Let \(}}^d\) and \(}}^d\) denote the hyperbolic space and the unit sphere, respectively, both d-dimensional. Let \(\Delta _d\), \(\Delta _d^\textrm\) and \(\Delta _d^\textrm\) denote the Laplace(–Beltrami) operators on \(}}^d\), \(}}^d\) and \(}}^d\), respectively. It is convenient to shift the hyperbolic Laplacian by \(-\frac\) and the spherical Laplacian by \(\frac\). Our paper is devoted to the operators

$$\begin \begin H_d&-\Delta _d, \\ H_d^\text &-\Delta _d^\text -\frac, \\ H_d^\text &-\Delta _d^\text +\frac, \end \end$$

(1.1)

possibly perturbed by a point potential.

The operators \(H_d\), \(H_d^\textrm\) and \(H_d^\textrm\) can be viewed as self-adjoint operators on \(L^2(}}^d)\), \(L^2(}}^d)\) and \(L^2(}}^d)\), respectively. For \(z\in }}\) outside of the spectrum of \(H_d\), \(H_d^\textrm\) and \(H_d^\textrm\), one can define their resolvent (Green’s operator)

$$\begin \begin G_d(z)&(-z+H_d)^, \\ G_d^\text (z)&(-z+H_d^\text )^, \\ G_d^\text (z)&(-z+H_d^\text )^. \end \end$$

(1.2)

The spectrum of \(H_d\) and \(H_d^\textrm\) is continuous and coincides with \([0,\infty [\). The spectrum of \(H_d^\textrm\) is discrete and equals \(\left\\right) ^2 \ |\ l=0,1,\dots \right\} \subset [0,\infty [\). Therefore, it is often convenient to represent the spectral parameter \(z\in }}\backslash [0,\infty [\) as \(z=-\beta ^2\) with \(\Re \beta >0\), so that

$$\begin G_d(-\beta ^2)&= (\beta ^2+H_d)^,\nonumber \\ G_d^\text (-\beta ^2)&= (\beta ^2+H_d^\text )^,\nonumber \\G_d^\text (-\beta ^2)&= (\beta ^2+H_d^\text )^. \end$$

(1.3)

Sometimes we will also write \(\zeta ^2\) for z.

For \(0\le a<b\), one can define the spectral projections onto [a, b]:

(1.4)

We can also introduce the spectral projections onto eigenvalues of \(H_d^\textrm\):

(1.5)

The integral kernels of the resolvents (1.3), denoted by \(G_d(-\beta ^2;x,x')\), \(G_d^}(-\beta ^2;x,x')\) and \(G_d^}(-\beta ^2;x,x')\), are often called Green’s functions. The integral kernels of the spectral projections (1.4) are denoted \(}}_d(a,b;x,x')\), \(}}_d^\textrm(a,b;x,x')\), \(}}_d^\textrm(a,b;x,x')\). The integral kernel of (1.5) is denoted \(}}_^\textrm(x,x')\). Explicit formulas for these in terms of special functions are known, and for convenience of the reader, we provide them in our paper.

The integral kernels related to \(H_d\) are expressed in terms of functions from the Bessel family. The integral kernels related to \(H_d^\textrm\) and \(H_d^\textrm\) are expressed in terms of Gegenbauer functions. Here are, for instance, the formulas for Green’s functions:

$$\begin G_d\big (-\beta ^2;x,x'\big )&=\frac}}\Big (\frac\Big )^-1}K_-1} \big (\beta r\big ), \end$$

(1.6)

$$\begin G_^\textrm\big (-\beta ^2;x,x'\big )&=\frac\Gamma (\frac+\beta )}(2\pi )^}2^ } \textbf_-1,\beta } \big (\cosh (r)\big ), \end$$

(1.7)

$$\begin G_^\textrm\big (-\beta ^2;x,x'\big )&=\frac+\textrm\beta ) \Gamma (\frac-\textrm\beta ) }} } \textbf_-1,\textrm\beta } \big (-\cos (r)\big ). \end$$

(1.8)

Above, r denotes the Euclidean, hyperbolic and spherical distance between x and \(x'\), respectively. \(K_\alpha \) is the Macdonald function (one of the functions from the Bessel family). \(\textbf_\) and \(\textbf_\) are two kinds of Gegenbauer functions, see Appendix C.

