Eigenvalue problems for slice functions

This paper addresses particular eigenvalue problems within the context of two quaternionic function theories. More precisely, we study two concrete classes of quaternionic eigenvalue problems, the first one for the slice derivative operator in the class of quaternionic slice-regular functions and the second one for the Cauchy-Riemann-Fueter operator in the class of axially monogenic functions. The two problems are related to each other by the four-dimensional Laplace operator and Fueter's Theorem. As an application of a particular case of second order eigenvalue problems, we obtain a representation of axially monogenic solutions for time-harmonic Helmholtz and stationary Klein-Gordon equations.


I
Complex function theory represents a powerful toolkit to study eigenvalue problems related to important differential operators arising in harmonic analysis and mathematical physics in two dimensions. Therefore, a strong motivation in mathematical analysis consists in developing higher dimensional analogues in order to tackle similarly corresponding spatial problems. The smallest division algebra that encompasses the three-dimensional space R 3 is the four-dimensional Hamiltonian skew field H which is not commutative anymore but still associative. It is the first algebra created beyond the complex numbers by applying the well-known Cayley-Dickson duplication process. The concept of holomorphic functions however can be generalized in a number of rather different ways to higher dimensional algebras, even in the simplest context of H. Following the Riemann approach one can consider quaternion-valued functions that are annihilated either from the left or from the right by the linear Cauchy-Riemann-Fueter operator := 1 2 0 + 1 + 2 + 3 . Here, , , are the quaternionic imaginary units, satisfying 2 = 2 = 2 = −1 and = , = , = as well as = − , = − , = − . Functions in the kernel of this operator are nowadays often called left (resp. right) monogenic functions or left (right) Fueter regular functions and have been studied by a constantly growing community for more than a century. As a classical reference we recommend for example [3]. This function theory is often called Clifford analysis. The Cauchy-Riemann-Fueter operator is nothing else than the Euclidean Dirac operator in R 4 associated to the Euclidean flat metric and is a first order square root of the Laplacian Δ 4 .
An alternative function theory, which actually is even more closely related to classical complex-analytic functions, is the theory of quaternionic slice-regular functions which was basically introduced in 2006-2007 by Gentili and Struppa [8,9]. This function theory exploits a particular slice-structure of H which is explained together with its most important definitions and relevant function classes of the so-called slice functions and slice-regular functions in Section 2 including the basic references. Now the aim of our paper is to investigate some eigenvalue problems for quaternionic slice functions, with particular emphasis on slice-regular functions and on axially monogenic functions.
The right-linear operator that we first consider is the so-called slice derivative operator acting on the classes of slice or slice-regular functions defined in Section 2. We study eigenvalue problems of the form = on an axially symmetric domain Ω of the quaternionic space H, with ∈ H. We show in Proposition 3 that in the class of slice functions, the solutions to this problem can be written as slice products of anti-slice-regular functions and a quaternionic exponential function. Slice-regular solutions are obtained when the first factor is a slice-constant function on Ω.
