Well-Posed Problems for the Laplace–Beltrami Operator

Abstract

Here, we study boundary value problems for the Laplace–Beltrami operator on a three-dimensional sphere with a circular cut, obtained by removing a smooth closed geodesic from $S^3$ embedded in ${\mathbb{R}}^4$. The presence of the cut introduces singular perturbations of the domain, and we develop an analytical framework to characterize well-posed problems in this setting. Our approach combines Green’s functions, spectral analysis, and Sobolev space methods to establish solvability criteria and uniqueness results. In particular, we identify explicit conditions for the existence of solutions with data supported near the cut, and extend the formulation to include delta-type perturbations supported on the removed circle. These results generalize earlier work on punctured two-dimensional spheres and provide a foundation for the study of PDEs on manifolds with localized singularities.

1. Introduction

Boundary value problems for elliptic operators on manifolds with singularities constitute an important and rapidly developing area of geometric analysis. The study of such problems combines techniques from partial differential equations, spectral theory, and differential geometry, with applications ranging from mathematical physics to geometric modeling. In particular, the Laplace–Beltrami operator on manifolds with punctures or cuts serves as a natural model for understanding the influence of localized geometric defects on the behaviour of harmonic and eigenfunctions.

The present paper focuses on the Laplace–Beltrami operator on the three-dimensional sphere $S^3$ with a smooth closed geodesic removed. This removal creates a “circular cut” in the manifold, introducing both nontrivial topological and analytic challenges. Our aim is to formulate well-posed boundary value problems in this setting, establish solvability conditions, and analyze the effect of delta-type perturbations supported on the removed curve. While similar problems have been extensively studied for two-dimensional manifolds, the extension to $S^3$ requires a more delicate treatment due to the higher-dimensional geometry, the structure of the Green’s function, and the singular nature of the cut.

The Laplace–Beltrami operator is defined on any smooth manifold $M^n.$ If the manifold $M^n$ is compact, then the Laplace–Beltrami operator has a purely discrete spectrum [1]. In particular, the Laplace–Beltrami operator defined on the sphere S [2] possesses this property.

If the Laplace–Beltrami operator is considered on the hemisphere $S_{+}^n=\{x\in {\mathbb{R}}^{n+1}:\left|x\right|=1,x_{n+1}>0\},$ then at the boundary of the hemisphere, it is necessary to additionally set boundary conditions in the form of Dirichlet or Neumann [2]. In this case, $C=\partial S_{+}^n$ the boundary of the hemisphere represents a closed hypersurface that divides $S^n$ into two connected parts $S_{+}^n$ and $S_{-}^n$. In the work [3] on the sphere $S^2$ in ${\mathbb{R}}^3$, a closed curve C is considered, which divides $S^2$ into two disjoint connected components $S_1$ and $S_2$. Using analogues of the potentials of the simple and double layers, which was solved in the work [3] the following Dirichlet problem for the Laplace–Beltrami operator on the connected part $S_2$ of the sphere $S^2$

$$-{\Delta }_{S^2}u=0\mathrm{in}{S}_2\mathrm{and}u=g\mathrm{on}C.$$

Let us also note the work [4], in which the following nonlocal boundary value problem is set on the domain $S_2$:

$$-{\Delta }_{S^2}u=f\left(x\right),\forall x\in S_2,$$

$$-{\displaystyle \frac{u\left(x\right)}{2}}+{\int }_{C}u\left(y\right){curl}_S\epsilon (x,y)t\left(y\right)d{S}_y-{\int }_C\epsilon (x,y){curl}_{S}u\left(y\right)t\left(y\right)d{S}_y=0,x\in C.$$

Note that the values ${curl}_S,\epsilon (x,y)$ and $t\left(y\right)$ were defined in [3]. In the work [4], the unique solvability of the above nonlocal boundary value problem is proved.

In the work [3,4] on the sphere $S^2$ in ${\mathbb{R}}^3$, a closed curve C is considered, which divides $S^2$ into two disjoint connected components $S_1$ and $S_2$. If C is a set of dimension zero (for example, if C consists of one point $x_0\in S^2$), then removing this point from $S^2$ leads to a punctured region $S^2\backslash \left\{x_0\right\}$. The paper [5] presents correctly solvable problems for the Laplace–Beltrami operator in the punctured domain $S^2\backslash \left\{x_0\right\}$. If we choose a non-closed curve $S^2$ as C, then examples of boundary value problems for the Laplace–Beltrami operator that are solvable in the domain $S^2\backslash C$ can be found in [5].

This paper considers the sphere $S^3$ in ${\mathbb{R}}^4.$ Now, the manifold C on the sphere $S^3$ can be shown in four significantly different ways:

- C is a manifold of dimension zero,

- C is a manifold of dimension one,

- C is a closed manifold of dimension two,

- C is a non-closed manifold of dimension two.