One should note that Bessel and Gegenbauer functions have special properties when their parameter \(\alpha \) is half-integer or integer. For half-integer \(\alpha \), Bessel and Gegenbauer functions can be expressed as elementary functions. For integer \(\alpha \), Bessel and Gegenbauer functions have a logarithmic singularity. From the point of view of Green’s operators, these values are important: half-integer \(\alpha \) is used in odd dimensions and integer \(\alpha \) in even dimensions.

All Green’s functions (1.6), (1.7) and (1.8) behave similarly for \(x,x'\) close to one another, which follows from well-known expansions of \(K_\alpha \), \(\textbf_\) and \(\textbf_\). However, for large distances they are rather different. This can be seen by comparing the expansions of (1.6), (1.7) and (1.8) for large distances, which we describe in (2.19), (3.14) and (4.13).

The main goal of this paper is to extend the above theory to the operators \(H_d\), \(H_d^\textrm\) and \(H_d^\textrm\) perturbed by a point potential (also called a contact or delta potential). It is a well-known fact that the one-dimensional Laplacian can be perturbed by a delta potential in the form sense [28]. In dimensions 2 and 3, the Laplacian can also be perturbed by a point-like perturbation; however, one cannot use the naive form formalism anymore [2,3,4]. Thus, in dimensions \(d=1,2,3\) we obtain one-parameter families of self-adjoint operators \(H_d^\gamma \), \(H_d^,\gamma }\) and \(H_d^,\gamma }\). We denote their resolvents by \(G_d^\gamma (z)\), \(G_d^,\gamma }(z)\) and \(G_d^,\gamma }(z)\). Their integral kernels have the form

$$\begin G_d^\gamma ( z ;x,x')&= G_d( z ;x,x')+ \frac, \end$$

(1.9)

$$\begin G_d^,\gamma }( z;x,x')&= G_d^\textrm( z;x,x')+ \frac( z;x,x_0)G_d^\textrm( z;x_0,x')}(z)}, \end$$

(1.10)

$$\begin G_d^,\gamma }( z;x,x')&= G_d^\textrm( z;x,x')+ \frac( z;x,x_0)G_d^\textrm( z;x_0,x')}(z)}, \end$$

(1.11)

where \(x_0\) is the position of the point potential (e.g., the origin of coordinates of \(}}^d\) or the north pole of \(}}^d\)). Here, the functions \(\Sigma _d\), \(\Sigma _d^\textrm\) and \(\Sigma _d^\textrm\) satisfy

$$\begin \partial _z\Sigma _d(z)=&-\int _}}^d}G_d(z;x_0,x)^2\textrmx, \end$$

(1.12)

$$\begin \partial _z\Sigma _d^\textrm(z)=&-\int _}}^d}G_d^\textrm(z;x_0,x)^2\textrmx, \end$$

(1.13)

$$\begin \partial _z\Sigma _d^\textrm(z)=&-\int _}}^d}G_d^\textrm(z;x_0,x)^2\textrmx. \end$$

(1.14)

The parameter \(\gamma \in }}\cup \\) is a real integration constant and describes the strength of the perturbation. The function \(\gamma +\Sigma _d^\bullet (z)\), where \(\bullet \) is empty, \(\textrm\) or \(\textrm\), will be called the full self-energy. \(\Sigma _d^\bullet (z)\) is the reference self-energy, fixed by imposing some additional conditions.

In dimensions 1 and 3, there exists a natural condition that allows us to fix the reference self-energy: \(\lim \limits _\Sigma _1^\bullet (z)=0\) and \(\lim \limits _(\Sigma _3^\bullet (z)-\frac})=0\).

