SPECTRAL THEORY OF THE INVARIANT LAPLACIAN ON THE DISK AND THE SPHERE A COMPLEX ANALYSIS APPROACH

A BSTRACT . The central theme of this paper is the holomorphic spectral theory of the canonical Laplace operator of the complement Ω : = { ( z , w ) ∈ (cid:98) C 2 : z · w ̸ = 1 } of the “complexified unit circle” { ( z , w ) ∈ (cid:98) C 2 : z · w = 1 } . We start by singling out a distinguished set of holomorphic eigenfunctions on the bidisk in terms of hypergeometric 2 F 1 functions and prove that they provide a spectral decomposition of every holomorphic eigenfunction on the bidisk. As a second step, we identify the maximal domains of definition of these eigenfunctions and show that these maximal domains naturally determine the fine structure of the eigenspaces. Our main result gives an intrinsic classification of all closed M ¨ obius invariant subspaces of eigenspaces of the canonical Laplacian of Ω . Generalizing foundational prior work of Helgason and Rudin, this provides a unifying complex analytic framework for the real–analytic eigenvalue theories of both the hyperbolic and spherical Laplace operators on the open unit disk resp. the Riemann sphere and, in particular, shows how they are interrelated with one another.


INTRODUCTION
Let C := C ∪ {∞} denote the Riemann sphere.The purpose of this paper is to explore the spectral theory of the complex invariant Laplace operator sphere C from a complex analytic point of view and, in addition, it also shows how they are interrelated to one another.Beyond that, the complex point of view taken in this paper offers several other useful advantages.In particular, it connects in a natural way the fine structure of the eigenspaces of the hyperbolic and spherical Laplacians as described by Helgason [21] and Rudin [37] with the maximal domain of existence of the corresponding holomorphic eigenfunctions of the invariant Laplacian ∆ zw .
As one instance, we analyze from a complex analysis point of view the building blocks of each λ -eigenspace of the hyperbolic Laplacian ∆ D , which have previously been identified e.g. by Helgason in [21] using real variable methods.It turns out that these so-called Poisson Fourier modes naturally extend to holomorphic eigenfunctions of ∆ zw which are (maximally) defined either on Ω or on one of three distinguished subdomains of Ω depending on the choice of the eigenvalue λ .This then allows a transparent proof that each holomorphic eigenfunction of ∆ zw defined on any rotationally invariant subdomain of Ω has a unique spectral decomposition in form of a locally uniformly and absolutely convergent infinite series composed of Poisson Fourier modes.In the special case of the bidisk D 2 and further restriction to the "diagonal" {(z, z) : z ∈ D} we recover the spectral decomposition of the smooth eigenfunctions of ∆ D on the unit disk D as described e.g. in [21] or [8].
As a second instance, we investigate the structure of the closed "Möbius invariant" subspaces of any fixed eigenspace X λ (D) of ∆ D from a complex analysis point of view.This topic has been investigated in detail by Rudin in [37] using purely "real" methods.We shall see that the distinguished subdomains of Ω mentioned above naturally lead to the same distinction between exceptional and non-exceptional eigenvalues which have been found by Rudin.In Rudin's work, the exceptional cases correspond to the eigenvalues λ = 4m(m + 1), m = 0, 1, 2 . .., and they are characterized by the existence of three non-trivial Möbius invariant closed subspaces of X λ (D), exactly one of which, X 0 λ (D) say, is finite dimensional.It turns out that a complex number λ ∈ C is an exceptional eigenvalue in the sense of Rudin if and only the invariant Laplacian ∆ zw has a globally (that is on Ω) defined holomorphic λ -eigenfunction.Moreover, in this case the unique finite dimensional invariant subspace X 0 λ (D) corresponds precisely to the full λ -eigenspace of all globally defined λ -eigenfunctions of the invariant Laplacian ∆ zw , which then, in fact, is invariant under the full group of all Möbius transformations.For the other two non-trivial invariant subspaces of the exceptional ∆ D -eigenspace X λ (D) discovered by Rudin as well as the full eigenspace X λ (D) itself we give a similar but more intricate description in form of Rungetype approximation results in terms of holomorphic λ -eigenfunctions defined precisely on one of the distinguished three subdomains of Ω, see Theorem 2.8.
In the next section we give an account of the main results of this work and their ramifications for the spectral theory of the hyperbolic and spherical Laplacian as well as an outline of the structure of the remaining sections.The accompanying papers [18,19,27,32] are related to other aspects of the function theory of the set Ω, the complement of the complexified unit circle, and its applications.Our interest in this set and its inhabitants, the holomorphic functions on Ω, first arose in connection with previous work [7,10,12,28,41,43] on canonical Wick-type star products in strict deformation quantization of the unit disk and the Riemann sphere, and from our desire to understand the somehow mysterious role played by Ω and in particular by its function-theoretic properties in this regard.A partial explanation was given in [19], where it was indicated that invariant differential operators of Peschl-Minda type, on the one hand, effectively facilitate and unify the study of the star products on the disk and the sphere, and on the other hand, are perhaps best understood as operators acting on the spaces of holomorphic functions on Ω and its three distinguished subdomains.We started wondering whether and how the most basic differential operator acting on Ω, the invariant Laplacian ∆ zw , and its spectral theory possibly fit into this emerging picture.This paper describes what we have found.In the forthcoming paper [32] of the second-named author these endeavours will come to full circle: it is shown that there are globally defined eigenfunctions of the Laplacian ∆ zw which form a Schauder basis of the Fréchet space H (Ω) of all holomorphic functions on Ω.The results of the present paper then imply that the algebra for which the Wick-star product on D in [28] is constructed, admits a spectral decomposition precisely into the finite dimensional invariant subspaces of the exceptional eigenspaces of the hyperbolic Laplace operator ∆ D on D discovered by Rudin [37] many years ago.This provides an intrinsic characterization of the algebra A (D) in terms of the natural hyperbolic geometry of the unit disk and its canonical invariant Laplacian ∆ D .