The most interesting case is when $dimC=1$. Other cases have already been encountered on $S^2$ and were studied in the works of [3,4,5].

Therefore, consider the sphere $S^3=\{(x_1, x_2, x_3, x_4):x_1^2+x_2^2+x_3^2+x_4^2=1\}$ in ${\mathbb{R}}^4=\{(x_1, x_2, x_3, x_4):x_1, x_2, x_3, x_4\in \mathbb{R}$}. Let us take a circle as the manifold C:

$$C=\{(x_1, x_2, x_3, x_4):x_1^2+x_2^2={\displaystyle \frac{1}{4}}, x_3={\displaystyle \frac{1}{2}}, x_4={\displaystyle \frac{\sqrt{2}}{2}}\}.$$

Problem statement: describe boundary value problems for the Laplace–Beltrami operator on the sphere $S^3$ from which the closed curve C has been removed.

To formulate an answer to the problem, the following notations and facts are needed.

It is convenient to introduce a class of functions:

$$W_{2,loc}^2\left(S_c^3\right)=\bigcup _{\delta >0}{W}_2^2(S^3\backslash {\Pi }_c\left(\delta \right)),$$

where $W_2^2(\Omega )$ is the Sobolev space on the domain $\Omega \subset S^3\subset {\mathbb{R}}^4$.

The meaning of the notation ${\Pi }_c\left(\delta \right)$ can be found in paragraph 4.

It is also necessary to define the following function class

$$W_{2,K}^2\left(S_c^3\right)=\{z\in W_{2,loc}^2\left(S_c^3\right):K_{i}z\in L_2(0,2\pi ),i=1, 2, 3, {\int }_{S^3}zdV=0\}.$$

The operators $K_1, K_2, K_3$ appear here, which are defined in paragraph 4 of this paper.

The structure of elements of the space $W_{2,K}^2\left(S_c^3\right)$ is clarified by the following statement.

Lemma 1. 

For any function $z\in W_{2,K}^2\left(S_c^3\right)$, the representation holds

$$z\left({\psi }^{\prime },{\theta }^{\prime },{\phi }^{\prime },\psi ,\theta ,\phi \right)=T\left({\psi }^{\prime },{\theta }^{\prime },{\phi }^{\prime }\right)-{\displaystyle \frac{3}{2}}{\int }_0^{2\pi }G\left({\psi }^{\prime },{\theta }^{\prime },{\phi }^{\prime },{\displaystyle \frac{\pi }{4}},{\displaystyle \frac{\pi }{4}},\phi \right)\left(K_{1}z\right)\left(\phi \right)d\phi -$$

$$-{\displaystyle \frac{3}{2}}{\int }_0^{2\pi }{\displaystyle \frac{\partial G\left({\psi }^{\prime },{\theta }^{\prime },{\phi }^{\prime };\frac{\pi }{4},\frac{\pi }{4},\phi \right)}{\partial \psi }}\left(K_{2}z\right)\left(\phi \right)d\phi +$$

$$+{\displaystyle \frac{3\sqrt{2}}{2}}{\int }_0^{2\pi }{\displaystyle \frac{\partial G\left({\psi }^{\prime },{\theta }^{\prime },{\phi }^{\prime };\frac{\pi }{4},\frac{\pi }{4},\phi \right)}{\partial \theta }}\left(K_{3}z\right)\left(\phi \right)d\phi ,$$

where $T\in W_2^2\left(S^3\right).$ This representation is unique.

It follows from Lemma 1 that the operator

$$B_{max}z=-{\Delta }_{S^3}T$$

is correctly defined for $\forall z\in W_{2K}^2\left(S_c^3\right).$

We formulate the main result of this paper.

Theorem 1. 

Let $i=1, 2, 3$ for the smooth functions $\forall {\alpha }_i\in W_2^2({(-{\displaystyle \frac{\pi }{2}},{\displaystyle \frac{\pi }{2}})}^2)$ satisfy the following requirements

$${\int }_{-{\displaystyle \frac{\pi }{2}}}^{{\displaystyle \frac{\pi }{2}}}{\int }_{-{\displaystyle \frac{\pi }{2}}}^{{\displaystyle \frac{\pi }{2}}}\left({\displaystyle \frac{\partial }{\partial \psi }}\left(cos\psi {\displaystyle \frac{\partial {\alpha }_i}{\partial \psi }}\right)cos\psi cos\theta +{\displaystyle \frac{\partial }{\partial \theta }}\left(cos\theta {\displaystyle \frac{\partial {\alpha }_i}{\partial \theta }}\right)\right)d\psi d\theta =0.$$

