Quaternionic metamonogenic functions in the unit disk

We construct a set of quaternionic metamonogenic functions (that is, in $\mbox{Ker}(D+\lambda)$ for diverse $\lambda$) in the unit disk, such that every metamonogenic function is approximable in the quaternionic Hilbert module $L^2$ of the disk. The set is orthogonal except for the small subspace of elements of orders zero and one. These functions are used to express time-dependent solutions of the imaginary-time wave equation in the polar coordinate system.


Introduction
We consider the first-order partial differential quaternionic operator (sometimes called the Moisil-Teodorescu operator) D + λ (λ ∈ R \ {0}) in planar domains, where Here i and j are two of the three basic quaternionic units, and the Dirac operator D acts on the left on smooth quaternion-valued functions of a complex variable z = x + iy.This is a special case of Clifford-type operators for which there is a vast literature covering functions defined on spaces of diverse dimensions, beginning with W. Hamilton [14], continued by R. Fueter [7,8] and followed by many others including [1,2,4,5,6,12,13,19]. Functions defined in R 2 taking values in Clifford algebras of dimension > 2 have been relatively less investigated.The case R 2 → R 4 ∼ = H was considered in [17], where a detailed investigation of D+λ was carried out for quaternionvalued functions in the particular situation of elliptical domains.In [20], the authors constructed a one-parameter family of reduced quaternion-valued (R 3 -valued) functions of a pair of real variables lying in an ellipse, termed λ-metamonogenic Mathieu functions.Returning to the context R 2 → H, we consider here the case of functions defined in the unit disk employing Bessel functions of the first kind in place of Mathieu functions.We produce a set of metamonogenic functions (that is, in Ker(D + λ) for diverse λ), which is orthogonal in the unit disk for orders ≥ 2 and in a certain sense complete in Ker(D + λ) ∩ L 2 for every λ.As an application, in the final section we use these functions to express time-dependent solutions of the imaginary-time wave equation in the disk.

Metamonogenic functions
We consider the quaternionic operator D defined by (1).This is interpreted as follows, in fairly standard notation and terminology, in which z = x + iy is a complex number, and a quaternion [16,18] is notated as a = a 0 +a 1 i+a 2 j+ a 3 k.Here a m ∈ R and i, j, k are the quaternionic imaginary units satisfying i 2 = j 2 = k 2 = ijk = −1.The set of real quaternions H = H(R) is naturally identified with R 4 , which determines the usual component-wise addition and also induces the absolute value on H. Thus D acts on H-valued functions defined in domains in the complex plane C applying the quaternionic multiplication rules, in principle, on the left or right, giving Df or f D. We will only consider the operator acting from the left, as the other case is analogous.
Let Ω be a domain in R 2 (open and connected).Let L 2 (Ω) = L 2 (Ω, H) denote the space of all H-valued functions f : Ω → H such that the components f m (m = 0, 1, 2, 3) are in the usual L 2 (Ω, R).It is easily seen that L 2 (Ω) is naturally a right H-linear module and admits the H-valued right inner product for f, g ∈ L 2 (Ω).Thus L 2 (Ω) is a quaternionic right Hilbert module with the associated norm R , where f, g R = Sc f, g H coincides with the usual L 2 -norm for f , viewed as an R 4 -valued function in Ω [10,11].
For functions taking values in the 2-dimensional subspace Ri ⊕ Rj ⊆ H, D echoes the classical Cauchy-Riemann operator 2∂/∂z = ∂/∂x + i∂/∂y, but it sends such functions to the complementary subspace R ⊕ Rk ⊆ H.
We thus have the spaces of left λ-metamonogenic functions It is well known [23] that metaharmonic functions are of class C ∞ , and so by the following factorization of the Laplacian via D (cf.[14, Section CVII] and [9]), metamonogenic functions are of class C ∞ also.
In polar coordinates x = ρ cos θ, y = ρ sin θ, one has and the Helmholtz operator in polar coordinates is From Proposition 2.2, it is clear that the components of any f ∈ M(Ω; λ) are λ-metaharmonic.The equation (D + λ)f = 0 is equivalent to the system of partial differential equations From this the following is immediate.
Indeed, take and observe that the system (5) is satisfied.Similarly, given f 1 , f 2 one has unique functions f 0 , f 3 completing the components of a λ-metamonogenic function.One may think of the relationship of pairs (f 1 , f 2 ) and (f 0 , f 3 ) as a generalization of the notion of harmonic conjugates.

Quaternionic metamonogenic functions
This section introduces a family of λ-metamonogenic functions in the real Hilbert space L 2 of the unit disk, which is the object of study of this paper.The factorization of Proposition 2.2 suggests that quaternionic λ-metamonogenic functions should play a role for the Laplace operator ∆, similar to the usual metaharmonic functions in two variables for the corresponding Helmholtz operator [22].