OVERVIEW AND MAIN RESULTS
In order to place the results of this paper into a broader context we begin by recalling in greater detail the striking distinction between exceptional and non-exceptional eigenvalues of the hyperbolic Laplace operator ∆ D and its relevance for the study of the invariant subspaces of the ∆ D -eigenspaces which has been discovered by Rudin [37].The Fréchet space of all twice continuously (real) differentiable functions f : D → C equipped with the standard compactopen topology is denoted by C2 (D).For each λ ∈ C we denote by X λ (D) the vector space of all λ -eigenfunctions f ∈ C 2 (D) of the hyperbolic Laplacian, that is, The hyperbolic Laplacian ∆ D is invariant under the full group of all conformal automorphisms (biholomorphic maps) T of the unit disk D in the sense that In order to emphasize that this group consists entirely of Möbius transformations, we call it the Möbius group of D and denote it by for some m = 0, 1, 2, . .., then X λ (D) has precisely three distinct nontrivial Möbius invariant subspaces.There is exactly one non-trivial Möbius invariant subspace of X λ (D) which is finite dimensional; its dimension is 2m + 1.
The alternative (E) in Theorem 2.1 will be called the exceptional case and the unique finite dimensional Möbius invariant subspace of X λ (D) will be denoted by X 0 λ (D).The alternative (NE) will be referred to as the non-exceptional case.
One of the main results of the present paper is a complete analogue of Theorem 2.1 with ∆ D replaced by the differential operator ∆ zw , see Theorem 2.5.Apart from being potentially interesting in its own right, it provides a concrete function-theoretic description of the exceptional eigenspaces in Theorem 2.1.This also adds a conceptual component to Rudin's handling of the invariant eigenspaces of the hyperbolic Laplace operator ∆ D .
The characteristic feature of our approach is to look for holomorphic solutions F of the eigenvalue equation defined on a subdomain D of Ω which we wish to choose as large as possible depending on the eigenvalue λ .These maximal domains of existence (see Definition 2.4) of the λ -eigenfunctions turn out to be the only essential ingredients which are needed to give a complete description of the (invariant) λ -eigenspaces of the operator ∆ zw and its offsprings ∆ D and ∆ C .
In order to state our main results we have to adapt the notation which we have introduced above for the hyperbolic Laplacian to the case of the differential operator Instead of working in the Fréchet space C 2 (D) we now fix a subdomain D of the set Ω := {(z, w) ∈ C : z • w ̸ = 1}, and work in the Fréchet space H (D) of all complex-valued holomorphic functions defined on D (again equipped with the topology of locally uniform convergence, this time on D).Our goal is determine the holomorphic solutions F : D → C of the eigenvalue equation (2.1), i.e. we are interested in the ∆ zw -eigenspaces With regard to Rudin's theorem (Theorem 2.1) a particularly natural choice for the domain D is the bidisk D 2 := D × D since it is easy to see that for every F ∈ X λ (D 2 ) the "restriction" of F to the "diagonal" {(z, z) of the hyperbolic Laplacian ∆ D for the eigenvalue λ .In fact, the following result shows that every f ∈ X λ (D) arises in this fashion, that is, it has an "extension" to an eigenfunction F of ∆ zw which is holomorphic on the bidisk D 2 : Then there is a uniquely determined function F ∈ H (D 2 ) such that ∆ zw F = λ F on D 2 and f (z) = F(z, z) for all z ∈ D.Moreover, the induced bijective linear map so there is trivially a holomorphic function F defined on some open neighborhood In view of a well-known variant of the identity principle (see Lemma 3.1) there is only one such holomorphic extension F : is called the hyperbolic restriction map.Its continuous inverse is called the hyperbolic extension map from X λ (D) to X λ (D 2 ).
The hyperbolic restriction and extension mappings provide the bridge between the holomorphic spectral theory of the invariant Laplacian ∆ zw and the smooth spectral theory of the hyperbolic Laplacian ∆ D .In particular, one can study the spectral properties of ∆ D on D from the viewpoint of complex analysis on the bidisk D 2 .Based on Theorem 2.2, we can now proceed to associate the fine structure of the eigenspaces X λ (D) with the maximal domains of existence of holomorphic eigenfunctions in X λ (D 2 ), a concept which is defined as follows, cf.[14, p. 97].

Definition 2.4
Let F be a holomorphic function on the bidisk D 2 .A subdomain D ⊆ Ω that contains the bidisk D 2 is called a maximal domain of existence of F if the function F has a holomorphic extension to D but to no strictly larger subdomain of Ω.
For our purposes, this definition is natural in several respects.First, the condition D ⊇ D 2 obviously comes from Theorem 2.2.Second, the condition D ⊆ Ω is natural in view of the fact that every function which is holomorphic on a subdomain of C 2 that is strictly larger than Ω is necessarily constant (see Theorem 5.3 in [18]).In particular, the largest possible maximal domain of existence of any eigenfunction of ∆ zw in H (D 2 ) is Ω.Notably, this includes the constant eigenfunctions.At first sight, such a linguistic subtlety might seem irritating, but it facilitates the statement of many of our results.
With this concept at hand, we can now give a function-theoretic characterization of the exceptional eigenvalues of the hyperbolic Laplacian ∆ D and the corresponding unique finitedimensional non-trivial Möbius invariant subspaces X 0 λ (D) of the ∆ D -eigenspaces X λ (D).In addition, the following theorem provides an equivalent condition in terms of existence of globally defined eigenfunctions of the spherical Laplacian ∆ C .(i) There exists a function F ∈ X λ (D 2 ) with Ω as maximal domain of existence.
(ii) There exists a function g ∈ C 2 ( C) such that ∆ C g = λ g on C.
(a) Note that X 0 0 (D 2 ) and X 0 (Ω) consist precisely of the constant functions.(b) In view of Theorem 2.5, an eigenvalue λ of ∆ D is exceptional if and only if there exists a holomorphic eigenfunction of ∆ zw on D 2 with the largest possible maximal domain of existence, the set Ω. (c) In Theorem 2.5 (a) we think of X λ (Ω) as a subspace of If λ ∈ C is an exceptional eigenvalue, then the spherical restriction map With Theorem 2.5 we have reached two of our goals, a conceptual characterization of exceptional eigenvalues and the finite dimensional non-trivial Möbius invariant subspaces of the eigenspaces X λ (D) of the hyperbolic Laplacian ∆ D .Next, we address the infinite dimensional non-trivial Möbius invariant subspaces of the ∆ D -eigenspaces X λ (D).This turns out to be more difficult, and first requires clarification of the invariance properties of the Laplacian ∆ zw .Implicitly, the underlying difficulty is already present in Theorem 2.5, and can be seen as follows.Let λ ∈ C be an exceptional eigenvalue of ∆ D .Then the finite-dimensional Möbius invariant subspace X 0 λ (D) is invariant under all automorphisms of D. However, the corresponding eigenspace X λ (Ω) for ∆ zw , which consists of functions holomorphic on Ω, is, loosely speaking, invariant under a much larger group of automorphisms.In fact, we first note that the invariant Laplacian ∆ zw of Ω is not invariant under all biholomorphic automorphisms of Ω in the sense that the invariance condition holds for all T ∈ Aut(Ω).The reason is simply that the automorphism group Aut(Ω) is much too large, see [18].However, ∆ zw is invariant under the subgroup M of Aut(Ω) defined by (2.4) where we write M ( C) for the group of all Möbius transformations ψ : C → C.This is easy to prove by direct verification.(Conversely, one can show that every T ∈ Aut(Ω) for which the invariance property (2.3) holds does necessarily belong to the subgroup M , see [18,Theorem 5.2], but we do not need such a result in this paper.)Since M is "induced" by the set M ( C) of all Möbius transformations, we call M the Möbius group of Ω.Note that M ( C) is strictly bigger than M (D) 3 , so the Möbius group M of Ω is strictly larger than the Möbius group M (D) of D. Now, while X 0 λ (D) is invariant under each element of M (D), the set X λ (Ω) is invariant under each element of M , simply because X λ (Ω) is the entire λ -eigenspace of the Laplacian ∆ zw on Ω and ∆ zw is invariant with respect to the Möbius group M .
In view of this discussion, it is now clear that a suitable concept of invariance for the eigenspaces X λ (D 2 ) of ∆ zw on the bidisk D 2 has to be based on the group which we call the Möbius group of the bidisk D 2 .It consists precisely of all automorphisms of the bidisk D 2 which have the invariance property (2.3).In fact, it is not difficult to show that ) which is invariant under the Möbius group M (D 2 ) will be called a Möbius invariant subspace.Theorem 2.2 implies immediately the following result.
We can now give a function-theoretic characterization of the Möbius invariant subspaces of X λ (D 2 ) and thereby, in view of Corollary 2.6, the Möbius invariant subspaces of X λ (D).The following subdomains of Ω play the essential role for this purpose: Note that each of these three subdomains of Ω contains the bidisk D 2 .Moreover, We note in passing that the subdomains Ω + and Ω − arise naturally in the study of the Fréchet space structure of H (Ω) (see [18]) and also for studying invariant differential operators of Peschl-Minda type acting on H (Ω) (see [19]).The following result shows that they are also useful for describing the Möbius invariant subspaces of the eigenspaces of the invariant Laplacian ∆ zw .We use the following terminology: Definition 2.7 Let U ⊆ V be subdomains of some complex manifold, and let Y ⊆ X be subsets of H (U).
We say that "X ∩ H (V ) is dense in Y " if X ∩ H (V ) ⊆ Y and if every function in Y can be approximated locally uniformly on U by functions in X, which have a holomorphic extension to V .
We are now, finally, in a position to formulate the main result of this paper.