For \(d=2\), one possible choice for the reference self-energy is to demand \(\Sigma _2^\bullet (-\beta ^2)\sim \frac\) for \(\beta \rightarrow \infty \), or equivalently, \(\Sigma _2^\bullet (-1)=0\). This, however, distinguishes a certain length scale corresponding to \(\beta =1\). In order to avoid such an a priori unphysical distinction, we treat all possible full self-energies on an equal footing as members of a family of reference self-energies parametrized by a real parameter \(\varepsilon =-2\pi \gamma \):

$$\begin \gamma +\Sigma _2^(z) =:\Sigma _2^(z). \end$$

(1.15)

\(\gamma \) (and \(\varepsilon \) in \(d=2\)) are closely related to the so-called scattering length a used in the physical literature. Here are the relations between these two parameters:

$$\begin d=1,&\qquad a=-2\gamma ; \end$$

(1.16)

$$\begin d=2,&\qquad a=\textrm^=\textrm^ ; \end$$

(1.17)

$$\begin d=3,&\qquad a=-\frac.\end$$

(1.18)

It is well known that the Laplacian is essentially self-adjoint on \(C_\textrm^\infty (}}^d\backslash \)\) in dimensions \(d\ge 4\) [28]. In other words, there are no point-like perturbations of the Laplacian in dimensions \(d\ge 4\), if we stick to the usual Hilbert space setting. This corresponds to the divergence of the integrals in (1.12), (1.13) and (1.14) defining the self-energies.

The description of Green’s functions for the Laplacians with a point potential in dimensions \(d\ge 4\) is probably the main novelty of our paper. Our starting points are Eqs. (1.9), (1.10) and (1.11). Hence, we need to give meaning to divergent self-energies. We will consider two different but consistent methods to do this. The first will be called the point-splitting method and the second the minimal subtraction method.

In the first method, we start with replacing the integrals (1.12), (1.13), (1.13) by their “point-split versions,” which are then repeatedly differentiated in z (the “energy”) until convergent integrals are obtained. Then we repeatedly integrate them to get the self-energy. Integration constants from multiple integrations can be gathered in a polynomial \(\gamma (z)\), which replaces the integration constant \(\gamma \) used in lower dimensions. \(\gamma (z)\) is a polynomial of degree \(\le n=\big \lfloor \frac\big \rfloor \), i.e., \(n=\frac\) if d is odd and degree \(n=\frac\) if d is even.

The second approach to define self-energies is to replace (1.12), (1.13), (1.14) with the corresponding generalized integrals. Then the self-energies \(\Sigma _d^\bullet (z)\) are well defined in all dimensions up to only one integration constant.

As we explain in Appendix A, the generalized integral is a natural extension of the classical integration to a certain class of not necessarily integrable functions. It resembles the minimal subtraction scheme in QFT. Clearly, it is only one of many linear extensions of the integration functional. Other extensions differ by an additional polynomial of degree \(\le \big \lfloor \frac\big \rfloor \), whose parameters can be viewed as arbitrary “renormalization constants.” Thus, both approaches to defining self-energies agree.

A generalized integral is said to have a scaling anomaly if it transforms inhomogeneously upon a rescaling of the integration variable. There is a big difference between non-anomalous and anomalous generalized integrals. In the non-anomalous case, the computation of a generalized integral essentially reduces to the analytic continuation of the usual integral in a certain parameter, which often (in particular, in our case) can be interpreted as the dimension. In the anomalous case, in addition to analytic continuation one has to perform an appropriate subtraction.

One could ask whether it is natural to fix a certain full self-energy, and to call it the reference self-energy. We would like our reference self-energies to be algebraically as simple as possible; in particular, they should be factorized in simple factors.

The generalized integral suggests a certain expression, which we denote \(\Sigma _d^}\), (where we fix a single integration constant in some natural way and \(\textrm\) stands for “minimal subtraction”). In odd dimensions \(d\ge 5\), there is an obvious choice of reference self-energy which is given by an algebraically simple expression. This reference self-energy is equal to \(\Sigma _d^}\) and can also be obtained by formally extending \(\Sigma _d^\bullet (z)\) to complex d in the region \(|\Re d-2|<2\) (\(d \ne 2\)), and then by using analytic continuation.

