracy curves l1 3 intersect. The eigenvalue at this point is Λ = √2π . 3 Conjecture 7 (8, 9 Eigenvalue Levels). b 1 (a) There are six 8, 9 level eigenvalue degeneracies for the rectangle at a = 7 , q 1 q 1 q 1 q 1 q 1 35 , 21 , 7 , 3 , and 2 . (b) There are ﬁve 8, 9 level eigenvalue degeneracies for the rhombus at l2 = 1 l1 and α = 17.6484 deg, 28.8459 deg, 48.2867 deg, 51.1805 deg, and 60 deg. (c) There are four 8, 9 level diabolical eigenvalue degeneracies for the general parallelogram at l2 = 0.5553, α = 22.4476 deg; l2 = 0.8685, α = 28.3905 deg; l1 l1 l2 = 0.3641, α = 38.1115 deg; and l2 = 0.2916, α = 36.1172 deg. l1 l1 (d) If 0 < l2 < 0.8684, then there are no 8, 9 level eigenvalue degeneracies l1 (except for those in (a) and (c)) DEGENERACY CURVES, GAPS, AND DIABOLICAL POINTS 9

(e) If 0.8684 ≤ l2 < 1, then for every value of l2 in this region there are two l1 l1 corresponding values of α that give 8, 9 level eigenvalue degeneracies.

Conjecture 8 (9, 10 Eigenvalue Levels). b 1 (a) There are seven 9, 10 level eigenvalue degeneracies for the rectangle at a = 8 , q 1 q 1 1 1 q 3 q 4 48 , 32 , 4 , 2 , 8 , and 5 . (b) There are four 9, 10 level eigenvalue degeneracies for the rhombus at l2 = 1 l1 and α = 15.4670 deg, 23.4978 deg, 37.6231 deg, and 48.9600 deg. l2 1 (c) If 0 < l < 8 , then there are no 9, 10 level eigenvalue degeneracies. 1 q (d) If 1 ≤ l2 < 1 , then for every value of l2 in this region there is one 8 l1 48 l1 corresponding value of α that gives a 9, 10 level eigenvalue degeneracy. q (e) If 1 ≤ l2 ≤ 0.1501, then for every value of l2 in this region there are two 48 l1 l1 corresponding values of α that give 9, 10 level eigenvalue degeneracies. (f) If 0.1502 ≤ l2 ≤ 0.1618 and if 64 deg < α < 72 deg (approximately), then for l1 every value of l2 in this region there are no corresponding values of α in the above l1 region that give 9, 10 level eigenvalue degeneracies. This is the first degeneracy gap of the 9, 10 eigenvalue levels. q (g) If 0.1619 ≤ l2 < 1 , then for every value of l2 in this region there are two l1 32 l1 corresponding values of α that give 9, 10 level eigenvalue degeneracies. q (h) If 1 ≤ l2 ≤ 0.1951, then for every value of l2 in this region there are 32 l1 l1 three corresponding values of α that give 9, 10 level eigenvalue degeneracies. (i) If 0.1952 ≤ l2 ≤ 0.2064 and if 59 deg < α < 64 deg (approximately), then for l1 every value of l2 in this region there are no corresponding values of α in the above l1 region that give 9, 10 level eigenvalue degeneracies. This is the second degeneracy gap of the 9, 10 eigenvalue levels. There is one total eigenvalue degeneracy for the value of l2 in this region for values of α < 59 deg . l1 (j) If 0.2065 ≤ l2 < 1 , then for every value of l2 in this region there are three l1 4 l1 corresponding values of α that give 9, 10 level eigenvalue degeneracies. (k) If 1 ≤ l2 ≤ 0.2813, then for every value of l2 in this region there are four 4 l1 l1 corresponding values of α that give 9, 10 level eigenvalue degeneracies. (l) If 0.2814 ≤ l2 ≤ 0.2980 and if 59 deg < α < 62 deg (approximately), then for l1 every value of l2 in this region there are no corresponding values of α in the above l1 region that give 9, 10 level eigenvalue degeneracies. This is the third degeneracy gap of the 9, 10 eigenvalue levels. There are two total eigenvalue degeneracies for the values of l2 in this region for values of α < 59 deg . l1 (m) If 0.2981 ≤ l2 < 1 , then for every value of l2 in this region there are four l1 2 l1 corresponding values of α that give 9, 10 level eigenvalue degeneracies. (n) If l2 = 1 , then there are five corresponding values of α that give 9, 10 level l1 2 eigenvalue degeneracies. q (o) If 1 < l2 < 3 , then for every value of l2 in this region there are four 2 l1 8 l1 corresponding values of α that give 9, 10 level eigenvalue degeneracies. q q (p) If 3 ≤ l2 ≤ 4 , then for every value of l2 in this region there are five 8 l1 5 l1 corresponding values of α that give 9, 10 level eigenvalue degeneracies. 10 P. L. OVERFELT

Levels l2 α(deg) Λ l1 6, 7 0.76241 46.4088 3.1558 6, 7 0.44509 36.6024 3.1613 8, 9 0.55533 22.4476 3.9510 8, 9 0.86851 28.3905 4.0760 8, 9 0.36413 38.1115 4.1902 8, 9 0.29160 36.1172 4.7011 10, 11 0.63135 16.1966 4.7442 10, 11 0.91887 19.9060 4.9060 10, 11 0.46988 23.4586 5.0126 10, 11 0.31144 39.7265 5.3436 10, 11 0.39143 22.1235 5.4795 10, 11 0.25257 36.8686 5.5611 10, 11 0.21721 35.9542 6.2566 10, 11 0.71840 44.5562 6.2838 Table 3. Diabolical Points of General Neumann Parallelograms

q (q) If 4 < l2 < 1, then for every value of l2 in this region there are four 5 l1 l1 corresponding values of α that give 9, 10 level eigenvalue degeneracies.

