Comment on “Dirac equation in the background of the Nutku helicoid metric” [J. Math. Phys. 48, 092301 (2007)]

Birkandan, T.; Hortaçsu, M.

doi:10.1063/1.2913520

The Dirac equation written on the boundary of the Nutku helicoid space consists of a system of ordinary differential equations. We tried to analyze this system and we found that it has a higher singularity than those of the Heun equations which give the solutions of the Dirac equation in the bulk. We also lose an independent integral of motion on the boundary. This facts explain why we could not find the solution of the system on the boundary in terms of known functions. We make the stability analysis of the helicoid and catenoid cases and end up with an Appendix which gives a new example wherein one encounters a form of the Heun equation.

I. INTRODUCTION

Although one usually needs only different forms of the hypergeometric equation or its confluent forms to describe many different phenomena in theoretical physics, functions with a higher singularity structure are seen more and more in literature.^1–12 A common form is the Heun function,¹³ which was extensively studied in Refs. 14 and 15, in the seminal book by Ince,¹⁶ as well as in several articles.^17–20 Although for linear equations both Ince and Slavyanov and Loy ended their singularity analysis with Heun-type functions, sometimes equations with even more singularities are needed for relatively simple situations, which lack some symmetries.

As an example of such a case, here we study the singularity structure of the Dirac equations, which written in the background of the Nutku helicoid solution²¹ and restricted to the boundary of the helicoid. This metric was formerly studied by Lorenz-Petzold.²² One can study the scalar field in this background and obtain the propagator in a closed form,²³ thanks to four integrals of motion allowed by the metric. One can also write the Dirac equation and obtain the solutions in terms of Mathieu functions.^24–26 One needs to study the eigenvalue problem on the boundary to impose the boundary conditions in this problem. The same, problem in the bulk, using partial differential equations, can be solved in terms of known functions. On the boundary, we get a system of ordinary differential equations. At first glance, this system seems to be easier to analyze. When we investigate the system further, we find that this system has a higher form of singularities and fewer integrals of motion. These turn out to be the reasons why we cannot express the solution in terms of the functions cited in Ince’s book.¹⁶

In this note, we comment on the symmetries of the new problem and study the singularity structure of the new system. We show that one gets an equation with a higher form of singularity. Since we do not have closed solutions, we then use stability analysis to see if this system describes a stable system. To our surprise, we find that although the answer is not affirmative for the helicoid case, we get a limit cycle for the related catenoid solution.

Below, we summarize our results. In the Appendix, we show a new example wherein one encounters a form of the Heun equation. This is the solution of the Laplacian in the background of the Eguchi–Hanson solution,²⁷ trivially extended to five dimensions.

II. OUR ANALYSIS

The Nutku helicoid metric is given as

d s^{2} = \frac{1}{\sqrt{1 + a^{2} ∕ r^{2}}} [d r^{2} + (r^{2} + a^{2}) d θ^{2} + (1 + \frac{a^{2}}{r^{2}} \sin^{2} θ) d y^{2} - \frac{a^{2}}{r^{2}} \sin 2 θ d y d z + (1 + \frac{a^{2}}{r^{2}} \cos^{2} θ) d z^{2}],

(1)

where $0 < r < \infty$ ⁠, $0 ⩽ θ ⩽ 2 π$ ⁠, and $y$ and $z$ are along the Killing directions and will be taken to be periodic coordinates on a 2-torus.²³ This is an example of a multicenter metric. This metric reduces to the flat metric if we take $a = 0$ ⁠,

d s^{2} = d r^{2} + r^{2} d θ^{2} + d y^{2} + d z^{2} .

(2)

If we make the following transformation:

r = a \sinh x,

(3)

the metric is written as

d s^{2} = \frac{a^{2}}{2} \sinh 2 x (d x^{2} + d θ^{2}) + \frac{2}{\sinh 2 x} [(\sinh^{2} x + \sin^{2} θ) d y^{2} - \sin 2 θ d y d z + (\sinh^{2} x + \cos^{2} θ) d z^{2}] .

(4)

The solutions of the Dirac equation, which is written in the background of the Nutku helicoid metric, can be expressed as a special form of Heun functions.^25,26 We could reduce the double confluent Heun function obtained for the radial equation to the Mathieu function with coordinate transformations. The Mathieu function is a related but much more studied function with a similar singularity structure. In the works cited above,^25,26 we tried to get the solution of the little Dirac equation, which is the name used for the equation restricted to a boundary of the helicoid. We also needed this solution to be able to calculate the index of the differential operator. For a fixed value of the radial coordinate, we had only a coupled system of ordinary differential equations, which, in general, should be much simpler to solve than the coupled system of partial differential equations obtained for the full Dirac equation, which is written for the bulk. We were successful in obtaining the solution in this latter case in terms of Mathieu functions. We were not able to identify the solutions for the little Dirac equation though.