In even dimensions \(d\ge 4\), we are in the anomalous case, which is much more complicated. The generalized integrals on the right-hand side of (1.12), (1.13) and (1.14) involve non-elementary functions: the logarithm or the digamma function \(\psi (z):\,=\,\frac\).

The anomalous generalized integral is not invariant under a change of variables. In the Euclidean case, the natural variable is r, the distance from the origin in some fixed units. Since the generalized integral is invariant under a change of variable \(r\rightarrow r^\alpha \) for any \(\alpha >0\), one can equivalently use the coordinate \(r^2\).

In the hyperbolic and spherical cases, the variables r (or \(r^2\)), now denoting the hyperbolic and spherical distance, respectively, seem not convenient to compute self-energies. Instead, in [10], to this end we used the variables \(w=2(\cosh (r)-1)\) in the hyperbolic case and \(w=2(1-\cos (r))\) in the spherical case. These variables are convenient in calculations involving resolvents of the Laplacian, and they seem to be a natural choice. Note that \(w=r^2 + \mathcal (r^4)\) is a function of \(r^2\) in both cases. Anyway, if we change the variable in the generalized integral according to (A.8), the resulting change in the self-energy is a polynomial of degree \(\le \big \lfloor \frac\big \rfloor \), which is consistent with the ambiguity in the point-splitting approach.Footnote 1

Thus, for even \(d\ge 4\), selecting in some way the integration constant, we can introduce the self-energy given by the generalized integral \(\Sigma _d^}\). All self-energies are given by \(\gamma (z)+\Sigma _d^}(z)\), where \(\gamma \) is of degree \(\le \frac\). In the hyperbolic and spherical cases, \(\Sigma _d^}\) is rather complicated and has no obvious factorization. There exists, however, a one-parameter family of factorized expressions \(\Sigma _d^(z)\), \(\varepsilon \in }}\), which one can use as reference self-energies. We absorb the highest term of the polynomial \(\gamma \) in \(\varepsilon \), so that now the remaining freedom consists of a polynomial \(\eta (z)\) of degree only \(\le \frac\). Thus, the general form of a full self-energy in even dimensions is now given by \(\eta (z)+\Sigma _d^(z)\).

Summarizing, for odd d we obtained the families of functions

$$\begin G_d^\gamma ( z ;x,x')&= G_d( z ;x,x')+ \frac, \end$$

(1.19)

$$\begin G_d^,\gamma }( z;x,x')&= G_d^\textrm( z;x,x')+ \frac( z;x,x_0)G_d^\textrm( z;x_0,x')}(z)}, \end$$

(1.20)

$$\begin G_d^,\gamma }( z;x,x')&= G_d^\textrm( z;x,x')+ \frac( z;x,x_0)G_d^\textrm( z;x_0,x')}(z)}, \end$$

(1.21)

parametrized by an arbitrary polynomial \(\gamma \) of degree \(\le \frac\).

For even d, we need to slightly modify (1.19), (1.20) and (1.21): We replace the superscript \(\gamma \) with \(\varepsilon ,\eta \) and \(\gamma (z)+\Sigma _d^\bullet (z)\) with \(\eta (z)+\Sigma _d^(z)\). Here \(\varepsilon \) is a real number and \(\eta \) is an arbitrary polynomial of degree \(\le \frac\).

In what follows, abusing the notation, we will sometimes write \(\gamma \) for a pair \(\varepsilon ,\eta \). \(G_d^(z)\) will be called Green’s functions. For \(d\ge 4\), they are not integral kernels of bounded operators. Hence, for such d, they are not resolvents of well-defined self-adjoint operators.