Conjecture 9 (10, 11 Eigenvalue Levels). b 1 (a) There are nine 10, 11 level eigenvalue degeneracies for the rectangle at a = 9 , q 1 q 1 q 1 1 q 3 q 4 q 3 63 , 45 , 27 , 3 , 16 , 15 , 5 , and 1. (b) There are three 10, 11 level eigenvalue degeneracies for the rhombus at l2 = 1 l1 and α = 13.6764 deg, 19.9984 deg, and 29.1706 deg . (c) There are eight 10, 11 level diabolical eigenvalue degeneracies for the general parallelogram (see Table 3). q (d) If 0 < l2 < 3 , then there are no 10, 11 level eigenvalue degeneracies l1 16 (except for those in (a) and (c)) q q (e) If 3 ≤ l2 ≤ 4 , then for every value of l2 in this region there is one 16 l1 15 l1 corresponding value of α that gives a 10, 11 level eigenvalue degeneracy (except for those in (a) and (c)). q (f) If 4 < l2 < 1, then there are no 10, 11 level eigenvalue degeneracies 15 l1 (except for those in (a) and (c))

5. Numerical Results Supporting the Conjectures for the Family of Neumann Parallelograms

Figures 3 - 16 show the numerical results of solving the Helmholtz equation on the family of parallelograms with Neumann boundary conditions based on the FEM from which the conjectures in Section 4 were generated. Each adjacent pair of eigenvalue levels is discussed separately. Throughout the remainder of the paper DEGENERACY CURVES, GAPS, AND DIABOLICAL POINTS 11 we refer to the rectangle degeneracies as given by b = l2 with α = 90 deg. We also a l1 assume that the entire parameter space is given by 0 < α ≤ 90 deg and 0 < l2 ≤ 1. l1