The first thing we check is whether we lose any of the three Killing vectors and one Killing tensor. For the metric in question, one has three integrals of motion,²³ namely, $p_{y}$ ⁠, $p_{z}$ ⁠, and $g^{μ ν} p_{μ} p_{ν} = μ^{2}$ ⁠, namely,

{(\frac{d S_{x}}{d x})}^{2} + {(\frac{d S_{θ}}{d θ})}^{2} + a^{2} (p_{y}^{2} + p_{z}^{2}) \sinh^{2} x - \frac{μ^{2} a^{2}}{2} \sinh 2 x + a^{2} {(\cos θ p_{y} + \sin θ p_{z})}^{2} = 0,

(5)

and an extra integral of motion, which is the Killing tensor,²³

K = - p_{θ}^{2} - a^{2} {(\cos θ p_{y} + \sin θ p_{z})}^{2},

(6)

which gives us the angular equation for a fixed value of the constant $λ$ ⁠. See Eqs. (46) and (47) of Ref. 23. When one restricts the solution to a fixed value of the radial coordinate, the value of $g^{μ ν} p_{μ} p_{ν}$ is not an independent constant of motion from the other two Killing vectors and the Killing tensor.

To try to see the effect of this result on our problem, we investigate the singularity structure of the equation we get for the little Dirac operator. Here, we study the simplest case, wherein the eigenvalue $λ$ is equal to zero, since the answer to our problem is already apparent here. For this case, instead of getting a system of four coupled equations, we get coupling only between two of them at a time. When we analyze the system by reducing them to a second order equation for a single dependent variable, we find an operator with two irregular and one regular singularities, which is one more than allowed for the equations considered among the Heun functions. The double confluent Heun function, which is the solution one obtains for the full Dirac equation, has two irregular singularities, missing the extra regular singularity of the case studied here. To make our discussion concrete, we explicitly perform the calculation in the next section.

A. Singularities

The Dirac equation written in the background of the Nutku helicoid metric is written as

(\partial_{x} + i \partial_{θ}) Ψ_{3} + i a k [\cos (θ - ϕ + i x)] Ψ_{4} = 0,

(7)

(\partial_{x} - i \partial_{θ}) Ψ_{4} - i a k [\cos (θ - ϕ - i x)] Ψ_{3} = 0,

(8)

(- \partial_{x} + i \partial_{θ}) f_{1} + i a k [\cos (θ - ϕ + i x)] f_{2} = 0,

(9)

(- \partial_{x} - i \partial_{θ}) f_{2} - i a k [\cos (θ - ϕ - i x)] f_{1} = 0 .

(10)

These equations have simple solutions,²⁴ which can also be expanded in terms of products of radial and angular Mathieu functions.^28,25 A problem arises when these solutions are restricted to the boundary.²⁶

To impose these boundary conditions, we need to write the little Dirac equation, which is the Dirac equation restricted to the boundary, where the variable $x$ takes a fixed value $x_{0}$ ⁠. We choose to write the equations in the forms

\frac{\sqrt{2}}{a} {i \frac{d}{d θ} Ψ_{3} + i k a \cos (θ - ϕ + i x_{0}) Ψ_{4}} = λ f_{1},

(11)

\frac{\sqrt{2}}{a} {- i \frac{d}{d θ} Ψ_{4} - i a k \cos (θ - ϕ - i x_{0}) Ψ_{3}} = λ f_{2},

(12)

\frac{\sqrt{2}}{a} {- i \frac{d}{d θ} f_{1} - i a k \cos (θ - ϕ + i x_{0}) f_{2}} = λ Ψ_{3},

(13)

\frac{\sqrt{2}}{a} {i \frac{d}{d θ} f_{2} + i a k \cos (θ - ϕ - i x_{0}) f_{1}} = λ Ψ_{4} .

(14)

Here, $λ$ is the eigenvalue of the little Dirac equation. We take $λ = 0$ as the simplest case. The transformation

ϴ = θ - ϕ - i x_{0}

(15)

can be used. Then, we solve $f_{1}$ in the latter two equations in terms of $f_{2}$ ⁠:

- \frac{d^{2}}{d ϴ^{2}} f_{2} - \tan ϴ \frac{d}{d ϴ} f_{2} + \frac{{(a k)}^{2}}{2} [\cos (2 ϴ) \cosh (2 x_{0}) - i \sin (2 ϴ) \sinh (2 x_{0}) + \cosh (2 x_{0})] f_{2} = 0 .

(16)

When we make the transformation

u = e^{2 i ϴ},

(17)

the equation reads

{4 (u + 1) u [u \frac{d^{2}}{d u^{2}} + \frac{d}{d u}] - 2 i u (u - 1) \frac{d}{d u} + \frac{{(a k)}^{2}}{2} (u + 1) [u e^{- 2 x_{0}} + \frac{1}{u} e^{2 x_{0}} + \cosh (2 x_{0})]} f_{2} = 0 .