Theorem 2.8
Let λ ∈ C and let Y be a non-trivial Möbius invariant subspace of X λ (D 2 ).Then one and only one of the following four alternatives holds.
for some non-negative integer m. (iii) None of the density statements (NE ), (E + ) and (E − ) may be improved to equalities.Remark 3. Recall that two domains U ⊆ V in C n form a Runge pair (U,V ) if every function in H (U) can be approximated locally uniformly in U by functions in H (V ).Identifying domains U ⊆ V as Runge pairs is a fundamental and in many cases challenging problem in complex analysis.Note that in our terminology (U,V ) is a Runge pair if and only if is a Runge triple.Theorem 2.8 can therefore be regarded as a Runge-type approximation theorem for the Möbius invariant spaces of eigenfunctions of the invariant Laplacian ∆ zw .
The plan of the paper is as follows.We introduce some basic concepts and notation in a preliminary Section 3. In Section 4 and Section 5 we develop the general spectral theory of the invariant Laplacian ∆ zw on Ω in analogy to the well-established spectral theory of the hyperbolic Laplacian ∆ D on D. In a sense, we rather closely follow the presentation Berenstein and Gay [8, Section 1.6] have given for the spectral theory of the hyperbolic Laplacian, but we have made an effort either to provide even more rigorous proofs or to give precise references to the literature for all auxiliary results which are needed.In contrast to [8] we completely work in the holomorphic setting.On the one hand, this makes it possible to take advantage of many efficient tools from complex analysis which are not available otherwise.On the other hand, we need to incorporate from the beginning the maximal domain of existence of eigenfunctions; an issue which does not even show up when working "only" on the unit disk.Here, our approach requires some finer analysis of the building blocks of the eigenfunctions, namely certain hypergeometric functions and their integral representations in terms of Poisson Fourier modes.
In Section 6 we prove Theorem 2.2 and show that it is in some sense best possible by providing an explicit example.This implies that the smooth spectral theory of the hyperbolic Laplacian ∆ D on the disk D and the holomorphic spectral theory of the Laplacian ∆ zw on the bidisk D 2 are essentially equivalent.In the same spirit we relate the smooth spectral theory of the spherical Laplacian ∆ C with the holomorphic spectral theory of the Laplacian ∆ zw on Ω as well as the exceptional eigenvalues of the hyperbolic Laplacian ∆ D , see Theorem 6.2.Section 7 is devoted to a study of the transformation behavior of the Poisson Fourier modes under precompositions with elements of the Möbius group M .These results are needed for Section 8 where we prove our main results, Theorem 2.5 and Theorem 2.8.By and large, we follow Rudin's [37] treatment of invariant subspaces of eigenfunctions of the hyperbolic Laplacian, but again completely working in the holomorphic setting; we briefly comment on the similarities and differences between our and Rudin's approach in Remark 13.We close the paper with Section 9 which connects the Poisson Fourier modes to the invariant differential operators of Peschl-Minda type studied in [19].
Four final preliminary remarks are in order.First, treating the hyperbolic eigenvalue equation ∆ D f = λ f and the spherical eigenvalue equation ∆ C f = λ f as special cases of the more general complex eigenvalue equation ∆ zw F = λ F has been a recurrent theme in the literature for a long time.To mention but a few of the many references, we refer for instance to the papers [4,5,6] and their bibliographies.What seems to be new is the systematic study of the maximal domains of existence of the holomorphic solutions of ∆ zw F = λ F and its ramifications for the study of the invariant subspaces of the ∆ D -eigenspaces.As a second remark, we should point out that Rudin's work [37] is in fact concerned with the invariant Laplace operator on the unit ball of C n , while our focus is exclusively on the complex one-dimensional case n = 1.Thirdly, even though we are superficially dealing with holomorphic functions of two complex variables, we only need very few and only elementary facts from the theory of several complex variables.Finally, Helgason [20,21,22] has systematically studied invariant differential operators and their eigenvalue problem in the setting of homogeneous spaces.In contrast to our holomorphic approach, he used entirely real methods.This Lie theoretic approach has since been generalized significantly.While providing a comprehensive list of references would go beyond what we can achieve here, we would like to mention [23,26], who generalized the theory to higher dimensional symmetric spaces, and Maaß [29], who initiated the vast and fruitful research of Maaß wave forms.For an open subset U of C or C 2 , we write ∂U for its boundary and U for its closure.The set of all twice resp.infinitely (real) differentiable functions f : and we write H (U) for set of all holomorphic functions f : an open submanifold of C 2 .In order to describe the complex structure of Ω only two charts are necessary, the standard chart and the flip chart In these local coordinates the invariant Laplace operator ∆ zw of the complex manifold Ω is then given by , and it is easily seen that ∆ zw is a well-defined object.Here, ∂ z and ∂ w denote the Wirtinger derivatives with respect to z and w, respectively.
One of the few elementary results from the theory of functions of several complex variables we need in this paper is the following simple lemma.Lemma 3.1 ("Two variable identity principle" (p.18 in [35])) Let U be a subdomain of C 2 which contains a point of the form (z, z) resp.(z, −z).Let f : U → C be a holomorphic function such that Then f ≡ 0.
Finally, we briefly recall some standard terminology from linear algebra.We denote by span M the collection of all finite linear combinations of elements of a subset M of a given vector space X.The vector spaces that occur in this paper are spaces of smooth or holomorphic functions defined on some open subset U of Ω which we equip with the standard topology of uniform convergence on compact subsets of U. If M denotes a set of smooth or holomorphic functions on U, we denote by clos U M the closure of M with respect to locally uniform convergence on U.