2, 3 Eigenvalue Levels: There is one 2, 3 level degeneracy which occurs for the square. (Recall that the Dirichlet parallelogram case also had one 2, 3 level degeneracy that occurred for the square). The 2, 3 level degeneracy is geometric in nature. For this level only one eigenvalue degeneracy has been determined and it is an isolated point. 3, 4 Eigenvalue Levels: The 3, 4 level eigenvalue degeneracies of the space of Neumann parallelograms consist of one degeneracy curve in α vs. l2 space (see l1 Figure 3). This curve is monotonically strictly decreasing. Figure 3 shows the values of α vs. l2 in two dimensions that give the degeneracies at the 3, 4 level. l1 This curve begins at the 3, 4 level rectangle degeneracy at l2 = 1 , α = 90 deg and l1 2 ends at the 3, 4 level rhombus degeneracy at l2 = 1 and α = 60 deg. It is bounded by l1 these two exact degeneracies. There appears to be an uncountably infinite number of degeneracies along this curve for 1 ≤ l2 ≤ 1 with 60 ≤ α ≤ 90 deg. The 3, 4 level 2 l1 degeneracy curve can be shown in three dimensions as in Figure 4. The values of l2 and α give the parallelogram geometries that are degenerate at this level while l1 Λ denotes the eigenvalue degeneracy magnitudes which run between 2√π ≤ Λ ≤ π . 3 3 2 Each point along this curve (to within the mesh size used) is a 3, 4 level eigenvalue degeneracy. In this case the lowest degeneracy occurs for the α = 60 deg rhombus; the highest occurs for the two to one rectangle. No degeneracy points off this curve have been determined. 4, 5 Eigenvalue Levels: The 4, 5 eigenvalue levels are similar to the 2, 3 levels. l2 1 There are two 4, 5 level degeneracies which occur for the rectangle at l = 3 and q 1 l2 = 1 (with α = 90 deg). There is one 4, 5 level degeneracy for the rhombus l1 3 at l2 = 1 and α = 40.2714 deg. No other parallelogram degeneracies have been l1 determined for the 4, 5 levels. These degeneracies are isolated points belonging to the special geometries. 5, 6 Eigenvalue Levels: The 5, 6 level eigenvalue degeneracies are a departure from the 3, 4 level degeneracies of Neumann parallelograms. There are two degeneracy curves that characterize this pair of eigenvalue levels. The shapes of both curves are significantly different from both the Dirichlet parallelogram curves and the 3, 4 level Neumann parallelogram curve discussed above. The α vs. l2 space l1 is again organized by the rectangle and rhombus degeneracies. Considering Figure 5 the curve for the smaller values of l2 is completely separated from the curve for l1 the larger values of l2 . There is a degeneracy gap between the two curves. This l1 gap occurs for 0.3853 ≤ l2 ≤ 0.4168. The lower curve begins at the first (smallest l1 in terms of l2 ) 5, 6 level rectangle degeneracy given by α = 90 deg, l2 = 1 . This l1 l1 4 curve runs to l2 = .3852 and then doubles back to end at the second rectangle l1 degeneracy given by α = 90 deg, l2 = √1 . Then the gap occurs and the curve l1 2 2 for the larger values of l2 begins on the other side of the gap at l2 = .4169. This l1 l1 second curve has two branches. One branch heads toward the rhombus degeneracy at l2 = 1, α = 31.1283 deg ; the other branch goes to the rhombus degeneracy at l1 12 P. L. OVERFELT l2 = 1, α = 73.2123 deg. There is a 5, 6 level degeneracy for the square but this l1 point is isolated from the two degeneracy curves. Figure 6 shows the 5, 6 level eigenvalue degeneracy curves in the three-dimensional (α, l2 , Λ)-space. They are well separated by the degeneracy gap. There are two l1 turning points - one occurs at (α, l2 , Λ) = (59.6159, 0.4169, 2.4832); the other occurs l1 at (64.2995, 0.3852, 2.5113). The eigenvalue degeneracies run from 2.1252 ≤ Λ ≤ π (without considering the gap) where π is the eigenvalue for the four to one rectangle and 2.1252 is the eigenvalue for the 31.1283 deg rhombus. This is the lowest eigenvalue degeneracy for the 5, 6 levels. There is a small region corresponding to the degeneracy gap in Figure 6, 2.4856 ≤ Λ ≤ 2.5081, where no eigenvalue degeneracies occur. These 5, 6 level degeneracy curves have the general form of bifurcation curves [14, 15, 16]. In particular Reference 15, Chapter 4 discusses using the domain of definition as the bifurcation parameter. From this perspective we see that the two dimensional degeneracy plots of the parameter space, (α, l2 ) for all paired l1 eigenvalue levels are actually parameter charts showing multiplicities of solutions (i, e., the different numbers of degeneracies) [15]. Of course with respect to changes in the number of degeneracies as the parameters change, most of the Dirichlet and Neumann j, j + 1 level degeneracy curves show abrupt changes in the numbers of degeneracies, often at the rectangle degeneracy points for those j, j + 1 levels, though not always. But the Neumann parallelogram 5, 6 level degeneracy curves are the first to show a degeneracy gap between curves with separate branches and the attendant turning points on either side of the gap which occur as cited above. Without regard to the degeneracy gap, the 5, 6 levels have values of 1 < l2 ≤ 1, 4 l1 while the values of alpha run between 31.1283 ≤ α ≤ 90 deg. From Figure 6 we see that the highest eigenvalue degeneracy occurs for the 4 − 1 rectangle and the smallest for the 31.1283 deg rhombus.

6, 7 Eigenvalue Levels: The degeneracies of the 6, 7 eigenvalue levels are shown in Figure 7. There are three rhombus degeneracies along l2 = 1 and four rectangle l1 degeneracies along α = 90 deg (see Tables 1 and 2). There is a single degeneracy √ curve that runs from the fourth rectangle degeneracy at l2 = 3 to the rhombus l1 2 degeneracy at α = 75.9545 deg. This curve appears to have inﬁnitely many points and is monotonically strictly decreasing. Notably this is the lowest pair of eigenvalue levels for which diabolical points have been determined. These are the red dots in Figure 7. Their numerical values are given in Table 3. A three dimensional plot of the 6, 7 level degeneracies is shown in Figure 8. The degeneracy with the lowest eigenvalue magnitude is the rhombus with α = 24.7088 deg and Λ = 2.5734. The Neumann parallelogram diabolical points have the next lowest eigenvalues while the degeneracies of the curve are higher. The ﬁve to one rectangle has the highest degeneracy value (Λ = 3.9270) for this case.