Let D be the right-linear operator defined by D = − . In Propositions 6, 8 and 18 we study the associated non-homogeneous eigenvalue problem D = ℎ for a polynomial right-hand side ℎ and for a wide class of entire sliceregular functions. These results in particular permit us to solve (Corollary 11) second-order eigenvalue problems D D = 0 for any choice of quaternionic eigenvalues , . In Corollary 15 we extend this result to the -th order eigenvalue problem D 1 · · · D = 0 by means of new generalized exponential functions (see Definition 12) associated with any ordered -tuple ( 1 , . . . , ) ∈ H .
The second right-linear operator that we consider is the (conjugated) Cauchy-Riemann-Fueter operator acting on the class of quaternionic monogenic (or Fueter-regular) slice functions, i.e., belonging to the kernel of the Cauchy-Riemann-Fueter operator . Using Fueter's Theorem (see e.g. [5]) as a bridge between the two function theories, we are able to apply the results described above to eigenvalue problems for axially monogenic functions. The crucial fact is the relation • Δ 4 = Δ 4 • on the space of slice-regular functions on Ω, where Δ 4 is the Laplacian of R 4 ≃ H. Applying the operator Δ 4 to the slice-regular solutions obtained above, we obtain in Proposition 28 the general axially monogenic solution of the eigenvalue problem = , with ∈ H, and more generally (Proposition 32) of the -th order eigenvalue problem L 1 · · · L = 0, where L is defined by L = − . We refer the reader at the beginning of Sect. 4 to references on the study of similar eigenvalue problems in quaternionic and Clifford analysis.
The final section is devoted to applications. We relate the solutions to L 1 L 2 = 0 to axially monogenic solutions of the three-dimensional time-harmonic Helmholtz and stationary massless Klein-Gordon equation on an axially symmetric domain (Proposition 34).
Let Δ 3 denote the Laplacian operator on R 3 ≃ Im(H). Suppose that 1 = and 2 = − , where is a quaternionic imaginary unit and where is an arbitrary non-zero real number. Then the solutions to L L − = 0 are axially monogenic solutions to the massless stationary Klein-Gordon equation (Δ 3 − 2 ) = 0 on Ω * := Ω ∩R 3 . If we take 1 = and 2 = − where again is supposed to be a nonzero real value, then the solutions to L L − = 0 are axially monogenic solutions to the time-harmonic Helmholtz equation (Δ 3 + 2 ) = 0 on Ω * . Finally, we establish an application (Remark 37) to the non-homogeneous equation associated to the Klein-Gordon equation, i.e., the Yukawa equation (Δ 3 − 2 ) = ℎ, with real and ℎ axially monogenic.
The paper is structured as follows: In Sect. 2 we recall the basic notions of slice function theory on H. Then, in Sect. 3, we present the eigenvalue problems for the slice derivative operator on slice and on slice-regular functions. Here we also introduce the generalized exponential functions Λ . This represents another essential novelty of this paper, and we give some examples to illustrate the method of solution. In Sect. 4 we recall Fueter's Theorem and present some new results about the Laplacian of a slice-regular function. Then we prove the commutativity relation linking Δ, and and obtain the axially monogenic solutions to the eigenvalue problems for , in terms of the generalized Δ-exponential functions Δ Λ . Finally, in Sect. 5, we present applications of the results of Sect. 4 to the Helmholtz, Klein-Gordon and Yukawa equations and round off our paper by presenting some explicit examples.