HOMOGENEOUS EIGENFUNCTIONS AND POISSON FOURIER MODES
By making a separation of variables approach Rudin [37] showed that every λ ∈ C is an eigenvalue of ∆ D .The analogous result is true for ∆ zw : let n ∈ Z and suppose f n : D → C is a holomorphic function defined on a domain D ⊆ C 2 containing the origin (0, 0).Further assume that f n is n-homogeneous, i.e., (4.1) f n (ηz, w/η) = η n f n (z, w) for all η ∈ ∂ D and for all (z, w) ∈ C2 belonging to the bidisk D 2 r = D r × D r for some (and hence all) r > 0 such that D 2 r ⊆ D. A consideration of the power series expansion of f n at (0, 0) in D 2 r implies in a straightforward way that there is a holomorphic function A power series ansatz shows that there is at most one solution y n of (4.3) which is holomorphic in a neighborhood of t = 0 and normalized such that y n (0) = 1.In order to find this solution, we convert (4.3) into a hypergeometric differential equation as follows.We choose µ ∈ C such that λ = 4µ(µ − 1) and let ŷn (t) := (1 − t) −µ y n (t).Then (4.3) is equivalent to It is well-known (see [33, §15.10]) that the only solution ŷn of (4.4) which is holomorphic at t = 0 and normalized at t = 0 by ŷn (0) = 1 is the hypergeometric series where denotes the (rising) Pochhammer symbol.This procedure leads to the conclusion that is the unique solution of (4.3) which is holomorphic in t = 0 and normalized by y n (0) = 1.Note that there are in fact two complex numbers µ ∈ C such that λ = 4µ(µ − 1).However, as we have seen, both necessarily lead to the same holomorphic solution y n of (4.3) with y n (0) = 1.
Remark 4. Note that if µ is one solution to λ = 4µ(µ − 1), then 1 − µ is the other one.As we have seen, both numbers induce the same function (4.6).This also follows from the transformation formula (see [1, Eq. 15.Returning to (4.2) with this choice of y n , we therefore see that is holomorphic at least on the domain and provides the unique n-homogeneous solution of ∆ zw f = λ f on Ω * up to a multiplicative constant.
Next, we relate the n-homogeneous eigenfunction F µ n of ∆ zw to a complexified version of the Poisson kernel of the unit disk.This slight change of perspective will turn out to be important in the sequel.In fact, a well-known integral representation formula for the hypergeometric function 2 F 1 (µ, µ + |n|; |n| + 1; zw), see [11, Section 2.5.1,Formula (10), p. 81], shows that for every n ∈ Z, for µ ∈ C, n ∈ N 0 .On the right-hand side of this identity one can recognize the (−n)-th Fourier coefficient of (a suitably defined power of order µ of) the generalized Poisson kernel (4.9) Note that if w = z ∈ D, then P(z, z; ξ ) is the standard Poisson kernel of the unit disk.These considerations motivate the following definition.
is called the n-th Poisson Fourier mode (PFM) of order µ.
Here, a µ is defined for a ∈ C \ (−∞, 0] as exp (µ log a), where log denotes the principal branch of the logarithm.We can now reformulate (4.8) in terms of PFMs as follows: (4.11) In particular, P In particular, f has a holomorphic extension to D ∪ Ω * .Moreover, the following dichotomy holds: (NE) (Non-exceptional case) If λ ̸ = 4m(m + 1) for all m ∈ N 0 , then Ω * is the maximal domain of existence of f .(E) (Exceptional case) If λ = 4m(m + 1) for some m ∈ N 0 , then the maximal domain of existence of f is is holomorphic in a neighborhood of the point x 0 := z 0 w 0 /(z 0 w 0 − 1) ≥ 1.However, see [33, 15.2.3], for all x > 1.By our assumption, the left-hand side of (4.13) has to vanish for all x ∈ R in some open interval (x 0 , x 0 + ε) with ε > 0 and hence the same is true for the right-hand side.
Since the right-hand side is a holomorphic function of x on the domain C \ (−∞, 0] this is clearly only possible if µ ∈ N. The remaining case n ≤ 0 follows from the case n ≥ 0 and we may assume n ≥ µ.In this case, the function G µ n in (4.14) is a polynomial in zw of degree µ − 1, and thus F µ n is the product of z n times a rational function in zw of numerator degree µ − 1 and of denominator degree µ − 1, and is therefore holomorphic on Ω − .Moreover, F µ n has no holomorphic extension to a point (∞, w 0 ) with w 0 ∈ C \ {0} since in view of (4.14) and (4.12) by [1, 15.1.20],so |F We close this section by collecting some elementary properties of the PFM P µ n which will be needed in the sequel.
This is (4.7) together with (4.11) resp.(4.12).(b) Using the series representation of 2 F 1 functions we see that (4.16) If −µ = m ∈ N 0 , then P −m ±n = 0, whenever n > m.Otherwise, (4.17 the vector space of polynomials of total degree m with dim(V m ) = m + 1.Then b a p(z, w) := p az − bw, −bz + aw for m ∈ N constitutes a complete list of the irreducible finite dimensional representations of SU(1, 1), see also [16,Prop. 4.11 & Sec. 4.6].Moreover, π m descends to the quotient PSU(1, 1) if and only if m is even, providing a description of all irreducible finite dimensional representations of PSU(1, 1).A computation then yields explicit formulas for (4.19), which completes the eigenvalue theory of the Laplacian on PSU(1, 1).Finally, parametrizing a copy of the rotation group U(1) ⊆ PSU(1, 1) by iη 0 0 −iη with η ∈ ∂ D yields that exactly the representative functions (π 2m ) m,d with 1 ≤ d ≤ 2m + 1 are invariant under the action of U(1) and thus pass to the quotient D ∼ = PSU(1, 1)/ U (1).By a computation, (π 2m ) m,d = P −m m−d with the exceptional Poisson Fourier modes from (4.17).Note that this only recovers the finite dimensional invariant eigenspace, i.e. the case (E 0 ) in Theorem 2.8.It would be interesting to study whether this approach generalizes to the other invariant subspaces by incorporating representations on infinite dimensional spaces.