7, 8 Eigenvalue levels: There are several 7, 8 level eigenvalue degeneracy curves, and two of them intersect (a new feature not seen previously). Considering Figure 9, the curve with the smallest l2 values begins at the first rectangle degeneracy l1 at l2 = 1 , runs to a turning point at l2 = 0.2153 with α = 69.1275 deg, then l1 6 l1 doubles back to end at the second rectangle degeneracy at l2 = √1 . There is a l1 2 6 degeneracy gap in the region 0.2154 ≤ l2 ≤ 0.2347 where no 7, 8 level degeneracies l1 DEGENERACY CURVES, GAPS, AND DIABOLICAL POINTS 13 occur. A second curve begins on the opposite side of the gap and has two branches. One branch runs from l2 = 0.2348 with α = 62.5372 deg and heads toward the l1 rhombus 7, 8 level degeneracy at α = 20.7292 deg. The other branch runs from l2 = 0.2348 with α = 62.5372 deg and runs to the fourth rectangle degeneracy at l1 l2 = 2 . The third degeneracy curve runs from the third rectangle degeneracy at l1 3 l2 = √1 and heads toward the rhombus 7, 8 level degeneracy at α = 36.2267 deg. l1 2 3 This third curve intersects with the second branch of the second curve at the point l2 = 1 , α = 60 deg (this intersection point is exact). This is the first time that two l1 3 degeneracy curves of the same pair of eigenvalue levels has intersected. As with the 5, 6 levels, these degeneracy curves look very much like bifurcation curves in both the two-dimensional (α, l2 )-space and the three-dimensional (α, l2 , Λ) - space. l1 l1 There is also a 7, 8 level degeneracy for the square but this is an isolated point. No other degeneracies have been determined near it. Figure 10 shows the three-dimensional form of the 7, 8 level degeneracy curves. The rhombus with opening angle α = 20.7292 deg with degeneracy eigenvalue Λ = 2.9788 has the lowest eigenvalue of the 7, 8 levels. The highest occurs for the rectangle geometry at l2 = 2 with Λ = 4.7124. l1 3 8, 9 Eigenvalue Levels: The 8, 9 eigenvalue levels contain one degeneracy curve, four Neumann parallelogram diabolical points (see the red dots in Figure 11) with values shown in Table 3, and a number of isolated points that occur for the rectangle and rhombus geometries. The set of parallelograms that comprises the 8, 9 level degeneracy curve is composed of those geometries with values of .8684 ≤ l2 ≤ 1 l1 and values of α that run from 51.1805 < α < 60 deg (see Figure 11 ). The shape of the curve is a skewed parabola with a minimum at l2 = 0.8684. This minimum l1 is not related to any of the rectangle and rhombus degeneracies. As far as we have determined, this curve does not contain gaps and there are an infinite number of points on the curve. The curve involves only the rhombus degeneracies at α = 60 deg and α = 51.1805 deg. All other 8, 9 level degeneracies are isolated points. Interestingly the other rectangle and rhombus degeneracies of this level do not help to organize the (α, l2 ) - space. l1 Considering the 8, 9 level eigenvalue degeneracies in three dimensional (α, l2 , Λ) l1 - space (see Figure 12), the eigenvalues of the degeneracy curve run from 4.8368 ≤ Λ ≤ 4.9915. The lowest eigenvalue degeneracy occurs for the rhombus with α = 17.6484 deg. The next lowest degeneracy occurs for one of the diabolical points at α = 22.4476 deg, l2 = 0.5553, and Λ = 3.9510. The other diabolical points l1 are a little higher than the one above, and the degeneracy curve has even higher eigenvalue degeneracies. The highest occurs for the seven to one rectangle with Λ = 5.4978. 9, 10 Eigenvalue Levels: The 9, 10 level eigenvalue degeneracy curves are quite complicated, much more so than for any previous pair of levels. The (α, l2 )- l1 parameter space consists of five degeneracy curves that are completely organized by the 9, 10 level rectangle and rhombus degeneracies. These curves also contain three degeneracy gaps as shown in Figure 13. The lowest (in terms of the value of l2 ) degeneracy curve begins at the first rectangle degeneracy and heads to a turning l1 point at l2 = 0.1502, α = 71.7486 deg. Then it doubles back to end at the second l1 rectangle degeneracy at l2 = 0.1443, α = 90 deg. At the turning point above, a l1 14 P. L. OVERFELT degeneracy gap appears for the region 0.1502 ≤ l2 ≤ 0.1618. For these values of l1 the side ratio and for 0 < α ≤ 90 deg, no degeneracies have been found inside this gap. The second lowest degeneracy curve begins at the third rectangle degeneracy, has a turning point at l2 = 0.1951, α = 63.2826 deg and runs to a second turning l1 point at l2 = .1618, then doubles back and heads toward the smallest (in terms of l1 α) 9, 10 level rhombus degeneracy at l2 = 1, α = 15.4670 deg and ends there. The l1 third lowest degeneracy curve begins at the fourth rectangle degeneracy, runs to a turning point at l2 = .2814 with α near 61 − 62 deg, heads to a second turning l1 point at l2 = .2064 with α near 60 deg and then runs to the second smallest l1 rhombus degeneracy at l2 = 1, α = 23.4978 deg. The fourth curve runs from l1 the third lowest rhombus degeneracy at l2 = 1, α = 37.6231 deg, heads toward l1 a turning point at l2 = .2981, α = 59.1864 deg, then runs to an inflection point l1 at l2 = 0.5070, α = 72.0600 deg, and then runs to the fourth lowest rhombus l1 degeneracy at l2 = 1, α = 48.9600 deg. The fifth degeneracy curve begins at the l1 sixth rectangle degeneracy, has a turning point at l2 = .707, α = 73.4058 deg, and l1 doubles back to end at the seventh (highest in terms of l2 ) rectangle degeneracy. l1 The three dimensional degeneracy curves for the 9, 10 eigenvalue levels are shown in Figure 14. The eigenvalue magnitudes for the 9, 10 levels run between 3.8132 ≤ Λ ≤ 6.4032. The lowest eigenvalue magnitude occurs for the rhombus with opening angle of 15.4670 deg and Λ = 3.813 while the highest magnitude occurs for the general parallelogram with l2 = .5004, α = 72.0222 deg, and Λ = 6.4032. Unlike l1 the 7, 8 levels, none of the five degeneracy curves for the 9, 10 levels intersects with any other. 10, 11 Eigenvalue Levels: The 10, 11 eigenvalue levels contain one degeneracy curve, a number of isolated point degeneracies that occur for the special geometries, and a number of diabolical parallelograms. In Figure 15 the degeneracy curve for q q this case is shown in two dimensions as the solid black line with 3 ≤ l2 ≤ 4 16 l1 15 and 76.4 ≤ α ≤ 90 deg. It runs only between the sixth and seventh rectangle degeneracies. This curve contains no gaps and is quite localized in terms of the total parameter space covered. Of special note are the eight diabolical Neumann parallelogram degeneracies as shown by the red dots in Figure 15. These points are given numerically in Table 3. The majority of the rectangle and rhombus degeneracies (shown by the black dots in Figure 15) do not help to organize the parameter space; only two of these special geometries do this. Figure 16 shows the 10, 11 degeneracies in three dimensions. The eigenvalue degeneracy magnitudes for this case run between 4.2356 ≤ Λ ≤ 7.2552. The lowest degeneracy occurs for the q 13.6764 deg rhombus, and the highest occurs for the rectangle with l2 = 3 . l1 16