(18)

This equation has irregular singularities at $u = 0$ and $\infty$ and a regular singularity at $u = - 1$ ⁠. If we try a solution in the form $\sum_{n = - \infty}^{\infty} a_{n} u^{n}$ around the irregular singularity $u = 0$ ⁠, we end up with a four-term recursion relation as

a_{n - 1} [4 (n^{2} - 2 n + 1) - 2 i (n - 1) + \frac{{(a k)}^{2}}{2} (\frac{3}{2} e^{- 2 x_{0}} + \frac{1}{2} e^{2 x_{0}})] + a_{n} [4 n^{2} + 2 i n + \frac{{(a k)}^{2}}{2} (\frac{3}{2} e^{- 2 x_{0}} + \frac{1}{2} e^{2 x_{0}})] + a_{n - 2} [\frac{{(a k)}^{2}}{2} e^{- 2 x_{0}}] + a_{n + 1} [\frac{{(a k)}^{2}}{2} e^{2 x_{0}}] .

(19)

As it is known, in the Heun equation case, this kind of series solution gives a three-term relation.²⁹

If we search for a solution of the Thomé type, we may try a solution of the form $f_{2} = e^{A ∕ \sqrt{u}} g (u)$ ⁠. This form does not allow us to get a Taylor series expansion around the irregular point $u = 0$ ⁠.^5,30

If we try a series solution around the regular singularity at $u = - 1$ as $\sum_{n = 0}^{\infty} a_{n} {(u + 1)}^{n + α}$ ⁠, we find a relation between five consecutive coefficients for the solution. Therefore, we may conclude that the solution of this equation cannot be written in terms of Heun functions or simpler special functions.

To check this further, we first set the coefficient of the $1 ∕ u$ term in Eq. (18) to zero to change our irregular singularity at zero to a regular one. Then, we keep this term and discard the $u e^{- 2 x_{0}}$ term to reduce the singularity structure of infinity. In both cases, one can check that the solution can be expressed in terms of confluent Heun functions. This shows that reducing one of the singularities yields a Heun function. Thus, we conclude that the full Eq. (18) is not one of the better known equations in the literature, which are included in computer packages such as MAPLE, as cited in the seminal book by Ince.¹⁶

To investigate the type of our equation, we try to get a confluent form of a new equation,

y^{″} (z) + (\frac{1 - μ_{0}}{z} + \frac{1 - μ_{1}}{z + 1} + \frac{1 - μ_{2}}{z - a}) y^{'} (z) + \frac{β_{0} + β_{1} z + β_{2} z^{2}}{z^{2} (z - a)} y (z) = 0,

(20)

with regular singularities at 0, $- 1$ ⁠, and $a$ and an irregular singularity at infinity. This equation differs from the generalized Heun equation:^7,17

y^{″} (z) + (\frac{1 - μ_{0}}{z} + \frac{1 - μ_{1}}{z + 1} + \frac{1 - μ_{2}}{z - a} - α) y^{'} (z) + \frac{β_{0} + β_{1} z + β_{2} z^{2}}{z (z + 1) (z - a)} y (z) = 0,

(21)

which also has regular singularities at 0, $- 1$ ⁠, and $a$ and an irregular singularity at infinity. These two equations both have four-term recursion relations. They, however, have different singularity ranks according to the classification given in Ref. 31. When we set $a = 0$ in Eq. (20), we get

y^{″} (z) + (\frac{2 - μ_{0} - μ_{2}}{z} + \frac{1 - μ_{1}}{z + 1}) y^{'} (z) + \frac{β_{0} + β_{1} z + β_{2} z^{2}}{z^{3}} y (z) = 0 .

(22)

We get a singularity structure as a regular singularity at $- 1$ and two irregular singularities at zero and infinity like Eq. (18).

Both Eqs. (18) and (22) have four-term recursion relations when a Laurent power series solution is attempted. We may name this equation as the confluent form of Eq. (20). It is in the same form as our original equation rewritten as

{\frac{d^{2}}{d u^{2}} + [\frac{1}{u} (1 + \frac{i}{2}) + \frac{i}{u + 1}] \frac{d}{d u} + \frac{{(a k)}^{2}}{2} [\frac{e^{- 2 x_{0}}}{u} + \frac{e^{2 x_{0}}}{u^{3}} + \frac{\cosh (2 x_{0})}{u^{2}}]} f_{2} = 0 .

(23)

Both of these equations have $s$ -rank multisymbols ${1, \frac{3}{2}, \frac{3}{2}}$ referring to the singularities at ${- 1, 0, \infty}$ ⁠.³¹

We could not obtain a confluent equation similar to Eq. (18) from the generalized Heun equation (21). If we simply set $a = 0$ in this equation, we get the confluent Heun solution. We can obtain an equation with the same singularity structure as our equation only if we write the equation as

y^{″} (z) + (\frac{1 - μ_{0}}{z} + \frac{1 - μ_{1}}{z + 1} + \frac{1 - μ_{2}}{z - a}) y^{'} (z) + \frac{(β_{- 1}) ∕ z + β_{0} + β_{1} z + β_{2} z^{2}}{z (z + 1) (z - a)} y (z) = 0

(24)

and make $a$ approach zero. Then, we end up with an equation with the same singularity structure as in Eq. (18).