Here is the list of the reference self-energies in various dimensions:

$$\begin d=1:\qquad & \Sigma _1(-\beta ^2) =-\frac,\nonumber \\ & \Sigma _1^\textrm(-\beta ^2) =-\frac,\nonumber \\ & \Sigma _1^\textrm(-\beta ^2)=-\frac; \end$$

(1.22a)

$$\begin d=2:\qquad & \Sigma _2^\varepsilon (-\beta ^2 )=\frac\big (\ln \beta -\varepsilon \big ),\nonumber \\ & \Sigma _2^,\varepsilon }(-\beta ^2) =\frac\Big (\psi (\tfrac+\beta )-\varepsilon \Big ),\nonumber \\ & \Sigma _2^,\varepsilon }(-\beta ^2)=\frac\Big (\psi (\tfrac+\textrm\beta )+\psi (\tfrac-\textrm\beta )-2\varepsilon \Big ); \end$$

(1.22b)

$$\begin d=3:\qquad & \Sigma _3(-\beta ^2 )=\frac,\nonumber \\ & \Sigma _3^\textrm(-\beta ^2) =\frac,\nonumber \\ & \Sigma _3^\textrm(-\beta ^2)=\frac; \end$$

(1.22c)

$$\begin \text d\ge 4:\qquad & \Sigma _d^\varepsilon (-\beta ^2) = \frac} \Gamma \big (\tfrac\big )} \big ( \ln (\beta ^2)-2\varepsilon \big ) (-\beta ^2)^} ,\nonumber \\ & \Sigma _d^,\varepsilon }(-\beta ^2)= \frac+\beta \big ) +\psi \big (\tfrac+\beta \big ) -2\varepsilon }} \Gamma \big (\tfrac\big )} \prod _^} \Big (-\beta ^2+\big (\tfrac + j\big )^2\Big ),\nonumber \\ & \Sigma _d^,\varepsilon }(-\beta ^2)= \frac+\textrm\beta \big )+\psi \big (\tfrac-\textrm\beta \big )-2\varepsilon }}\Gamma (\frac)} \prod _^}\big (-\beta ^2-(\tfrac+j)^2\big ) ;\nonumber \\ \end$$

(1.22d)

$$\begin \text d\ge 5:\qquad & \Sigma _d(-\beta ^2)=\frac}\Gamma (\frac)}\beta (-\beta ^2)^},\nonumber \\ & \Sigma _d^\textrm(-\beta ^2) = \frac} \, \Gamma \big (\tfrac\big )} \beta \prod _^} \big ( -\beta ^2 +k^2 \big ),\nonumber \\ & \Sigma _^\textrm(-\beta ^2) = \frac}\Gamma \big (\tfrac\big )} \beta \prod _^} \big ( -\beta ^2 -k^2 \big ).\end$$

(1.22e)

Of course, some items of the above list are well known. The self-energy in the Euclidean case for \(d=1,2,3\) belongs to standard knowledge of contemporary quantum physics. The Euclidean self-energy for \(d\ge 4\) obtained with help of the generalized integral is partially covered in the literature, see, e.g., [21] for odd dimensions. The self-energies for the hyperbolic and spherical Laplacian appear to be new.

As we stressed above, for \(d\ge 4\), the functions (1.9), (1.10) and (1.11) do not define bounded operators and their inverses do not define self-adjoint operators. It is natural to ask what is their meaning.

One approach that can be found in the literature is to extend the Hilbert space, typically, to a Pontryagin space (with an indefinite metric product). This approach is described, e.g., in [20].

One can also consider a different interpretation. Fix a point \(x_0\) in \(}}^d\), \(}}^d\) or \(}}^d\). Suppose that \(H_d^\bullet +V\) is a self-adjoint operator obtained by perturbing \(H_d\) in a ball around \(x_0\) of small radius r. We expect that far away from that ball, the integral kernel of \((H_d^\bullet +V-z)^\) is well approximated by \(G_^\) for some \(\gamma \) determined by V. Thus, coefficients of \(\gamma \) summarize universal long distance properties of V. We will discuss this idea further in a separate paper, which is in preparation.

The reference self-energies (corresponding to \(\gamma =0\) in odd dimensions and \(\eta =0\) in even dimensions) are in some sense distinguished—the poles of the corresponding Green’s functions can be easily computed. In the Euclidean case, they are also distinguished by their scaling property. (They are “fixed points of the renormalization group.”)