6. Generalizations for Adjacent Eigenvalue Levels

Within the lowest eleven eigenvalue levels of Neumann parallelograms, there are a few generalizations to be made concerning the degeneracies of adjacent j, j + 1 levels. These generalizations fall under the categories of (1) parameter space organizing centers, (2) presence or absence of diabolical points, and (3) appearance of degeneracy curve gaps. DEGENERACY CURVES, GAPS, AND DIABOLICAL POINTS 15

Parameter Space Organizing Centers: With respect to the rectangle and rhombus degeneracies at each level, these special geometries organize the (α, l2 ) -parameter l1 space differently based on the oddness or evenness of the adjacent j, j + 1 levels. For j odd, j + 1 even, the degeneracy curves for these levels all begin and end on associated rectangle/rhombus degeneracies of the same levels. There are three possibilities for the odd/even eigenvalue levels. A degeneracy curve can begin on a rectangle degeneracy and end on a rhombus degeneracy (e.g., the 3, 4 level curve); a curve can begin on a rectangle degeneracy and double back to end on a different rectangle degeneracy (e.g., the lower (in terms of l2 ) degeneracy curve of the 5, l1 6 levels); and a degeneracy curve can begin on a rhombus degeneracy and double back to end on a different rhombus degeneracy (e.g., the higher curve of the 5, 6 levels). As the level numbers increase we see that all combinations of the above are allowed (see Figure 13 for the 9, 10 levels). For these odd/even levels, the parameter space is characterized by degeneracy curves whose numbers increase as j increases. For this case the majority of rectangle and rhombus degeneracies are parts of degeneracy curves, not isolated points. This organization of the parameter space by the special geometry degeneracies is much less in evidence for the j even, j +1 odd levels. These levels are characterized by isolated point rectangle/rhombus degeneracies with at most one degeneracy curve. Recall that no degeneracy curves have been determined for the 2, 3 and 4, 5 levels. But there is one degeneracy curve for each of the 6/7, the 8/9, and the 10/11 levels. In each case these degeneracy curves are tied to the larger values of l2 . In contrast to the degeneracy curves l1 of the odd/even levels, the even/odd level curves are much more localized in the parameter space. Diabolical Points: The appearance or absence of diabolical points (isolated points that occur for general parallelogram geometries) so far seems related to the oddness or evenness of adjacent eigenvalue levels. Diabolical points have been determined for the 6/7, 8/9, and 10/11 levels (see Table 3 and Figures 7, 11, and 15), but we have been unable to determine them for any j odd, j + 1 even levels. This may be related to the odd/even parity of parallelogram wave functions with respect to a 180 deg rotation as suggested by Korsch [12]. Also we speculate that this lack of diabolical points may simply be an effect of masking by the presence of the larger size and number of degeneracy curves that extend throughout the parameter space for the odd/even levels. Gaps in the Degeneracy Curves: The odd/even adjacent eigenvalue levels also exhibit a phenomenon that is unique to the Neumann parallelogram case. With the exception of the 3, 4 levels, all other odd/even levels contain multiple degeneracy curves that are separated by degeneracy gaps, i.e., areas of l2 and α where no l1 eigenvalue degeneracies occur. No gaps have been determined for any of the curves of the even/odd levels. For the odd/even levels, the number of gaps increases as the level numbers increase. The 5, 6 and the 7, 8 level degeneracy curves each contain only one degeneracy gap, while the 9, 10 level curves contain three gaps. These gaps are what contribute to the degeneracy curves of the parameter space of the odd/even levels looking like bifurcation curves. 16 P. L. OVERFELT