If we want to compare Eq. (18) to Eq. (21), we have to form a confluent form of the latter equation. To coalesce the singularities at zero, we make a detour and then use standard techniques.³² We first translate the singularity at zero to a singularity at $b$ ⁠,

y^{″} (z) + (\frac{1 - μ_{0}}{z - b} + \frac{1 - μ_{1}}{z + 1} + \frac{1 - μ_{2}}{z - a} - α) y^{'} (z) + \frac{β_{0} + β_{1} z + β_{2} z^{2}}{(z - b) (z + 1) (z - a)} y (z) = 0 .

(25)

This equation has regular singularities at $z = - 1, a, b$ and an irregular singularity at infinity. We make the transformation $z = 1 ∕ v$ ⁠. Then, Eq. (25) becomes

y^{″} (v) + (\frac{2 - (1 - μ_{0}) - (1 - μ_{1}) - (1 - μ_{2})}{v} + \frac{α}{v^{2}} - \frac{1 - μ_{0}}{1 ∕ b - v} - \frac{1 - μ_{1}}{v + 1} - \frac{1 - μ_{2}}{1 ∕ a - v}) y^{'} (v) + \frac{β_{0} ∕ v + β_{1} ∕ v^{2} + β_{2} ∕ v^{3}}{(1 - b v) (1 + v) (1 - a v)} y (v) = 0 .

(26)

We set $μ_{0} = - μ_{1} = 1 ∕ b = 2 ∕ a = ϵ$ and take the limit $ϵ \to \infty$ ⁠. Then, we transform back to the original variables by using $v = 1 ∕ z$ to obtain

y^{″} (z) + (\frac{3 - μ_{1}}{z} - α - \frac{μ_{1} - 1}{z (z + 1)} + \frac{1}{z^{2}}) y^{'} (z) + \frac{β_{0} + β_{1} z + β_{2} z^{2}}{z^{2} (z + 1)} y (z) = 0 .

(27)

This equation is a confluent form of Eq. (25). It has a regular singularity at $- 1$ and two irregular singularities at zero and infinity and a four-way recursion relation when expanded around zero by using a Laurent expansion. This equation has $s$ -rank multisymbols {1,2,2} referring to singularities at ${- 1, 0, \infty}$ ⁠.³¹ Here we get the rank-2 irregular singularities at zero and infinity only from the coefficient of the first derivative, whereas in Eqs. (18) and (22), the coefficient of the term without derivatives gives us these singularities. They also have different ranks. Even if we set $α = 0$ in Eq. (27), we get $rank = {1, 2, \frac{3}{2}}$ ⁠, which is different from the rank of our original equation. Hence, our Eq. (18) may be a confluent form only of a variation of Eq. (21), which is like Eq. (24).

B. Stability analysis for the little Dirac equation

The little Dirac equation is a system of linear differential equations with periodic coefficients. Then, we can write the system as^33,34

\partial_{θ} ψ = P (θ) ψ .

(28)

Here, $P (θ + τ) = P (θ)$ and $τ$ is the period of the coefficients $(τ \neq 0)$ ⁠. According to Bellman, if $ψ (0) = I$ ⁠, we can write

ψ (θ) = Q (θ) e^{B θ} .

(29)

The matrix $Q (θ)$ is also periodic with the period $τ$ ⁠. Then, we have

ψ (θ + τ) = Q (θ + τ) e^{B θ} e^{B τ} = Q (θ) e^{B θ} e^{B τ} = ψ (θ) e^{B τ} .

(30)

We can define $e^{B τ} \equiv C$ and use

C = ψ {(θ)}^{- 1} ψ (θ + τ)

(31)

to obtain $C$ ⁠. The Jordan normal form of $C$ ⁠, $T$ being the transformation matrix,

C = T (\begin{matrix} L_{1} \\ ⋱ \\ L_{r} \end{matrix}) T^{- 1},

(32)

gives us the $B$ matrix:

B = \frac{1}{τ} \ln L .

(33)

The eigenvalues of $B$ give us the characteristic roots. We will use these characteristic roots with different parameters in our stability analysis. The characteristic roots are given by $α_{i}$ $(i = 1 - 4)$ ⁠.

Our calculations indicate that the $f_{1}$ and $f_{2}$ solutions are not stable (positive characteristic root), while the $Ψ_{3}$ and $Ψ_{4}$ solutions are stable (negative characteristic root). As can be seen in Table I, when we keep all the other parameters constant and vary only $a$ ⁠, the value of the roots are influenced most, whereas the effect of the variation in the value of $λ$ changes the value of the roots least. We also find that when these parameters exceed unity in absolute value, we encounter inconsistencies in the numerical values. The separation between consecutive roots increase and some negative roots go to positive values for large values of the parameters. If we keep the values of the parameters in the range $[- 1, 1]$ ⁠, we seem to have no such problems.

Table I.

The change in the characteristic roots with respect to parameters (helicoid case).