SPECTRAL DECOMPOSITION OF EIGENSPACES
In this section we show that for every rotationally invariant domain D ⊆ Ω containing the origin each holomorphic eigenfunction of the invariant Laplacian ∆ zw on D has a unique representation as a Poisson Fourier series, a doubly infinite series with Poisson Fourier modes as building blocks.We shall also see that if D = Ω this series corresponds to a finite sum, and when D is one of the distinguished domains Ω + or Ω − , then the series is one-sided infinite.
Theorem 5.1 Let D be a rotationally invariant subdomain of Ω containing the origin, and let f ∈ H (D) be such that ∆ zw f = λ f for some λ ∈ C of the form λ = 4µ(µ − 1) with Re µ ≥ 1/2.Then there are uniquely determined coefficients c n ∈ C such that Here, both series converge absolutely and locally uniformly in D.
Proof.(i) For each n ∈ Z and all (z, w) ∈ D we consider Since D is rotationally invariant, f n is well-defined.Clearly, f n is holomorphic on D and nhomogeneous.By Theorem 4.2, there are complex numbers c n ∈ C such that We fix (z, w) ∈ D and choose positive constants r < 1 < R such that (ηz, w/η) ∈ D for all r < |η| < R.This is possible as D is a rotationally invariant domain.Then η → f (ηz, w/η) is holomorphic in the annulus r < |η| < R and therefore has a representation as the Laurent series which converges locally uniformly in r < |η| < R. In particular, This series converges in fact uniformly on each compact set K ⊆ D. In order to see this, let K be such a compact subset of D. We can then choose positive constants 1 for all n ≥ 0, and this ensures the absolute and uniform convergence of the two series on the compact set K.
(iii) In view of Step (ii) the coefficients f n (z, w) are exactly the Laurent coefficients of η → f (ηz, w/η) in an annulus containing the unit circle and are thus uniquely determined by f .Hence, f n (z, w) = c n P µ −n (z, w) shows that the coefficients c n are uniquely determined by f .□ In fact, the previous proof provides the following more precise information.□ Since all PFM P µ n are holomorphic on Ω * , it is natural to inquire whether the series (5.1) in Theorem 5.1 converges on some bigger domain than D. In Section 6 we shall see that in general this is not the case.

COMPARISON WITH THE EIGENVALUE EQUATION OF THE LAPLACIAN ON THE UNIT DISK AND THE RIEMANN SPHERE
In this section we relate the spectral theory of the Laplacian on the unit disk D resp. the Riemann sphere C with the spectral theory of the invariant Laplacian on Ω which we have developed so far.In particular, we show that all eigenfunctions of ∆ D on D resp.∆ C on C do have holomorphic extensions to eigenfunctions of ∆ zw on the bidisk D 2 resp.Ω.Our approach is similar to the one employed in [8, Section 1.6] which deals exclusively with the hyperbolic Laplacian ∆ D on the unit disk D. However, we need some fine properties of hypergeometric functions, in addition to those which have been employed in [8].We begin with the following lemma which is crucial for our approach.
Proof.Fix m ∈ N 0 such that m > Re µ − 2 and let n ∈ N 0 .Then by [11,Formula (11), p. 76] or [30, p. 84] there are complex numbers Note that our choice of the non-negative integer m guarantees that the integral in (6.2) converges.To prove (6.1) it therefore suffices to show that ρ m+1 (n, ω) → 0 uniformly on every compact subset |1 − stω| > 0 , and since −Re µ − 1 − m < 0, we have where An application of Stirling's formula (6.4) to each of the first three quotients in (6.3) shows that there is a constant γ > 0 depending only on µ, m and K such that In particular, ρ m+1 (n, ω) → 0 as n → ∞ uniformly for ω ∈ K.This completes the proof of (6.1).□ We are now in a position to prove Theorem 2.2.
Proof Theorem 2.2.Let f ∈ X λ (D), and write λ = 4µ(µ − 1) for some complex number µ ∈ C with Re µ ≥ 1/2.By [8,Theorem 16.18] there are uniquely determined coefficients c n ∈ C such that the series converges absolutely and pointwise for each z ∈ D. As it is shown in the proof of [8, Theorem 1.6.18], the coefficients c n do have the additional property that We proceed to show that (6.5) and Lemma 6.1 together guarantee that the series (6.6) converges locally uniformly for (z, w) ∈ D 2 , and hence defines a function F ∈ H (D 2 ) with the required properties.The identity principle shows further that F is then uniquely determined.
It remains to prove the local uniform convergence of the series (6.6) in D 2 .Fix r ∈ (0, 1).We begin by noting that (4.11) and (4.7) lead to for all |z|, |w| ≤ r.Since the Möbius transformation Combining this with Stirling's formula (6.4) we see from inequality (6.7) that there is a constant γ > 0 such that (6.8) |P µ n (z, w)| ≤ γ • (|n| + 1) Re µ−1 r |n| for all |z| ≤ r, |w| ≤ r and every n ∈ Z.This estimate together with (6.5) implies that the series (6.6) converges uniformly for |z|, |w| ≤ r, as required.In particular, we have shown that the restriction map ) and X λ (D) are both Fréchet spaces with respect to the topology of locally uniform convergence on D 2 resp.D (see [38, Corollary 1 to Theorem 4.2.4] for the fact that X λ (D) is a Fréchet space) and the restriction map R h is obviously continuous, its inverse is continuous as well by the Open Mapping Theorem.This completes the proof of Theorem 2.2.□ Theorem 6.2 (Smooth eigenfunctions of ∆ C on C vs. holomorphic eigenfunctions of ∆ zw on Ω) Let λ ∈ C. Then the following are equivalent: Theorem 6.2 is a special case of Theorem 2.5.
Proof.The implication (ii) ⇒ (iii) is Corollary 5.2, and (iii) ⇒ (ii) is Theorem 4.2.Clearly, (ii) implies (i), so we only need to prove that (i) implies (ii).Accordingly, we write λ = 4µ(µ − 1) with µ ∈ C and Re µ ≥ 1/2.For n ∈ Z consider Arguing in a similar way as in the proof of Theorem 4.2 we see that there is a constant c n ∈ C such that Therefore, the behavior of f n as z → ∞ depends essentially on the value 2 F 1 (µ, We conclude that either c n = 0 for every n ∈ Z and then f ≡ 0, or c n ̸ = 0 for at least one n ∈ Z and then λ = 4m(m + 1) for some m ∈ N 0 .In the latter case, we see that We see that the spectrum of the hyperbolic Laplacian ∆ D is C whereas the spectrum of the spherical Laplacian ∆ C is notably smaller as it only consists of the scalars 4m(m + 1) for m ∈ N 0 .Furthermore, by Theorem 6.2 every eigenfunction of the spherical Laplacian can be extended to the whole domain Ω.This is different to the hyperbolic case where the extension to D 2 provided by Theorem 2.2 is maximal at least for the category of rotationally invariant domains as the following example shows.