Our analysis of point interactions in dimensions \(d\ge 4\) resembles renormalization in quantum field theory. In QFT, especially in the Wilsonian approach, one does not worry too much whether the quantities computed by renormalization techniques correspond to a well-defined Hamiltonian. They should reproduce the “infrared behavior of correlation functions.” We apply a similar philosophy to Green’s functions. Note in particular that the borderline case when the perturbed Green’s functions do not correspond to self-adjoint operators is \(d=4\)—the physical dimension of our space-time. (Our space-time has a Lorentzian signature; however, using the Wick rotation it can often be replaced by the Euclidean \(}}^4\).)

Our analysis can be viewed as a toy model illustrating various aspects of renormalization in QFT. As explained above, we use two methods to define self-energies. The first applies the so-called point splitting and then regularization by differentiation in the energy. The second method, using generalized integrals, resembles the minimal subtraction method. To compute them, we use dimensional regularization. Both methods have their widely used counterparts in QFT.

Let \(R>0\) and let \(}}_R^d\), \(}}_R^d\) denote the rescaled hyperbolic space of curvature \(-\frac\) and the rescaled sphere of curvature \(\frac\) (that means, of radius R). Intuitively it is clear that in some sense \(}}_R^d\), \(}}_R^d\) converge to \(}}^d\) as \(R\rightarrow \infty \).

Green’s functions on the rescaled spaces are

$$\begin G_^\textrm(-\beta ^2;x,x')&=R^G_d^\textrm\Big (-(\beta R)^2,\frac,\frac\Big ), \end$$

(1.23)

$$\begin G_^\textrm(-\beta ^2;x,x')&=R^G_d^\textrm\Big (-(\beta R)^2,\frac,\frac\Big ). \end$$

(1.24)

We describe the convergence of these Green’s functions to the Euclidean ones \(G_d(-\beta ^2;x,x')\). This is of course well known, see, e.g., [5].

On the rescaled spaces, the reference self-energies are defined as follows:

$$\begin \Sigma _^\textrm(-\beta ^2)&: =R^ \Sigma _^\textrm\big (-(\beta R)^2\big ),&\text d, \nonumber \\ \Sigma _^,\varepsilon }(-\beta ^2)& R^\Sigma _^,\varepsilon +\ln R}\big (-(\beta R)^2\big ) ,&\text d, \nonumber \\ \Sigma _^\textrm(-\beta ^2)&: =R^ \Sigma _^\textrm\big (-(\beta R)^2\big ),&\text d, \nonumber \\ \Sigma _^,\varepsilon }(-\beta ^2)&R^\Sigma _^,\varepsilon +\ln R}\big (-(\beta R)^2\big ) ,&\text d. \end$$

(1.25)

Note that in even dimensions we need an additional additive renormalization, which can be traced back to rescaling of the variable in a generalized integral.

Using the above self-energies, we define the corresponding Green’s functions. We prove that they converge to the Euclidean Green’s function with a point potential and the same parameters. That is, in odd dimensions \(G_^\gamma }(-\beta ^2;x,x')\) and \(G_^,\gamma }(-\beta ^2;x,x')\) converge to \(G_d^\gamma (-\beta ^2;x,x')\), and in even dimensions, \(G_^,\varepsilon ,\eta }(-\beta ^2;x,x')\) and \(G_^,\varepsilon ,\eta }(-\beta ^2;x,x')\) converge to \(G_d^(-\beta ^2;x,x')\). This convergence is, perhaps, not very surprising. However, it requires a rather careful treatment of the self-energy (including the choice of renormalization), especially for even \(d\ge 4\), when there is the scaling anomaly.

In dimensions 1,2,3, the singularities of Green’s functions \(G_d^( z )\) and \(G_d^( z )\) are located at the spectrum \(H_d^( z )\) and \(H_d^( z )\). In the Euclidean and hyperbolic case, the continuous spectrum remains \([0,\infty [\), but the point potential may introduce an additional eigenvalue. In the spherical case, the point potential shifts the old eigenvalues and may introduce a new one. For example, in the Euclidean case we have the following new eigenvalues:

$$\begin H_1^\gamma :\quad&-\frac,\quad \text a<0, \end$$

(1.26)

$$\begin H_2^\varepsilon :\quad&-\frac, \end$$

(1.27)

$$\begin H_3^\gamma :\quad&-\frac,\quad \text a>0,\end$$

(1.28)

where we use the scattering length a (see Subsect. 2.2 for its relation to \(\gamma \) and \(\varepsilon \)).