$a$	$k$	$x_{0}$	$λ$	Real parts of thecharacteristic roots $(\times 2 π)$	Signs of thecharacteristic roots
1	1	1	1	$α_{1, 2, 3, 4} = 9.250 29$	+ + − −
0.5	1	1	1	$α_{1, 2, 3, 4} = 4.039 26$	+ + − −
0.8	1	1	1	$α_{1, 2, 3, 4} = 7.216 3$	+ + − −
1.1	1	1	1	$α_{1, 2, 3, 4} = 10.254 2$	+ + − −
1.5	1	1	1	$α_{1, 2, 3, 4} = 14.221 5$	+ + − −
1	0.5	1	1	$α_{1, 2, 3, 4} = 5.832 7$	+ + − −
1	0.8	1	1	$α_{1, 2, 3, 4} = 7.754 21$	+ + − −
1	1.1	1	1	$α_{1, 2, 3, 4} = 10.031 9$	+ + − −
1	1.5	1	1	$α_{1, 2, 3, 4} = 13.274 5$	+ + − −
1	1	0.5	1	$α_{1, 2, 3, 4} = 6.562 06$	+ + − −
1	1	0.8	1	$α_{1, 2, 3, 4} = 7.915 86$	+ + − −
1	1	1.1	1	$α_{1, 2, 3, 4} = 10.063 2$	+ + − −
1	1	1.5	1	$α_{1, 2, 3, 4} = 14.464 8$	+ + − −
1	1	1	0.5	$α_{1, 2, 3, 4} = 8.301 64$	+ + − −
1	1	1	0.8	$α_{1, 2, 3, 4} = 8.813 42$	+ + − −
1	1	1	1.1	$α_{1, 2, 3, 4} = 9.492 49$	+ + − −
1	1	1	1.5	$α_{1, 2, 3, 4} = 10.588 8$	+ + − −

$a$	$k$	$x_{0}$	$λ$	Real parts of thecharacteristic roots $(\times 2 π)$	Signs of thecharacteristic roots
1	1	1	1	$α_{1, 2, 3, 4} = 9.250 29$	+ + − −
0.5	1	1	1	$α_{1, 2, 3, 4} = 4.039 26$	+ + − −
0.8	1	1	1	$α_{1, 2, 3, 4} = 7.216 3$	+ + − −
1.1	1	1	1	$α_{1, 2, 3, 4} = 10.254 2$	+ + − −
1.5	1	1	1	$α_{1, 2, 3, 4} = 14.221 5$	+ + − −
1	0.5	1	1	$α_{1, 2, 3, 4} = 5.832 7$	+ + − −
1	0.8	1	1	$α_{1, 2, 3, 4} = 7.754 21$	+ + − −
1	1.1	1	1	$α_{1, 2, 3, 4} = 10.031 9$	+ + − −
1	1.5	1	1	$α_{1, 2, 3, 4} = 13.274 5$	+ + − −
1	1	0.5	1	$α_{1, 2, 3, 4} = 6.562 06$	+ + − −
1	1	0.8	1	$α_{1, 2, 3, 4} = 7.915 86$	+ + − −
1	1	1.1	1	$α_{1, 2, 3, 4} = 10.063 2$	+ + − −
1	1	1.5	1	$α_{1, 2, 3, 4} = 14.464 8$	+ + − −
1	1	1	0.5	$α_{1, 2, 3, 4} = 8.301 64$	+ + − −
1	1	1	0.8	$α_{1, 2, 3, 4} = 8.813 42$	+ + − −
1	1	1	1.1	$α_{1, 2, 3, 4} = 9.492 49$	+ + − −
1	1	1	1.5	$α_{1, 2, 3, 4} = 10.588 8$	+ + − −

The periodicity of the defined $Q$ can be checked by using numerical means. We use

ψ (θ + τ) = Q (θ + τ) e^{B θ} C

(34)

for $θ = 0$ ⁠, $ψ (θ) = Q (θ) e^{B θ}$ to give

ψ (τ) C^{- 1} = Q (τ) = Q (0) = ψ (0) = I .

(35)

We numerically check that this equation is satisfied; hence, $Q$ is periodic.

For the catenoid case, one replaces $a$ with $i a$ in the metric. The same stability procedure is performed for this case and we find the characteristic roots given in the Table II.

Table II.

The change in the characteristic roots with respect to parameters (catenoid case).