Note that asymptotically
This follows from Lemma 6.1 and Stirling's formula (6.4).In view of (6.8) this shows that the series converges locally uniformly in D 2 .Hence F ∈ H (D 2 ) and, since every function P µ n is an eigenfunction of ∆ zw , we have , where D is a rotationally invariant domain such that D 2 ⊊ D ⊆ Ω and (z 0 , w 0 ) ∈ D \ D 2 .By Theorem 5.1 there are coefficients ( bn for all (z, w) ∈ D. Since the coefficients are uniquely determined, we conclude bn = b n .This is a contradiction because

Corollary 6.4
Let λ ∈ C and D a rotationally invariant subdomain of Ω such that D 2 ⊊ D. Then there exists a function f ∈ X λ (D 2 ) that cannot be analytically continued to D.

POISSON FOURIER MODES AND THE M ÖBIUS GROUP
Recall the Möbius group M (D 2 ) of the bidisk from (2.5).In order to give a characterization of the Möbius invariant eigenspaces of ∆ zw (e.g.proving Theorem 2.8), it will turn out that all we need to understand are precompositions of PFM with automorphisms in M (D 2 ), that is Remark 9. Lemma 7.1 is exactly Lemma 2.2 in [19] where we have replaced the automorphisms Φ z,w by the T z,w = Φ z,w • ρ −1 automorphisms in (7.1).The reason will become apparent in (7.3) and (7.4).Essentially, the automorphisms T z,w in (7.1) are precisely the automorphisms of Ω interchanging a given point (z, w) ∈ Ω ∩ (C × C) with (0, 0) -instead of only sending (0, 0) to (z, w).See [32,Sec. 2.3] for more details on this.
Therefore, understanding precompositions of a PFM P µ n with elements T ∈ M (D 2 ) breaks down to understanding precompositions with the above generators.Note that the mappings F and T z,w for w ̸ = z are not elements of M (D 2 ).However, the precompositions with these mappings still make sense for those PFM defined on all of Ω, that is, in view of Theorem 4.2 the precompositions P −m n • T with m ∈ N 0 and T ∈ M are well-defined.In the case that T = ρ γ we have P 3) and (7.4) then take the form Proof.The way of reasoning is as follows: first, fix z, u ∈ D and let v := u.Then ψ u,u ∈ M (D) which means, in particular, that T u,u (z, z) ∈ {(t,t) : t ∈ D}.By the two variable identity principle, Lemma 3.1, if we can show (7.5) in this special case, then, keeping u ∈ D fixed, (7.5) also holds for points (z, w) ∈ Ω since both sides of the equation are holomorphic functions in Ω. Next, fix (z, w) ∈ Ω and note that both sides of (7.5) are holomorphic as functions of (u, v) ∈ Ω ∩ C 2 .Assuming (7.5) for v = u ∈ D then implies the claim, again by Lemma 3.1.It remains to show (7.5) for w = z and v = u with z, u ∈ D. For this purpose we compute 2π 0 P z, z; e is −m P u, u; e is m+1 ds .
In the last step we used (7.3).Now we can interpret the resulting integral as the inner product on L 2 ([0, 2π], C), the space of square integrable functions f : [0, 2π] → C. Using Parseval's identity in (P) we obtain The previous proof can be modified to establish the following result.
for all (z, w) ∈ D 2 .Moreover, both series converge locally uniformly and absolutely w.r.t.(z, w) in D 2 .
Proof.Using (7.3) and (7.4) we compute Note that taking complex powers is unproblematic since the appearing Poisson kernels are positive real quantities.Applying Lemma 3.1 and, additionally, the generalized binomial theorem leads to 2π 0 P z, w; e is µ P u, u; e is 1−µ e is(k−n−ℓ) ds .
Interpreting the above integral as a L 2 ([0, 2π], C) inner product as it was done in the proof of Proposition 7.2 shows (7.6a).A similar computation yields (7.6b).
We have as u ∈ D is fixed.Since K is compact, we find a non-negative real number r < 1 such that |z|, |w| ≤ r for all (z, w) ∈ K. Hence, using (6.8) we can ensure that where γ r > 0 depends on r (and µ).Thus, since the binomial series over k and ℓ converge absolutely.□ Let m, n ∈ N 0 , n ≤ m.For µ = −m the j-series in (7.6) terminate.Moreover in this case, when replacing (u, u) by (u, v) ∈ D 2 in (7.6), a similar argument as in the proof of Theorem 7.3 shows that the series in (7.6) are absolutely and locally uniformly convergent with respect to (u, v) in D 2 , too.Further, we know that P −m ±n • T u,v ∈ H (Ω). Thus, both sides of (7.6a) resp.(7.6b) (with (u, u) replaced by (u, v)) define holomorphic functions w.r.t.(u, v) ∈ D 2 which allows us to apply the two variable identity principle, Lemma 3.1.We obtain: Lemma 8.2 Let D ⊆ C 2 be a subdomain, f ∈ H (D). For all (z, w) ∈ D it then holds that What makes these operators useful is that they act by shifts of the Fourier index on the PFM.which is the Euler vector field corresponding to the homogeneity (4.1), they generate the Lie algebra of M .By (8.3), all three operators may moreover be regarded as fundamental vector fields of the natural action of M as automorphisms of Ω.
Proof of Theorem 8.1.Write λ = 4µ(µ − 1) for µ ∈ C, Re µ ≥ 1/2, and let Y be a non-trivial Möbius invariant subspace of X λ (D 2 ).The way of reasoning is as follows: we first show that Y contains a PFM P µ n for some n ∈ Z.If the assumption of (NE) holds, then this will imply that P µ n ∈ Y for all n ∈ Z. Conversely, if the assumption of (E) holds for m ∈ N 0 , the existence of P −m n for some n ∈ Z will imply that certain other PFM need to be contained in Y , too.Here, we will need to distinguish three cases which will lead to the three specific spaces described on the right-hand side of (8.1).This shows that there are at most three possible choices for Y .Then, in a second step, we show that the three spaces found in the first step are in fact Möbius invariant, which then proves the existence of exactly three non-trivial proper Möbius invariant subspaces.
Step 1: (i) Using (8.(ii) Since Y is Möbius invariant, Y is, in particular, rotationally invariant.Thus, for every n ∈ Z, we have f n ∈ Y where f n is defined by By Theorem 4.2, f n is a multiple of the PFM P , q < −m or q > m, does not necessarily need to be contained in Y .However, if P m+1 q ∈ Y for some q < −m, then necessarily all P m+1 p , −∞ < p ≤ m, need to be contained in Y since P m+1 q can be shifted to each of these function without producing the zero function.The same argument works for q > m.In summary, these considerations show that every non-trivial closed Möbius invariant subspace of X λ (D 2 ) contains one of the three spaces  is invertible.However, it is true in general that given linearly independent complex valued functions F 1 , . . ., F M for some M ∈ N on a set containing at least M elements, it is possible to find the same number of points x 1 , . . ., x M in the same set such that the matrix (F j (x k )) 1≤ j,k≤M is invertible, see [15,Proof of Prop. 7.28].The second equality in (8.5) follows similarly.□