For dimensions \(d\ge 4\), the point potential may introduce additional poles of Green’s functions located at z satisfying

$$\begin \gamma (z)+ \Sigma _d^\bullet (z)&=0, \qquad \text d; \end$$

(1.29)

$$\begin \eta (z)+ \Sigma _d^(z)&=0, \qquad \text d. \end$$

(1.30)

The interpretation of these singularities is less clear. We may call them eigenvalues of \(H_d^\), \(H_d^\), even though strictly speaking these Hamiltonians do not exist in the Hilbert space sense. For \(d\ge 4\), these poles may appear outside of the real line. (After all, they are not eigenvalues of a true self-adjoint operator.)

For pure reference self-energies, the additional singularities are easy to determine. If \(d\ge 3\) is odd, the singularities originating from (1.29) with \(\gamma =0\) are as follows:

$$\begin H_d^:\qquad&z=0; \nonumber \\ H_d^,0}:\qquad&z=-k^2,\quad k=0,1,\dots ,\frac;\nonumber \\ H_d^,0}:\qquad&z=\big (k+\tfrac)^2,\quad k\in }}_0. \end$$

(1.31)

If \(d\ge 2\) is even, the singularities originating from (1.30) with \(\eta =0\) are

$$\begin H_d^:\qquad&z=-\textrm^,\quad \text d\ge 4 \text z=0;\nonumber \\ H_d^,0}:\qquad&z \text \psi \big (\tfrac+\sqrt\big ) +\psi \big (\tfrac+\sqrt\big ) =2\varepsilon ,\nonumber \\&\text \quad z=-(k+\tfrac)^2,\quad k=0,1,\dots ,\frac;\nonumber \\ H_d^,0}:\qquad&z\text \psi \big (\tfrac+\textrm\sqrt\big )+\psi \big (\tfrac-\textrm\sqrt\big )=2\varepsilon \nonumber \\&\text \quad z=(k+\tfrac)^2,\quad k=0,1,\dots ,\frac. \end$$

(1.32)

What is especially interesting are eigenvalues (or poles of Green’s functions) in the spherical case inside \([0,\infty [\) for a general point potential, which we discuss in Subsect. 4.4. In the unperturbed case for the sphere of radius R, they are located at

$$\begin \frac)^2},\quad \text \frac,\quad l=0,1,\dots .\end$$

(1.33)

One effect of the perturbation is that the multiplicity of each of these eigenvalues is decreased by one (in particular \(\left( \frac \right) ^2\) is not an eigenvalue) and a shifted eigenvalue appears.

Let \(E_^\) be the lth shifted eigenvalue in the odd case. Below we give formulas for \(E_^\) in the generic case \(\gamma (z) \ne 0\). If \(\nu \) denotes the order of vanishing of \(\gamma (z)\) at \(z=0\), we find:

$$\begin E_^&= \frac)^2} + \frac)^2} + \mathcal \left( \frac \right) , \end$$

(1.34)

$$\begin E_^&=\frac)^2} -\frac)^2 \prod _^}\big ((l+\frac)^2-k^2\big )}}\Gamma (\frac)R^d \gamma \left( \frac)^2} \right) }\nonumber \\&\quad +\mathcal (R^),\quad \text d\ge 3 .\end$$

(1.35)

Note that \(d=1\) is special.

Now consider the lth shifted eigenvalue \(E_^\) in the even-dimensional case. We have

$$\begin \begin E_^&=\frac)^2}+\frac}^)} +\mathcal \Big (\frac^)}\Big ), \end\end$$

(1.36)

$$\begin \beginE_^&= \frac)^2} -\frac)\prod _^}\big ((l+\frac)^2-(j+\frac)^2\big )}}\Gamma (\frac)R^d \eta \left( \frac)^2} \right) } \\ &\quad + \mathcal \Big ( \ln (\text ^ R) R^\Big ),\qquad \text d\textrm\!\!4. \end \end$$

(1.37)

Note that for \(d\ge 3\) we have a systematic shift of the lth eigenvalue asymptotically proportional to \(\gamma (0)^\) or \(\eta (0)^\), respectively, and inversely proportional to the volume of \(}}^d\). The scaling of the shift with R is changed if \(\gamma (0) =0\) and \(\eta (0)=0\), respectively.