$a$ $(\times i)$	$k$	$x_{0}$	$λ$	Characteristic roots $(\times 2 π)$	Signs of thecharacteristic roots
1	1	1	1	$α_{1, 2} = - 5.845 43 \times 10^{- 8} + 2.084 26 i$ $α_{3, 4} = - 6.409 67 \times 10^{- 8} + 0.722 66 i$	+ − + −
0.5	1	1	1	$α_{1, 2} = - 2.405 53 \times 10^{- 7} - 1.197 24 i$ $α_{3, 4} = - 2.492 25 \times 10^{- 7} - 2.372 1 i$	− + − +
0.8	1	1	1	$α_{1, 2} = - 5.630 46 \times 10^{- 7} - 0.233 835 i$ $α_{3, 4} = - 5.781 78 \times 10^{- 7} - 2.426 96 i$	− + − +
1.1	1	1	1	$α_{1, 2} = - 7.828 56 \times 10^{- 7} + 2.998 2 i$ $α_{3, 4} = - 7.883 58 \times 10^{- 7} - 0.186 969 i$	+ − + −
1.5	1	1	1	$α_{1, 2} = - 7.535 39 \times 10^{- 7} - 0.526 163 i$ $α_{3, 4} = - 7.725 5 \times 10^{- 7} - 2.359 78 i$	− + − +
1	0.5	1	1	$α_{1, 2} = 3.490 81 \times 10^{- 8} + 0.661 356 i$ $α_{3, 4} = 2.027 94 \times 10^{- 8} - 1.317 26 i$	+ − − +
1	0.8	1	1	$α_{1, 2} = - 4.049 57 \times 10^{- 7} + 0.864 554 i$ $α_{3, 4} = - 4.228 23 \times 10^{- 7} + 2.363 8 i$	+ − + −
1	1.1	1	1	$α_{1, 2} = - 2.257 7 \times 10^{- 7} + 2.732 8 i$ $α_{3, 4} = - 2.309 29 \times 10^{- 7} + 0.120 058 i$	+ − + −
1	1.5	1	1	$α_{1, 2} = - 1.339 16 \times 10^{- 7} + 0.715 427 i$ $α_{3, 4} = - 1.389 04 \times 10^{- 7} - 2.750 07 i$	+ − − +
1	1	0.5	1	$α_{1, 2} = - 4.686 25 \times 10^{- 8} - 0.232 525 i$ $α_{3, 4} = - 5.347 71 \times 10^{- 8} - 2.024 31 i$	− + − +
1	1	0.8	1	$α_{1, 2} = - 4.699 01 \times 10^{- 7} - 1.069 77 i$ $α_{3, 4} = - 4.729 86 \times 10^{- 7} - 2.117 09 i$	− + − +
1	1	1.1	1	$α_{1, 2} = - 6.064 25 \times 10^{- 7} - 2.741 53 i$ $α_{3, 4} = - 6.141 88 \times 10^{- 7} - 0.143 986 i$	− + − +
1	1	1.5	1	$α_{1, 2} = - 4.263 75 \times 10^{- 7} + 0.299 098 i$ $α_{3, 4} = - 4.421 4 \times 10^{- 7} + 1.558 22 i$	+ − + −
1	1	1	0.5	$α_{1, 2} = - 4.677 73 \times 10^{- 7} + 0.761 149 i$ $α_{3, 4} = - 4.696 27 \times 10^{- 7} + 0.741 915 i$	+ − + −
1	1	1	0.8	$α_{1, 2} = - 2.822 56 \times 10^{- 7} - 1.520 8 i$ $α_{3, 4} = - 2.865 71 \times 10^{- 7} - 0.798 92 i$	− + − +
1	1	1	1.1	$α_{1, 2} = - 4.144 79 \times 10^{- 7} + 2.380 47 i$ $α_{3, 4} = - 4.198 5 \times 10^{- 7} + 0.650 337 i$	+ − + −
1	1	1	1.5	$α_{1, 2} = - 3.339 14 \times 10^{- 7} - 2.625 03 i$ $α_{3, 4} = - 3.420 67 \times 10^{- 7} - 0.145 628 i$	− + − +