7. Conclusions

Using the family of parallelograms with Neumann boundary conditions, we have determined the degeneracies occurring in the spectra of these shapes. Some of the degeneracies are expected and determined for the special rectangle, square, and rhombus geometries. We have determined two types of isolated point degeneracies - those occurring for the special geometries that are not part of a degeneracy curve, and those Neumann parallelogram diabolical points that occur for the even/odd eigenvalue levels. The overwhelming number of degeneracies appear as as points of degeneracy curves - they comprise an extended seam between adjacent eigenvalue levels of slightly differing geometries where the eigenvalue magnitudes of the paired levels vary in value but preserve the level degeneracy. The odd/even eigenvalue levels are characterized by degeneracy curves that extend throughout the parameter space with the number of such curves increasing with level number. For these levels the degeneracy curves have the rectangle and rhombus degeneracies as the organizing centers of the parameter space. These curves also show degeneracy gaps, values of l2 and α where no eigenvalue degeneracies occur. These gaps contribute l1 to the form of certain degeneracy curves’ likeness to bifurcation curves. The interesting appearance of diabolical Neumann parallelograms has so far been determined only for the j even, j+1 odd eigenvalue levels. So far the even/odd levels contain isolated point special geometry degeneracies, one localized degeneracy curve confined to the larger values of l2 and α (excepting the 2/3 and 4/5 eigenvalue l1 levels), and a number of diabolical points. The lowest eigenvalue of these diabolical Neumann parallelograms occurs for the 6/7 levels at α = 46.4088 deg, l2 = .7624, l1 with Λ = 3.1558. As an added benefit stemming from the numerical solution of the Helmholtz equation on parallelograms with Neumann boundary conditions, we have also determined a number of exact eigenvalues and eigenvalue degeneracies for certain parallelograms with opening angle of α = 60 deg (see Appendix A). While some exact eigenvalues for the 60 deg rhombus are well known, we have also determined some exact values for general parallelograms with rational side ratios and α = 60 deg. Exact eigenvalues for certain degeneracies of these more general parallelograms were found to be multiples of the lowest-order nonzero normalized Neumann eigenvalue of the equilateral triangle. In fact the amount of degeneracy encountered for each adjacent pair of eigenvalue levels in the family of parallelogram shapes seems surprising, especially in view of a number of general results proving that the eigenvalues of the Helmholtz equation for most C2-regular bounded regions are simple for both Dirichlet boundary conditions [16, 17] and for Neumann boundary conditions [16, 18]. However as pointed out by Dan Henry [16], Appendix 1, “Given this result, it may be surprising to find higher multiplicity in many specific examples... It seems that higher multiplicity appears whenever a detailed analysis can be done. Now, a common characteristic in these cases is the symmetry of the regions.” In conclusion eigenvalue degeneracies in the spectra of the Neumann Laplacian on the space of all parallelograms have been determined. These degeneracies consist of the degeneracies of the special geometries (the square, rectangle, and rhombus), the degeneracy curves that appear for most eigenvalue levels, and the diabolical Neumann parallelograms. The degeneracy curves of the odd/even levels exhibit DEGENERACY CURVES, GAPS, AND DIABOLICAL POINTS 17 degeneracy gaps and such curves with gaps have forms similar to those of bifurcation curves. The even/odd levels (with the exception of the 2, 3 and 4, 5 levels) exhibit the Neumann diabolical points which are easily determined for that case but which have not been found (at least so far) for the odd/even levels.

References [1] J. von Neumann and E. P. Wigner, ”On the behavior of eigenvalues in adiabatic processes”, Physik. Z. 30, p. 467 (1929). Translated in R. S. Knox and A. Gold, Symmetry in the Solid State, New York: Benjamin (1964), p.167. [2] M. V. Berry and M. Wilkinson, Diabolical points in the spectra of triangles, Proc. R. Soc. Lond. A 392, 15-43 (1984). [3] M. V. Berry, Aspects of Degeneracy, in Chaotic Behavior in Quantum Systems, G. Casati, ed., Plenum Press, New York (1985), pp. 123 - 140. [4] V. I. Arnol’d, Mathematical Methods of Classical Mechanics, New York: Springer (1978). [5] P. L. Overfelt, Degeneracies in the Spectra of Neumann Triangles, (2018) (hal - 01875118). [6] P. L. Overfelt, Degeneracy Curves in the Spectra of Dirichlet Parallelograms, (2019) (hal - 02113056). [7] G. B. Shaw, Degeneracy in the particle in a box problem, J. Phys. A 7 (13) 1537-1546 (1973). [8] P. L. Overfelt, Rings, quadratic forms, and complete degeneracy for a subclass of highly over- moded waveguides, J. Math. Phys., 34, 2975 (1993). [9] M. A. Pinsky, The eigenvalues of an equilateral triangle, SIAM J. Math. Anal. 11, 819-827 (1980). [10] P. L. Overfelt and D. J. White, TE and TM modes of some triangular cross-section waveguides using superposition of plane waves, IEEE Transactions on Microwave Theory and Tech- niques 34, 161-167 (1986). [11] B. J. McCartin, Laplacian Eigenstructure of the Equilateral Triangle (Hikari Ltd, 2011). [12] H. J. Korsch, On the nodal behavior of eigenfunctions, Phys. Lett. A 97, 77-80 (1983). [13] Wolfram Research, Inc., Mathematica, Version 11.0, Champaigne, IL (2017). [14] J. Hale and H. Kocak, Dynamics and Bifurcations, New York: Springer-Verlag, (1991). [15] R. Seydel, Practical Bifurcation and Stability Analysis, New York: Springer-Verlag, (1994). [16] D. Henry, Perturbation of the Boundary in Boundary Value Problems of Partial Diﬀerential Equations, Cambridge: London Mathematical Society Lecture Notes Series 318 (2005). [17] A. L. Pereira, Eigenvalues of the Laplacian on symmetric regions, Nonlinear Diﬀerential Equations Applications 2, 63-109 (1995). [18] M. A. M. Morrocos and A. L. Pereira, Eigenvalues of the Neumann Laplacian in symmetric regions, arXiv:1310.5178v1 [math.AP] 18 October (2013). [19] B. J. McCartin,On polygonal domains with trigonometric eigenfunctions of the Laplacian under Dirichlet or Neumann boundary conditions, Applied Mathematical Sciences 2, 2891- 2901 (2008). 18 P. L. OVERFELT