In particular, for \(d=3\) the \(l=0\) eigenvalue moves up by

$$\begin \approx -\frac}}^3|\gamma }=\frac}}^3|},\end$$

(1.38)

where \(|}}^3|\) is the volume of \(}}^3\). It is interesting to ask whether a similar formula is true for other compact manifolds.

Explicit formulas for the Euclidean, hyperbolic and spherical Green’s functions in any dimension are known in the literature, see, e.g., [5, 13]. In our presentation, we made an effort to describe various facets of these Green’s functions in a (hopefully) complete and transparent way. In particular, we use the Gegenbauer functions with the conventions of Appendix C, because they yield much simpler expressions than the so-called associated Legendre functions, which are commonly found in the literature [5, 26] in this context.

Point potentials in dimension \(d=3\) go back to Fermi [14], and since then, they have been often used in the physics literature. Berezin and Faddeev [4] seem to have been the first who interpreted them in a rigorous way. They are the subject of an extensive mathematical literature, confer, for example, [2, 3]. Point potentials in dimension \(d=1,2,3\) are special cases of singular perturbations, that is, perturbations which cannot be interpreted as operators. As we mentioned above, the case \(d=1\) can be interpreted as a form perturbation, so that one can use the so-called KLMN theorem [28]. For \(d=2,3\), the form technique is not applicable; therefore, in these dimensions point potentials belong to the class of form singular perturbations.

The formula for the resolvent of the form (1.9) is often called the Krein formula [19]. One can also find the name Aronszajn–Donoghue theory for this kind of treatment of singular rank one perturbations, see, e.g., [9]. (1.9) is essentially the singular version of the formula for the resolvent of an operator with a rank one perturbation, sometimes called the Sherman–Morrison formula.

There exist a large literature about point potentials on \(}}^d\) for \(d\ge 4\). These potentials are examples of supersingular perturbations. In order to interpret them as true linear operators, one needs to extend the Hilbert space by adding additional dimensions, see, e.g., [21]. This is reviewed in [20]. Note that our approach is different: We do not look for the perturbed operator, and we try to compute Green’s function associated with point potentials in all dimensions. It seems that the formulas (1.22e), (1.22d) are new, at least in the hyperbolic and spherical case.

It is clear that point potentials can be defined on a manifold of dimension 1,2,3, and have then similar properties as on the Euclidean space. Nevertheless, we have never seen their analysis on hyperbolic and spherical spaces including an explicit formula for their resolvent. So we think that also the identities (1.22b) and (1.22c) in the hyperbolic and spherical case are new.

The concept of a generalized integral goes back to independent considerations of Hadamard [16, 17] and Riesz [29]. The generalized integral is a linear extension of the integration functional to not necessarily integrable functions. It is closely related to the extension of homogeneous distributions [18]. More recent accounts are given in [22, 27]. In a parallel work [10], we revisited this concept in a manner that is well suited for our applications.

The flat limit of hyperbolic and spherical Green’s functions (without point potentials) is discussed in [5]. Note that the latter reference uses the associated Legendre equation instead of the Gegenbauer equation. The two equations are equivalent. The relation between Gegenbauer functions in our convention and associated Legendre functions can be found in [10].

Our results about the energy shift of eigenvalues of the spherical Laplacian in dimensions \(d\ge 3\) seem to be new. They are consistent with the following known fact, implicit, e.g., in [24]: In dimension 3, in a large box the ground state energy of the Schrödinger Hamiltonian with a short-range potential characterized by scattering length a has the ground state energy \(\approx \frac}\), where \(\textrm\) is the volume of the box (compare with (1.38)). This fact plays an important role in the well-known asymptotics of the bound state energy of the three-dimensional N-body Bose gas.

The ground state energy of a dilute Bose gas as in dimensions \(d\ge 4\) was studied in [