$a$ $(\times i)$	$k$	$x_{0}$	$λ$	Characteristic roots $(\times 2 π)$	Signs of thecharacteristic roots
1	1	1	1	$α_{1, 2} = - 5.845 43 \times 10^{- 8} + 2.084 26 i$ $α_{3, 4} = - 6.409 67 \times 10^{- 8} + 0.722 66 i$	+ − + −
0.5	1	1	1	$α_{1, 2} = - 2.405 53 \times 10^{- 7} - 1.197 24 i$ $α_{3, 4} = - 2.492 25 \times 10^{- 7} - 2.372 1 i$	− + − +
0.8	1	1	1	$α_{1, 2} = - 5.630 46 \times 10^{- 7} - 0.233 835 i$ $α_{3, 4} = - 5.781 78 \times 10^{- 7} - 2.426 96 i$	− + − +
1.1	1	1	1	$α_{1, 2} = - 7.828 56 \times 10^{- 7} + 2.998 2 i$ $α_{3, 4} = - 7.883 58 \times 10^{- 7} - 0.186 969 i$	+ − + −
1.5	1	1	1	$α_{1, 2} = - 7.535 39 \times 10^{- 7} - 0.526 163 i$ $α_{3, 4} = - 7.725 5 \times 10^{- 7} - 2.359 78 i$	− + − +
1	0.5	1	1	$α_{1, 2} = 3.490 81 \times 10^{- 8} + 0.661 356 i$ $α_{3, 4} = 2.027 94 \times 10^{- 8} - 1.317 26 i$	+ − − +
1	0.8	1	1	$α_{1, 2} = - 4.049 57 \times 10^{- 7} + 0.864 554 i$ $α_{3, 4} = - 4.228 23 \times 10^{- 7} + 2.363 8 i$	+ − + −
1	1.1	1	1	$α_{1, 2} = - 2.257 7 \times 10^{- 7} + 2.732 8 i$ $α_{3, 4} = - 2.309 29 \times 10^{- 7} + 0.120 058 i$	+ − + −
1	1.5	1	1	$α_{1, 2} = - 1.339 16 \times 10^{- 7} + 0.715 427 i$ $α_{3, 4} = - 1.389 04 \times 10^{- 7} - 2.750 07 i$	+ − − +
1	1	0.5	1	$α_{1, 2} = - 4.686 25 \times 10^{- 8} - 0.232 525 i$ $α_{3, 4} = - 5.347 71 \times 10^{- 8} - 2.024 31 i$	− + − +
1	1	0.8	1	$α_{1, 2} = - 4.699 01 \times 10^{- 7} - 1.069 77 i$ $α_{3, 4} = - 4.729 86 \times 10^{- 7} - 2.117 09 i$	− + − +
1	1	1.1	1	$α_{1, 2} = - 6.064 25 \times 10^{- 7} - 2.741 53 i$ $α_{3, 4} = - 6.141 88 \times 10^{- 7} - 0.143 986 i$	− + − +
1	1	1.5	1	$α_{1, 2} = - 4.263 75 \times 10^{- 7} + 0.299 098 i$ $α_{3, 4} = - 4.421 4 \times 10^{- 7} + 1.558 22 i$	+ − + −
1	1	1	0.5	$α_{1, 2} = - 4.677 73 \times 10^{- 7} + 0.761 149 i$ $α_{3, 4} = - 4.696 27 \times 10^{- 7} + 0.741 915 i$	+ − + −
1	1	1	0.8	$α_{1, 2} = - 2.822 56 \times 10^{- 7} - 1.520 8 i$ $α_{3, 4} = - 2.865 71 \times 10^{- 7} - 0.798 92 i$	− + − +
1	1	1	1.1	$α_{1, 2} = - 4.144 79 \times 10^{- 7} + 2.380 47 i$ $α_{3, 4} = - 4.198 5 \times 10^{- 7} + 0.650 337 i$	+ − + −
1	1	1	1.5	$α_{1, 2} = - 3.339 14 \times 10^{- 7} - 2.625 03 i$ $α_{3, 4} = - 3.420 67 \times 10^{- 7} - 0.145 628 i$	− + − +

We see that the real parts of these roots are compatible with assigning to zero within numerical errors. This corresponds to a limit cycle.^35,36

III. CONCLUSION

Here, we performed a systematic analysis of the Dirac equation restricted to the boundary when it is written in the background of the Nutku helicoid solution.²³ We find that the resulting system of ordinary differential equations has a singularity which is higher than those of the Heun functions, which are solutions for the bulk. We also lose an independent integral of motion. This fact explains why we could not obtain the solution of the system on the boundary in terms of well known functions.

The stability analysis we performed shows that although this system is not stable, a related system, which is the catenoid solution, is. We can thus give a meaning to its solutions, although we cannot get explicit solutions for the little Dirac equation obtained from it, too.

ACKNOWLEDGMENT

We would like to thank Professor Ayşe Bilge for correspondence and discussions. This work is supported by TÜBİTAK, which is The Scientific and Technological Council of Turkey. The work of M.H. is also supported by TÜBA, which is the Academy of Sciences of Turkey.

APPENDIX: SCALAR FIELD IN THE BACKGROUND OF THE EXTENDED EGUCHI–HANSON SOLUTION

To go to five dimensions, we can add a time component to the Eguchi–Hanson metric,²⁷ so that we have

d s^{2} = - d t^{2} + \frac{1}{1 - a^{4} ∕ r^{4}} d r^{2} + r^{2} (σ_{x}^{2} + σ_{y}^{2}) + r^{2} (1 - \frac{a^{4}}{r^{4}}) σ_{z}^{2},

(A1)

where

σ_{x} = \frac{1}{2} (- \cos ξ d θ - \sin θ \sin ξ d ϕ),

(A2)

σ_{y} = \frac{1}{2} (\sin ξ d θ - \sin θ \cos ξ d ϕ),

(A3)

σ_{z} = \frac{1}{2} (- d ξ - \cos θ d ϕ) .

(A4)

This is a vacuum solution.

If we take

Φ = e^{i k t} e^{i n ϕ} e^{i (m + 1 ∕ 2) ξ} φ (r, θ),

(A5)

we find the scalar equation as

H φ (r, θ) = (\frac{r^{4} - a^{4}}{r^{2}} \partial_{r r} + \frac{3 r^{4} + a^{4}}{r^{3}} \partial_{r} + k^{2} r^{2} + \frac{4 a^{4} m^{2}}{a^{4} - r^{4}} + 4 \partial_{θ θ} + 4 \cot θ \partial_{θ} + \frac{8 m n \cos θ - 4 (m^{2} + n^{2})}{\sin^{2} θ}) φ (r, θ) .