P
The theory of quaternionic slice-regular functions was introduced in 2006-2007 by Gentili and Struppa [8,9]. We refer the reader for instance to [4,7,12,13,14] and also to the references therein for precise definitions and for more results on this class of functions. Slice function theory is based on the "slice" decomposition of the quaternionic space H. For each imaginary unit in the sphere we denote by C = 1, ≃ C the subalgebra generated by . Then it holds [9] on the open set Ω if, for each ∈ S, the restriction |Ω∩C : Ω ∩ C → H is holomorphic with respect to the complex structure defined by left multiplication by .

Slice functions.
A different approach to slice regularity was introduced in [11,12], making use of the concept of slice function. These are exactly the quaternionic functions that are compatible with the slice structure of H. Given a set ⊆ C that is invariant with respect to complex conjugation, a function : → H ⊗ C that satisfies ( ) = ( ) for every ∈ is called a stem function on . Here the conjugation in H ⊗ C is the one induced by complex conjugation of the second factor.
Let be open and let Ω = ∪ ∈S H Φ ( ) ⊂ H, where for any ∈ S, the map Φ : C → C is the canonical isomorphism defined by Φ ( + ) := + . Open domains in H of the form Ω = Ω are called axially symmetric domains. Note that every axially symmetric domain Ω is either a slice domain if Ω ∩ R ≠ ∅, or it is a product domain, namely if Ω ∩ R = ∅. Moreover, any axially symmetric open set can be represented as a union of a family of domains of these two types.
The stem function = 1 + 2 on (with 1 and 2 H-valued functions on ) induces the slice function = I ( ) : Ω → H as follows: if = + = Φ ( ) ∈ Ω ∩ C , then The tensor product H ⊗ C can be equipped with the complex structure induced by the second factor. The slice function is then called slice-regular if is holomorphic. If a domain Ω in H is axially symmetric and intersects the real axis, then this definition of slice regularity is equivalent to the one proposed by Gentili and Struppa [9]. We will denote by SR (Ω) the right quaternionic module of slice-regular functions on Ω. If ∈ SR (H), then is called a slice-regular entire function. In particular, every polynomial ( ) = =0 ∈ H[ ] with right quaternionic coefficients , is an entire slice-regular function. The function = I ( ) is called slice-preserving if the H-components 1 and 2 of the stem function are real-valued. This is equivalent to the condition ( ) = ( ) for every ∈ Ω . If is slice-preserving, then · coincides with the pointwise product of and . If , are slice-regular on Ω, then also their slice product · is slice-regular on Ω. We recall that the slice product has also an interpretation in terms of pointwise quaternionic product: For every ∈ S (Ω), the slice function = I ( ) = I ( 1 + 2 ) is the sliceconjugate of and ( ) = · = · is the normal function of . If ∈ SR (Ω), then also , ( ) do belong to SR (Ω). Assume that ∈ S (Ω) is not identically zero on Ω. Then the zero set ( ( )) = { ∈ Ω | ( ) = 0} of the normal function does not coincide with the whole set Ω. The slice product permits us to introduce on Ω \ ( ( )) the slice reciprocal −• of , as the function −• := ( ) −1 · (see [13,Prop. 2.4]). Again, if is slice-regular, then −• is slice-regular, too.
The slice derivatives , of a slice functions = I ( ) are defined by means of the Cauchy-Riemann operators applied to the inducing stem function : Note that is slice-regular on Ω if and only if = 0 and if is slice-regular on Ω then also is slice-regular on Ω. Moreover, the slice derivatives satisfy the Leibniz product rule w.r.t. the slice product.
A slice-regular function is called slice-constant if = 0 on Ω. We will denote by SC (Ω) the right H-module of slice-constant functions on Ω. If Ω is a slice domain, then every ∈ SC (Ω) is a quaternionic constant. If Ω is a product domain, then other possibilities arise: for any imaginary unit ∈ S, a slice-constant function on H \ R can be defined by setting, such as indicated in [12,Remark 12], By the representation formula (see [12,Prop. 5] and [1,Prop. 15]), any sliceconstant ∈ SC (H \ R) is determined by its values on two arbitrarily chosen half-slices C + , C + , with ≠ . If |C + = 1 ∈ H and |C + = 2 ∈ H, then can be expressed as

E
Eigenvalue problem for slice-regular functions. Let Ω be an axially symmetric domain of H. Consider the following eigenvalue problem for the slice derivative operator in the class of slice-regular functions: with ∈ SR (Ω), ∈ H. Observe that whenever is not real, then in general exp ( ) ≠ := +∞ =0 ( ) ! . The exponential function exp ( ) is slice-preserving if and only if is real. In general, if ∈ C , then exp ( ) is one-slice-preserving, i.e., it maps the slice C into C .
We consider also a more general eigenvalue problem in which the solutions are searched in the space of slice functions on Ω which are not necessarily slice-regular.

Eigenvalue problem for slice functions.
Let Ω be an axially symmetric domain of H. Consider the following eigenvalue problem for the slice derivative operator in the class S 1 (Ω) of slice functions induced by stem functions of the class 1 on Ω: with ∈ S 1 (Ω), ∈ H.

Proposition 3. Let Ω ⊂ H be an axially symmetric domain. A function ∈ S 1 (Ω) is a solution of (4) if and only if = · exp ( ), with ∈ SR (Ω) anti-sliceregular. The solution is slice-regular, and then a solution of (3), if and only if
Let be the operator of right multiplication by and let D = − denote the linear operator mapping a slice function ∈ S 1 (Ω) to the continuous slice function Note that D is slice-regular for any slice-regular function . Another useful property of the operator D is the following: for every slice-constant ∈ SC (Ω) and every ∈ S 1 (Ω), it holds