Appendix A. Exact Eigenvalues for Neumann Parallelograms with 60 deg Opening Angle and Rational Side Ratio There are a number of eigenvalues and eigenvalue degeneracies for certain parallelogram opening angles, notably α = 60 deg and α = 45 deg, that can be determined exactly. While it is well known that the spectrum of the 60 deg rhombus contains an incomplete set of exact eigenfunctions and eigenvalues [19], the fact that this is also true for certain eigenvalues of more general parallelograms with α = 60 deg and rational side ratios is a little surprising. The following table (Table 4) gives some of these exact eigenvalues and eigenvalue degeneracies and com- pares this result with the FEM approximation assuming a 10−6 mesh size. The eigenvalues are given from smallest to largest for those parallelogram geometries having normalized eigenvalues with magnitude less than ten. Note that all the exact eigenvalues in Table 4 are multiples of the lowest order nonzero Neumann equilateral triangle eigenvalue. Parallelogram geometries with 60 deg opening angle and rational side ratio can be comprised of an even number of congruent equilateral triangles. They contain some trigonometric modes, and these form incomplete sets of eigenfunctions of the above shapes that can be determined exactly. For example the 60 deg rhombus can be decomposed into two congruent equilateral triangles and the 3, 4 level eigenvalue degeneracy magnitude is twice that of the normalized√ lowest-order nonzero Neumann equilateral triangle eigenvalue given by π/(3 3) = .6046... The α = 60 deg, l2 = 1 parallelogram can be decomposed l1 2 into four congruent equilateral triangles and has a 5, 6 level eigenvalue degeneracy magnitude that is four times that of the lowest-order nonzero Neumann equilateral triangle eigenvalue. The α = 60 deg, l2 = 1 parallelogram can be decomposed l1 3 into six congruent equilateral triangles and has a 7, 8 level eigenvalue degeneracy magnitude that is six times that of the lowest-order nonzero Neumann equilateral triangle eigenvalue, and so on. Thus McCartin’s Theorem 3 [19] can be used to explain these partial sets of trigonometric eigenfunctions and exact eigenvalues found for all parallelograms with α = 60 deg and l2 ∈ with prototiles that are equilateral triangles. l1 Q It is also possible to obtain some exact eigenvalues for the α = 45 deg parallelogram case with l2 = √1 (even though irrational). Some of the exact eigenvalues for l1 2 this case are given by the eigenvalues of the isosceles right triangle. This case does not appear to contain any degeneracies within the lowest eleven eigenvalue levels. These exact eigenvalues and eigenvalue degeneracies allow us to obtain a direct relative error comparison with the FEM method used [13]. The agreement between the approximate and exact values is very good. DEGENERACY CURVES, GAPS, AND DIABOLICAL POINTS 19 l2 Levels Λ (FEM) Exact Λ Exact Λ(num) Relative Error l1 1 3, 4 Degen 1.209199576156205 2√π 1.209199576156145 4.97636 × 10−14 3 3 1 5, 6 Degen 2.418399152343515 4√π 2.418399152312290 1.29115 × 10−11 2 3 3 1 7 Eigenval 3.627598728494261 √2π 3.627598728468436 7.11894 × 10−12 3 1 7, 8 Degen 3.627598728706618 √2π 3.627598728468436 6.56583 × 10−11 3 3 1 8, 9 Degen 4.836798304686590 8√π 4.836798304624581 1.28203 × 10−11 3 3 1 9, 10 Degen 4.836798304822762 8√π 4.836798304624581 4.09736 × 10−11 4 3 3 1 11, 12 Degen 6.045997881385475 10√π 6.045997880780726 1.00025 × 10−10 5 3 3 2 11, 12 Degen 7.255197457413547 √4π 7.255197456936871 6.57013 × 10−11 3 3 1 11 Eigenval 7.255197457787633 √4π 7.255197456936871 1.17262 × 10−10 2 3 1 13, 14 Degen 7.255197458443055 √4π 7.255197456936871 2.07601 × 10−10 6 3 1 13, 14 Degen 8.464397033429448 14√π 8.464397033093017 3.97466 × 10−11 3 3 1 15, 16 Degen 8.464397034123818 14√π 8.464397033093017 1.21781 × 10−10 7 3 3 1 14, 15 Degen 9.673596611258041 16√π 9.673596609249162 2.07666 × 10−10 2 3 3 1 17, 18 Degen 9.673596609646006 16√π 9.673596609249162 4.10235 × 10−11 8 3 3 Table 4. Exact Eigenvalues for Neumann Parallelograms with 60 deg Opening Angle and Rational Side Ratio (Λ < 10) 20 P. L. OVERFELT