(A6)

If we take $φ (r, θ) = f (r) g (θ)$ ⁠, the solution of the radial part is expressed in terms of confluent Heun $(H_{C})$ functions,

f (r) = {(- a^{4} + r^{4})}^{(1 ∕ 2) m} H_{C} (0, m, m, \frac{1}{2} k^{2} a^{2}, \frac{1}{2} m^{2} - \frac{1}{4} λ - \frac{1}{4} k^{2} a^{2}, \frac{a^{2} + r^{2}}{2 a^{2}}) + {(a^{2} + r^{2})}^{- (1 ∕ 2) m} {(r^{2} - a^{2})}^{(1 ∕ 2) m} H_{C} (0, - m, m, \frac{1}{2} k^{2} a^{2}, \frac{1}{2} m^{2} - \frac{1}{4} λ - \frac{1}{4} k^{2} a^{2}, \frac{a^{2} + r^{2}}{2 a^{2}}) .

(A7)

The angular solution is in terms of hypergeometric solutions,

g (θ) = \frac{1}{\sin θ} {\sqrt{2 - 2 \cos (θ)} {(\frac{1}{2} \cos (θ) - \frac{1}{2})}^{(1 ∕ 2) m} {(\frac{1}{2} \cos (θ) - \frac{1}{2})}^{- (1 ∕ 2) n} [{(2 \cos (θ) + 2)}^{1 ∕ 2 - (1 ∕ 2) n - (1 ∕ 2) m} {}_{2}F_{1} ([- n + \frac{1}{2} \sqrt{λ + 1} + \frac{1}{2}, - n - \frac{1}{2} \sqrt{λ + 1} + \frac{1}{2}], [1 - n - m], \frac{1}{2} \cos (θ) + \frac{1}{2}) + {(2 \cos (θ) + 2)}^{1 ∕ 2 + (1 ∕ 2) n + (1 ∕ 2) m} {}_{2}F_{1} ([m + \frac{1}{2} \sqrt{λ + 1} + \frac{1}{2}, m - \frac{1}{2} \sqrt{λ + 1} + \frac{1}{2}], [1 + n + m], \frac{1}{2} \cos (θ) + \frac{1}{2})]} .

(A8)

If the variable transformation $r = a \sqrt{\cosh x}$ is made, the solution can be expressed as

f (x) = \frac{1}{\sinh x} {{(\sinh (x))}^{m + 1} H_{C} (0, m, m, \frac{1}{2} k^{2} a^{2}, \frac{1}{2} m^{2} - \frac{1}{4} λ - \frac{1}{4} k^{2} a^{2}, \frac{1}{2} \cosh (x) + \frac{1}{2}) + {(2 \cosh (x) + 2)}^{- (1 ∕ 2) m + 1 ∕ 2} {(2 \cosh (x) - 2)}^{(1 ∕ 2) m + 1 ∕ 2} H_{C} (0, - m, m, \frac{1}{2} k^{2} a^{2}, \frac{1}{2} m^{2} - \frac{1}{4} λ - \frac{1}{4} k^{2} a^{2}, \frac{1}{2} \cosh (x) + \frac{1}{2})} .

(A9)

We tried to express the equation for the radial part in terms of $u = (a^{2} + r^{2}) ∕ 2 a^{2}$ to see the singularity structure more clearly. Then the radial differential operator reads

4 \frac{d^{2}}{d u^{2}} + 4 (\frac{1}{u - 1} + \frac{1}{u}) \frac{d}{d u} + k^{2} a^{2} (\frac{1}{u - 1} + \frac{1}{u}) + \frac{m^{2}}{u^{2} {(1 - u)}^{2}} .

(A10)

This operator has two regular singularities at zero and one, and an irregular singularity at infinity, which is the singularity structure of the confluent Heun equation. This is different from the hypergeometric equation, which has regular singularities at zero, 1 and infinity.

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American Institute of Physics

Comment on “Dirac equation in the background of the Nutku helicoid metric” [J. Math. Phys. 48, 092301 (2007)]

I. INTRODUCTION

II. OUR ANALYSIS

A. Singularities

B. Stability analysis for the little Dirac equation

III. CONCLUSION

ACKNOWLEDGMENT

APPENDIX: SCALAR FIELD IN THE BACKGROUND OF THE EXTENDED EGUCHI–HANSON SOLUTION

REFERENCES

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Comment on “Dirac equation in the background of the Nutku helicoid metric” [J. Math. Phys. 48, 092301 (2007)] Free

I. INTRODUCTION

II. OUR ANALYSIS

A. Singularities

B. Stability analysis for the little Dirac equation

III. CONCLUSION

ACKNOWLEDGMENT

APPENDIX: SCALAR FIELD IN THE BACKGROUND OF THE EXTENDED EGUCHI–HANSON SOLUTION

REFERENCES

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Comment on “Dirac equation in the background of the Nutku helicoid metric” [J. Math. Phys. 48, 092301 (2007)]