Remark 4.
When Ω is a slice domain, in view of Proposition 3 it holds If ∈ Ω ∩ R, the solution of (3) is uniquely determined by its value at , and takes the form If Ω is a product domain, then the solutions of (3) have more degrees of freedom (eight real d.o.f. instead of four). If ∈ SC (H \ R) is the function defined in (1), then formula (2) for = − = implies that := 2 + − 1 is the unique slice-constant function on H \ R with |C + = 1 , |C − = 2 . We then obtain, for every ∈ S, the representation H} for the kernel of the operator D on slice-regular functions on Ω. A function = · exp ( ) = ( 2 ) · exp ( ) + ( − 1 ) · exp ( ) ∈ Ker(D ) is uniquely determined by its values at two points, for example by and − . Since ·exp − ( ) = , a direct computation shows that where we used the fact that exp − (− ) = exp ( ) for every , ∈ H.
The eigenvalue problem (3) is the homogeneous problem associated to the following generalized eigenvalue problem. Given ℎ ∈ SR (Ω) and ∈ H, find ∈ SR (Ω) such that In view of the linearity of (5), the previous solution is uniquely determined up to solutions of the homogeneous problem (3). From Proposition 3 we can infer that the general solution is for any 0 ∈ H. On product domains the general solutions have similar forms, with any slice-constant function ∈ SC (Ω) in place of the constant 0 .
Remark 5 is a hint for finding solutions of (5) for every polynomial right-hand side ℎ ∈ H[ ] and ≠ 0. In this case we can take for Ω the whole quaternionic space H (a slice domain) or the product domain H \ R.
If Ω = H \ R, a product domain, then the solution of (5) is uniquely determined by its values at two points.
The preceding Proposition means that one can define a right inverse of D on the space of quaternionic polynomials of degree at most . It is the operator , then Now we consider equation (5) with an exponential right-hand side ℎ( ) = · exp ( ), with ∈ H and slice-constant.
be the slice-regular polynomial of degree Then the series of functions +∞

=0
, ( ) converges uniformly on the compact sets of H to an entire slice-regular function , ∈ SR (H) such that D ( , ) = exp ( ). As a consequence, D ( · , ) = · exp ( ) for every slice-constant on Ω. If and commute, then the function , has the expected explicit form From the estimate follows the uniform convergence of the series on compacts and then the equality To prove the last statement, we can assume that , ∈ C for a unit ∈ S and write We then compute the sums and conclude observing that Proof. Assume for the moment that the series is uniformly convergent. Then Let Since the estimate is symmetric in and , we can assume that | | < | |. Then and the uniform convergence of the series follows also in this case. If and commute, the following equality holds: The last statement follows immediately from the series expansion Observe that the slice-regular functions cos = I (cos ) and sin = I (sin ) are exactly the functions defined in [17,Def.11.23] in the more general context of Clifford algebras.
Eigenvalue problem of the second order for slice-regular functions. Propositions 3 and 8 permit us to study eigenvalue problems of the second order. Given , ∈ H, we consider the equation Notice that two operators D and D commute if and only if and commute, i.e., they belong to the same complex slice C , ∈ S.
The statement of the third case addressing = follows from Proposition 8 and Proposition 3.
Eigenvalue problem of the th-order for slice-regular functions. Now we generalize Corollary 11 to eigenvalue problems of any order . Let 1 , . . . , ∈ H and consider the equation We begin by introducing a family of slice-regular functions which generalize the exponential functions exp ( ) of Remark 1. Let := max{| 1 |, . . . , | |}. Then we have from which follows the uniform convergence of the series Λ ( ) = ≥ −1 ( ) on the compacts sets of H.
The canonical solution ( 1 ,..., ) ∉ ker(D 2 · · · D ) corresponds to the choice of parameters 1 = 1, 2 = · · · = = 0. (3) If all the eigenvalues 1 , . . . , commute and are distinct, then a simple computation shows that Then is another solution of (8). Indeed, using the preceding remark it can be verified that under the commutativity assumption the following equality holds in accordance with formula (9) of Corollary 15. In the preceding formula the product is assumed to be equal to 1 when = 1. Since the operators D 1 , . . . , D commute, the general form of the solutions of (8) can be written in the form = =1 ℎ · exp ( ) with ℎ ∈ SC (Ω) for every . Since 1 , . . . , belong to a complex slice C (unique is the eigenvalues are not all real), this result can be obtained also from the complex case.