A class of λ-metamonogenic functions
First we define a continuous family of quaternionic metamonogenic functions.Let J n (z), z ∈ C denote the n-th Bessel function of the first kind, n = 0, 1, 2, . . .[15].We recall that and for z = 0 the limiting value, In particular, F 0 [λ](z) = −J 1 (λρ) cos θ + J 0 (λρ) i − J 1 (λρ) sin θ k because of the second of the recurrence relations with Note that the i and j components of F n [λ](z) are the classical solutions J n (λρ) cos nθ, J n (λρ) sin nθ for the Helmholtz equation in polar coordinates, which are indeed complete in the space of all solutions in L 2 (Ω 0 , R), where Ω 0 = {z ∈ C : |z| < 1} denotes the unit disk in the complex plane [23].It follows directly from Proposition 2.3 that all F n [λ](z) are λ-metamonogenic.We also note that F n [λ] may be constructed as in terms of the reduced-quaternionic valued functions where we write Φ + n (z) = cos nθ, Φ − n (z) = sin nθ.

Basic metamonogenics
Next we introduce a special subset of the λ-metamonogenic functions defined in the previous section.It is well known [15] that J n has a countable collection of simple real zeros j n,m , The basic metamonogenic functions are defined by for n ≥ 0, m ≥ 1.Thus Some examples of F n,m in Ω 0 are given in Figure 1.Our main result is as follows. while (ii) The norms of these metamonogenic functions are given by Proof.Since F n,m is continuous in fact in the whole plane, it is clearly in L 2 (Ω 0 ).The proof divides naturally into parts.
(i) Orthogonality.We must show that and {n 1 , n 2 } = {0, 1}.We break down the integrand into quaternionic components as follows, For convenience, let us write With this notation, one finds after a great deal of cancellation that Now it is best to group the parts as follows: using Φ and then we can integrate, For n 1 = n 2 the θ-integral is zero, and so is the scalar product.Now suppose n 1 = n 2 = n, and consider the ρ-integral.We will use the following orthogonality property [15] for Bessel functions scaled by distinct zeros, when m 1 = m 2 , as well as (7).Within the ρ-integral at the end of ( 15) we find which by ( 7) is equal to Also in (15) we have again with the help of (7).Combining these we find that which is zero by ( 16) as we are assuming m 1 = m 2 .
(iii) Completeness.Now fix λ and suppose that f in the sense of (2).Since every well defined function in Ω 0 is periodic in θ in polar coordinates, it follows from Definition 3.1 that We apply integration by parts to the second and third terms of the first integral: when n > 0 by (6), and for the sixth and seventh terms, also when n > 0. Integrating the remaining integrals by parts, we have when n > 0. Similar arguments using the i, j, and k components enable one to show that in fact for n > 0. By the completeness of the set {J n (j n,m ρ)Φ ± n } in L 2 (Ω 0 , R) it follows that f 1 and f 2 are in the linear span of {J 0 (j 0,m ρ)Φ ± 0 }.Since Φ + 0 = 1, Φ − 0 = 0, we have the series representations converging in L 2 for real constants c 1,m , c 2,m .By Proposition 2.3, Let m ≥ 1.Using these series representations, first we look at the scalar part of the hypothesis Thus f 1 is orthogonal to J 0,m and hence is orthogonal in fact to all J n,m Φ ± n , which implies f 1 = 0.When one expands the k component of the inner product it is seen similarly that f 2 = 0.In consequence, f = 0 identically as required.
The information ( 12)-( 13) permits one to orthogonalize (say via the Gram-Schmidt process) the subspace generated by {F 0,m , F 1,m : m ≥ 1}, which by (11) will combine with the remaining F n,m to give a full orthogonal basis.The resulting functions are not particularly interesting, so we will omit the details.

Time-dependent solutions
Consider the partial differential equation for v(z, t) ∈ H, z ∈ Ω 0 , t ≥ 0. This can be interpreted as a wave equation using imaginary time it.(cf. the Wick transformation [3]).
We consider the natural quaternionic extensions of the real-valued solutions of (21).Since we are led to consider the companion equation Since the operator ∆ + K 2 (∂ 2 /∂t 2 ) has only real ingredients, it operates independently on each component of v = v 0 + v 1 i + v 2 j + v 3 k.
Because of (10), a time-dependent function given by a series of the form converging in L 2 (Ω 0 ) clearly satisfies (23).One may propose a boundary value problem for this equation with an initial condition given by an arbitrary v(z) ∈ M 2 (Ω 0 ; λ), whose coefficients c n,m ∈ H are given according to (A similar result for reduced-quaternion-valued functions in elliptical domains is worked out in detail in [20]).An example of the evolution of a wave function (25) is given in Figure 2.

Figure 1 :
Figure 1: The functions F n,m for assorted values of (n, m).The scalar parts are shown in the left column.The symmetries due to the presence of the functions Φ ± n are clearly visible.

FFigure 2 :
Figure 2: The initial condition (t = 0, top row) contains high order terms which are not visible in the graphics until approximately t > 0.3, when the exponential terms in time in (24) become sufficiently large.