POISSON FOURIER MODES AND PESCHL-MINDA OPERATORS
In [19] the classical Peschl-Minda differential operators, which were introduced by Peschl [34] and studied e.g. by [2,17,24,25,31,39,40,42], have been extended to differential operators acting on holomorphic functions defined on subdomains of Ω ∩ C 2 .It is the purpose of this section to put these (generalized) Peschl-Minda operators into the context of the present paper.In particular, we relate the Poisson Fourier modes with the Peschl-Minda operators.
We first briefly recall the definition from [19].Let U be an open subset of Ω ∩ C 2 and f ∈ C ∞ (U).The Peschl-Minda derivative D m,n f at the point (z, w) ∈ U is defined by . We write D n z := D n,0 and D n w := D 0,n and refer to these operators as pure Peschl-Minda operators.Comparing with (8.3), we note that the roles of (z, w) and (u, v) have been swapped, which yields the operators D 1 z and D 1 w instead of D + and D − , respectively.
Given a linear mapping T : V −→ W we write ker(T ) := {x ∈ V : T x = 0} for its kernel.We observe that the pure Peschl-Minda operators reproduce the generalized Poisson kernel from (4.9) in the following way: We note in passing that Lemma 9.1 for w = z says that up to multiplication with a unimodular constant the (classical) Poisson kernel z → P(z, z; ξ ) is a joint eigenfunction of the (classical) Peschl-Minda operators studied e.g. in [24].Remarkably, the Peschl-Minda operators also act as weighted shifts when applied to the zeroth PFM.

Theorem 2 . 5 (
Exceptional smooth eigenfunctions ∆ D on D vs. holomorphic eigenfunctions of ∆ zw on Ω vs. smooth eigenfunctions of ∆ C on C) Let λ ∈ C. Then the following conditions are pairwise equivalent:

Figure 1 FIGURE 1 .
Figure1provides a schematic view of Ω and its distinguished subsets.

3 .
NOTATION AND PRELIMINARIES We denote the open unit disk in C by D, the bidisk D 2 := D × D and the Riemann sphere by C. The open disk of radius r > 0 centered at the origin is denoted by D r .Moreover, we write N := {1, 2, . ..} for the set of positive integers, N 0 := N ∪ {0} and Z for the set of all integers.
3.3]) By a standard fact about hypergeometric functions ([1, 15.3.1]) the function y n has a holomorphic extension at least to the slit plane C \ [1, ∞) .

Remark 5 .
If w = z ∈ D, then P µ n (z, z) is the n-th Fourier coefficient of the µ-power of the (realvalued and, in fact, non-negative) Poisson kernel of the unit disk D. Further, if µ = m ∈ Z, then P m n (z, w) also is the n-th Fourier coefficient of the m-power of the generalized Poisson kernel from (4.9).

µn
has a holomorphic extension from D 2 to Ω * , which we continue to denote by P µ n .In analogy with (4.1) we call a subdomain D ⊆ Ω rotationally invariant if (ηz, w/η) ∈ D for all (z, w) ∈ D and all η ∈ ∂ D. Summarizing our considerations leads to the following complete description of the n-homogeneous eigenfunctions of ∆ zw : Theorem 4.2 Let D be a rotationally invariant subdomain of Ω containing the origin and n

Proof.
It remains to prove that for any nonconstant n-homogeneous eigenfunction F µ n of ∆ zw in H (Ω * ) the dichotomy "(NE) vs. (E)" holds.(i)Let µ ̸ ∈ N and assume that Ω * is not the maximal domain of existence of F µ n which is contained in Ω, see Definition 2.4.Then F µ n has a holomorphic extension to some point (z 0 , w 0 ) ∈ Ω such that z 0 w 0 ∈ R ∪ {∞} and z 0 w 0 > 1.We first consider the case n ≥ 0. By definition of F µ n , see (4.7), and in view of[1, 15.3.4],we have(4.12) is a polynomial in zw of degree µ − n − 1 ≥ 0, see[1, 15.3.3 and 15.1.1].Hence F µ n is the product of z n and a rational function in zw of numerator degree µ − n − 1 and of denominator degree µ − 1 with pole only at the point 1.Therefore,

µnCorollary 4 . 3
(z, w 0 )| → ∞ as z → ∞.This implies that Ω − is the maximal domain of existence of F µ n .□For every λ ∈ C and each n ∈ Z there is an n-homogeneous holomorphic solution of ∆ zw f = λ f on the domain Ω * .More precisely, if λ = 4µ(µ − 1) ∈ C with Re µ ≥ 1/2, then every such solution has the form cP µ −n for some c ∈ C.