A direct computation shows that
≤ | | |2 | 2 −1 we directly observe the uniform convergence of the series (10). It holds For every ∈ H \ {0}, we introduce an H-submodule of the right H-module of entire slice-regular functions on which it is possible to extend the operator E . Let A ⊂ SR (H) be defined as There exists a positive real ′ such that for every ∈ N, i.e., D ∈ A . Now let and this means that also E ( ) ∈ A . The last statement is a consequence of Proposition 18.
Example 20. Let ℎ( ) = sin . Then Since ℎ is slice-preserving (its series coefficients are all real), the solution E (ℎ) can also be obtained from the complex solution.
Remark 21. The previous example suggests an alternative method to obtain the entire function E (ℎ) for a quaternionic function ℎ ∈ A . In view of [10, Lemma 6.1], given a real basis {1, , , } of H, every quaternionic slice function ℎ can be written as ℎ = ℎ 0 +ℎ 1 +ℎ 2 +ℎ 3 , with ℎ slice-preserving functions for = 0, 1, 2, 3. Moreover, ℎ is slice-regular on a domain if and only all the functions ℎ are. It is also easy to verify that ℎ belongs to A if and only if all the components ℎ ∈ A . If ≠ 0, given any real basis { , 1 , 2 , 3 } of H, also the set {1, 1 −1 , 2 −1 , 3 −1 } is a real basis of H. Therefore for any function ℎ ∈ A one can write with slice-preserving functions ℎ in A . If ∈ C , the solution := E (ℎ ) of the equation (5) can be obtained for every = 0, . . . , 3 from the complex solution in C . Observe that the functions are one-slice preserving, since (C ) ⊆ C . We then have Here we point out that the second equality holds since the functions ℎ are slicepreserving. We then obtain that = 0 + 3 =1 ( −1 ) · is the uniquely determined entire solution E (ℎ) of (5) that vanishes at the origin.