1 9 6

2 10

3 11

4 Λ 4

6 2 7

0.2 0.4 0.6 0.8 1.0 b/a

Figure 1. Eigenvalues Versus Side Ratio for the Eleven Lowest Levels of Neumann Rectangles DEGENERACY CURVES, GAPS, AND DIABOLICAL POINTS 21

6 1 9

2 10

3 11

4 Λ 4

6 2 7

20 40 60 80 α(deg)

Figure 2. Eigenvalues Versus Opening Angle for the Eleven Low- est Levels of Neumann Rhombi 22 P. L. OVERFELT

Rhombus Degeneracy 1.0

0.8

0.6 l2/l1

Rectangle Degeneracy

0.4

0.2

20 30 40 50 60 70 80 90 α(deg)

Figure 3. Two Dimensional Degeneracy Curve for the 3, 4 Ei- genvalue Levels DEGENERACY CURVES, GAPS, AND DIABOLICAL POINTS 23

Figure 4. Three Dimensional Degeneracy Curve for the 3, 4 Ei- genvalue Levels 24 P. L. OVERFELT

Rhombus Degeneracies 1.0

0.8

0.6 l2/l1

. 0.4 Gap 2nd Rectangle Degeneracy

1st Rectangle Degeneracy

0.2

30 40 50 60 70 80 90 α(deg)

Figure 5. Two Dimensional Degeneracy Curves for the 5, 6 Ei- genvalue Levels DEGENERACY CURVES, GAPS, AND DIABOLICAL POINTS 25

Figure 6. Three Dimensional Degeneracy Curves for the 5, 6 Ei- genvalue Levels 26 P. L. OVERFELT

Rhombus Degeneracies 1.0

4th Rectangle Degeneracy

0.8

l2/l1 0.6

3rd Rectangle Degeneracy

0.4

2nd Rectangle Degeneracy

1st Rectangle Degeneracy 0.2

0 20 40 60 80 α(deg)

Figure 7. Two Dimensional Degeneracies for the 6, 7 Eigenvalue Levels DEGENERACY CURVES, GAPS, AND DIABOLICAL POINTS 27

Figure 8. Three Dimensional Degeneracies for the 6, 7 Eigenvalue Levels 28 P. L. OVERFELT

Rhombus Degeneracies 1.0

0.8

4th Rectangle Degeneracy

l2/l1 0.6

0.4

3rd Rectangle Degeneracy

. Gap 2nd Rectangle Degeneracy 0.2 1st Rectangle Degeneracy

20 30 40 50 60 70 80 90 α(deg)

Figure 9. Two Dimensional Degeneracy Curves for the 7, 8 Ei- genvalue Levels DEGENERACY CURVES, GAPS, AND DIABOLICAL POINTS 29

Figure 10. Three Dimensional Degeneracy Curves for the 7, 8 Eigenvalue Levels 30 P. L. OVERFELT

Rhombus Degeneracies 1.0

Start of Degeneracy Curve

0.8

0.6 l2/l1

0.4

0.2

20 30 40 50 60 70 80 90 α(deg)

Figure 11. Two Dimensional Degeneracies for the 8, 9 Eigenvalue Levels DEGENERACY CURVES, GAPS, AND DIABOLICAL POINTS 31

Figure 12. Three Dimensional Degeneracies for the 8, 9 Eigen- value Levels 32 P. L. OVERFELT

Rhombus Degeneracies 1.0

7th Rectangle Degeneracy

0.8

6th Rectangle Degeneracy 0.6 l2/l1

5th Rectangle Degeneracy

0.4

4th Rectangle Degeneracy

0.2 3rd Rectangle Degeneracy 2nd Rectangle Degeneracy 1st Rectangle Degeneracy

0 20 40 60 80 α(deg)

Figure 13. Two Dimensional Degeneracy Curves for the 9, 10 Eigenvalue Levels DEGENERACY CURVES, GAPS, AND DIABOLICAL POINTS 33

Figure 14. Three Dimensional Degeneracy Curves for the 9, 10 Eigenvalue Levels 34 P. L. OVERFELT

1.0

0.8

0.6 l2/l1

0.4

0.2

20 40 60 80 α(deg)

Figure 15. Two Dimensional Degeneracies for the 10, 11 Eigen- value Levels DEGENERACY CURVES, GAPS, AND DIABOLICAL POINTS 35

Figure 16. Three Dimensional Degeneracies for the 10, 11 Eigen- value Levels