Remark 6 (
Elementary properties of Poisson Fourier modes).Let µ ∈ C, n ∈ N 0 and z, w ∈ D. (a) The Poisson Fourier modes are related to the hypergeometric function 2 F 1 via

( 1 −Remark 7 (
zw) m .(c)The PFM are symmetric in the sense that P µ n (z, w) = P µ −n (w, z).This allows us to simplify our proofs in the following: we will often prove identities for P µ n only which then implies the corresponding result for P µ Invariant representative functions and the finite dimensional invariant eigenspaces).We consider D as a symmetric space M (D)/∂ D over its automorphism group M (D) with ∂ D ∼ = U(1) acting by rotations.This yields an alternative way of deriving the restrictions to D of those Poisson Fourier modes from (4.17), which are defined on all of Ω, using finite dimensional representation theory.Such considerations are very much in the spirit of[21].The group M (D) is isomorphic to the projective split unitary group PSU(1, 1) = a b b a : |a| 2 − |b| 2 = 1 ± Lie group.By Schur's Lemma every irreducible representation π : M (D) −→ GL n (C) as invertible (n × n)-matrices induces eigenfunctions of the Laplacian on PSU(1, 1) via (4.19)

Corollary 5. 2 .
Let D be a subdomain of Ω containing the origin, and let f ∈ H (D) be such that ∆ zw f = λ f for some λ ∈ C. (i) If D = Ω, then λ = 4m(m + 1) for some m ∈ N 0 and there are uniquely determined coefficients c n ∈ C such that f = If D = Ω + , then λ = 4m(m + 1) for some m ∈ N 0 and there are uniquely determined coefficients c n ∈ C such that f = If D = Ω − , then λ = 4m(m + 1) for some m ∈ N 0 and there are uniquely determined coefficients c n ∈ C such that f = Proof.(i) If f ∈ H (Ω), then the functions f n (z, w) n+1) dη are holomorphic and n-homogeneous on Ω and f n (z, w) = c n P µ −n (z, w) with µ ∈ C such that λ = 4µ(µ − 1) and Re µ ≥ 1/2.Hence, λ = 4m(m + 1) for some m ∈ N 0 since otherwise P µ −n is not holomorphic on Ω by Theorem 4.2.If λ = 4m(m + 1), then for each |n| > m the function P m+1 −n is not holomorphic on Ω again by Theorem 4.2 which forces c n = 0 for those n.Parts (ii) and (iii) follow in the same way.

µn
• T = γ −n P µ n by the (−n)-homogeneity of the PFM for all µ ∈ C. Next, if T = F , it is easily seen by direct verification based on (4.17) that (P −m n • F )(z, w) = (−1) m P −m n (z, w) for m ∈ N 0 and |n| ≤ m.In order to treat the case that T = T z,w resp.T = T z,z , some preliminary observations are useful.First, every automorphism T z,w is induced by a Möbius transformation ψ z,w of the form(7.2) ψ z,w (u) := z − u 1 − wu , in the sense that T z,w (u, v) = (ψ z,w (u), 1/ψ z,w (1/v)).Remark 10.Choosing w = z in (7.2) yields all self-inverse automorphisms of D except for the identity, and, similarly, the choice w = −z ∈ D yields all self-inverse rigid motions of C.Further, we will employ the definition of the PFM via integrals from (4.10).By Remark 5, when considering points (z, z) ∈ D 2 , this definition coincides with the n-th Fourier mode of the µ-power of the Poisson kernel on the unit disk.The Poisson kernel on D satisfies the wellknown properties(7.3)P (z, z; ξ ) = ψ ′ z,z (ξ ) ψ z,z(ξ ) ξ and (7.4) P ψ z,z (u), ψ z,z (u); ψ z,z (ξ ) = P(u, u; ξ ) P(z, z; ξ ) where z, u ∈ D, ξ ∈ ∂ D and ψ z,z ∈ M (D), see [38, Th. 3.3.5].Remark 11.One can define P on Ω×∂ D. Then P is never zero, and for each ξ ∈ ∂ D the function P(•; ξ ) is meromorphic in the sense of [13, Chapter VI.2].The identities (7.

Lemma 8. 3 2 D
The operators D + and D − commute with ∆ zw .Furthermore, for µ ∈ C and n ∈ Z it holds that D + P µ n = (µ − n − 1)P µ n+1 and D − P µ n = (µ + n + 1)P µ n−1 .Proof.The operator D + commutes with ∆ zw in view of the latter's M -invariance, (8.3) and commutativity of partial derivatives.Moreover, D + lowers homogeneity in the sense of (4.1) by one degree.Since P µ n is a (−n)-homogeneous eigenfunction of ∆ zw to the eigenvalue 4µ(µ − 1), this implies that D + P µ n is a −(n + 1)-homogeneous eigenfunction of ∆ zw to the same eigenvalue.Hence, Theorem 4.2 yields D + P µ n = cP µ n+1 for some c ∈ C. Applying (8.2) and evaluating (4.16) in (z, w) = (0, 1) yields c = µ − n − 1.Note that one has to distinguish the cases n ≥ 0 and n < 0. The second equality in (8.3) may be derived analogously.□ Remark 12.One may thus regard the operators D + and D − as ladder operators and together with their commutator 1 − • D + − D + • D − = z∂ z − w∂ w , 2) it follows from Y being closed and Möbius invariant that the differential quotients D + f , D − f ∈ Y .Therefore, Y is invariant with respect to D + and D − , and, by iteration, (D + ) k f , (D − ) k f ∈ Y for every k ∈ N.

µn
. Moreover, by Theorem 5.1, there are c n ∈ C for all n ∈ Z depending on f such that f has the representation Since Y ̸ = {0}, we may assume f ̸ ≡ 0, i.e. c n ̸ = 0 for some n ∈ Z.Thus, the above observations imply P µ n ∈ Y for some n ∈ Z. (iii) By Part (ii) there exists P µ n ∈ Y for some n ∈ Z, and by Part (i) we conclude (D + ) k P µ n , (D − ) k P µ n ∈ Y for every k ∈ N. Lemma 8.3 shows that these functions are multiples of PFM again, so (µ − n − k) k P µ n+k , (µ + n − k) k P µ n−k ∈ Y for every k ∈ N. Recall that the prefactors denote (rising) Pochhammer symbols, see (4.5).For µ ̸ ∈ Z the factors (µ ± n − k) k never vanish.In this case, P µ n ∈ Y for every n ∈ Z.Using Theorem 5.1 again, this implies Y = X λ (D 2 ) which proves (NE).For µ ∈ N the factors (µ ± n − k) k vanish for appropriate k ∈ N.This leads to the dichotomy in case (E): assume µ = m + 1 ∈ N, and P m+1 n ∈ Y .Then necessarily P m+1 k ∈ Y for −m ≤ k ≤ m.Since shifting P m+1 k with D ± eventually produces the zero function, P m+1 q
[20,h are known as representative functions or matrix elements.For a systematic discussion we refer to the textbook[9, Chapter III].The corresponding eigenvalues may be computed from representation theoretic data, see e.g.[16, Prop.10.6].Note that in the literature, many results are formulated for general invariant differential operators or the corresponding Casimir elements and joint eigenfunctions thereof instead of the Laplacian, which is always invariant and thus constitutes a special case.However, the disk D is a two-point homogeneous space, so all M (D)-invariant differential operators are polynomials in the Laplacian, see[20, Theorem 11].