Let denotes the Cauchy-Riemann-Fueter operator
and let be the conjugated operator Let Ω be an axially symmetric domain of H. Our aim is to apply the results of the previous sections to the following eigenvalue problem for the (conjugated) Cauchy-Riemann-Fueter operator in the class of monogenic slice functions (also called Fueter-regular functions in the present quaternionic case): on Ω with slice function s.t.
Eigenvalue problems of the similar form = have been studied rather extensively within the very general context where is some arbitrary function simply belonging to the function space 1 (Ω) without claiming any further conditions or properties on the considerable neither on the domain Ω.
The solutions of ( − ) = 0 where is the classical quaternionic Cauchy-Riemann-Fueter operator can be described in the form 0 where is an element of the kernel of , cf. for example [15], where the most general context of polynomial Cauchy-Riemann equations of general integer order and general multiplicity of the eigenvalues has been addressed extensively in the Clifford analysis setting, but under the assumption that is a general function from (Ω). This general treatment also includes the special case of -monogenic functions considered more than two decades earlier by F. Brackx in [2].
Notice that above we consider slightly differently its conjugated operator and the action of from the right hand-side.
F. Sommen and Xu Zhenyuan also studied in [25,23] the analogous equation in the vector formalism where the Dirac operator is considered instead of the Cauchy-Riemann-Fueter operator exploiting decompositions in terms of axially monogenic functions which have been considered in the preceding work [22]. The particular three-dimensional case has already been treated by K. Gürlebeck in [16]. The conjugated Dirac operator coincides with the Dirac operator up to a minus sign. Also this operator is a first order operator factorizing the Laplacian. The connection to the Helmholtz operator has been explicitly addressed in [16]. See also [19] where this topic has been investigated more extensively. Even more generally, one also considered the case where belongs to a general Sobolev spaces, for instance to 1 (Ω), cf. [18], however again without considering any further (symmetric) conditions.
In our framework we now address the special situation where has additionally the property of being a slice function defined over an axially symmetric domain Ω. These analytic and geometric aspects are new. This particular context allows us to use special series representations and the special properties of slice functions which do not hold in the general context in the above mentioned papers, as it will be explained in the sequel.
In order to tackle problem (11), we apply Fueter's Theorem, which can be seen as a bridge between the classes of slice-regular functions and monogenic functions.
We recall a useful concept introduced in [12]. Given a slice function on Ω, the function ′ : Ω \ R → H, called spherical derivative of , is defined as (b) The following generalization of Fueter's Theorem holds: In the following we will use also the following result.
since is holomorphic. Therefore, and using Theorem 22(1) we get formula (12). Since is slice-regular, we have = 0 (see e.g. [9]). This equality and (12) give formula (13). Let AM (Ω) denote the class of axially monogenic functions, i.e., of monogenic slice functions on Ω: In view of the generalized Fueter's Theorem, the Laplacian maps the space SR (Ω) into AM (Ω). It is known that this map is surjective (see e.g. [5]). Now we construct an operator L : AM (Ω) → AM (Ω) which makes the following diagram Here Z ( ) := ( +1 ) ′ is a harmonic homogeneous polynomial of degree − 1 in the four real variables 0 , 1 , 2 , 3 . The polynomials Z are called zonal harmonic polynomials with pole 1, since they have an axial symmetry with respect to the real axis (see [20,21]). Observe that the polynomials P are axially monogenic but not zonal. According to our knowledge, zonal monogenics were introduced in [24]. See also [6] where further basic properties have been studied. Proposition 24. The polynomials P are axially monogenic homogeneous polynomial of degree in 0 , 1 , 2 , 3 . They are slice functions on H, given by the explicit formula In particular, P is slice-preserving and the restriction of P to the real axis is the monomial P ( 0 ) = +2 Proof. The first statement is a consequence of Fueter's Theorem. Formula (15) follows from [20, Corollary 6.7], since To prove (16), we observe that

Example 25. The first four axially monogenic polynomials
We Proof. By direct computation, using Theorem 22(2c) and Definition (14), we get that is L (P ) = ( + 2)P −1 . On the other hand, since P is in the kernel of , it holds Since +2 is slice-regular, we have +2 0 = +2 (see e.g. [9]) and then as before for every integer ≥ 1.
Remark 27. Definition (14) can be extended to negative indices ∈ Z. One obtains axially monogenic functions P ∈ AM (H \ {0}) that satisfy the same relation as in the case ≥ 0: P = ( + 2)P −1 for every < 0. Observe that P −1 = P −2 = 0. The functions P and P − are related through the Kelvin transform of R 4 (see [20,Prop.5.1(c)]). It follows that also for negative the P 's are homogeneous of degree .
Proof. The first two statements follow immediately from Corollary 15: The same computation holds for every function of the form (18).

A
As a very interesting bi-product we can relate the solutions to L 1 L 2 = 0 to axially monogenic solutions to the 3D and 4D time-harmonic Helmholtz and stationary massless Klein-Gordon equation considered in an axially symmetric domain. In the sequel we abbreviate for convenience the purely quaternionic part of the Cauchy-Riemann-Fueter operator by D := 1 + 2 + 3 . Actually, D is often called the Euclidean Dirac operator and it satisfies D 2 = −Δ 3 , where Δ 3 is the ordinary Euclidean Laplacian in R 3 = Im(H). Notice that = 1 4 Δ 4 , where Δ 4 now represents the Laplacian in R 4 , cf. for example [18] and many other classical textbooks on quaternionic and Clifford analysis, such as also [3] and others. In this section we use the symbol Δ 4 instead of Δ as in the previous section to avoid confusion with Δ 3 . As a direct consequence of Proposition 32 we can establish Proposition 34. Suppose that is an axially monogenic solution to L 1 L 2 = 0 on some axially symmetric domain Ω ⊂ H.
(a) Let ∈ S be any imaginary unit. In the particular case where we take