Then, for $\forall f\in L_2\left(S^3\right)$ the boundary value problem has a unique solution in the space $W_{2,K}^2\left(S_c^3\right):$

$$B_{max}z=f\left(x\right),x\in S^3\backslash C,$$

(1)

$$\left(K_{i}z\right)\left(\phi \right)={\int }_{-{\displaystyle \frac{\pi }{2}}}^{{\displaystyle \frac{\pi }{2}}}{\int }_{-{\displaystyle \frac{\pi }{2}}}^{{\displaystyle \frac{\pi }{2}}}{\alpha }_i{\displaystyle \frac{1}{cos\theta }}{\displaystyle \frac{{\partial }^{2}Jz}{\partial {\phi }^2}}d\psi d\theta ,i=1, 2, 3,0<\phi <2\pi .$$

(2)

Note that the problem (1) and (2) can be written as

$${\Delta }_{S^3}u\left(x\right)+q\left(x\right)u\left(x\right)=f\left(x\right),x\in S^3,$$

where ${\Delta }_{S^3}$ is the Laplace–Beltrami operator on the sphere $S^3,$$q\left(x\right)={\delta }_c\left(x\right)$ is the delta function with support on curve $C.$ Delta-shaped perturbations of the Laplace operator on standard domains were studied in the work [6].

2. Main Results

2.1. On the Fundamental Solutions of the Laplace–Beltrami Operator on a Three-Dimensional Sphere

It is known [7] that the ${\Delta }_{S^3}$ Laplace–Beltrami operator on a three-dimensional sphere is defined through the $\Delta$ ordinary Laplace operator in ${\mathbb{R}}^4.$ More precisely, the representation is held

$$\Delta \equiv {\displaystyle \frac{\partial }{\partial r^2}}+{\displaystyle \frac{3}{r}}{\displaystyle \frac{\partial }{\partial r}}+{\displaystyle \frac{1}{r^2}}{\Delta }_{S^3},$$

where ${\Delta }_{S^3}$ is applied on the unit sphere, with subsequent continuation along zero homogeneity, and we designated

$${\widehat{\Delta }}_{\theta ,\phi }^{\prime }={\displaystyle \frac{1}{}}{\displaystyle \frac{\partial }{\partial \psi }}(cos\psi {\displaystyle \frac{\partial u}{\partial \psi }})+{\displaystyle \frac{1}{cos\theta {cos}^2\psi }}{\displaystyle \frac{\partial }{\partial \theta }}(cos\theta {\displaystyle \frac{\partial u}{\partial \theta }})+{\displaystyle \frac{1}{{cos}^2\psi {cos}^2\theta }}{\displaystyle \frac{{\partial }^{2}u}{\partial {\phi }^2}}.$$

We need a formulation for Green’s formula.

Let $f\left(u\right)$ and $g\left(u\right)$ be continuously differentiable functions over ${\mathbb{R}}^4$. Let us assume that these functions for $u\ne 0$ are homogeneous of degrees p and q, respectively. We will choose numbers $r_1$ and $r_2$, such that $0<r_1<1\le r_2$. Let E denote the unit ball in ${\mathbb{R}}^4$. Let us write the Gauss formula

$${\int }_{r_{2}E\backslash r_{1}E}{\displaystyle \frac{\partial }{\partial u_i}}\left(f\left(u\right)g\left(u\right)\right)dV\left(u\right)=({\int }_{r_2{S}^3}-{\int }_{r_1{S}^3})f\left(u\right)g\left(u\right){\displaystyle \frac{u_i}{\left|\right|u\left|\right|}}dS\left(u\right),$$

(3)

where $dV\left(u\right)$ is the volume element in ${\mathbb{R}}^4$ at the point u, $dS\left(u\right)=r_{\nu }^{3}d\omega ({\displaystyle \frac{u}{\left|\right|u\left|\right|}}),\nu =1, 2$ is a 3-dimensional volume element, and $d\omega$ is a volume element on the sphere $S^3$.

We differentiate (3) with respect to $r_2$ and set $r_2=1.$ Then, due to the homogeneity of f and g and the nature of $dS\left(u\right)$, we obtain the relation

$${\int }_{S^3}f\left(u\right)·{\displaystyle \frac{\partial g\left(u\right)}{\partial u_i}}·d\omega \left(u\right)={\int }_{S^3}g\left(u\right)\{[3+p+q]f\left(u\right)u_i-{\displaystyle \frac{\partial f\left(u\right)}{\partial u_i}}\}d\omega \left(u\right).$$

(4)

The relation (4) represents the formula for integration by parts for homogeneous continuously differentiable functions. Following the work of [8], functions that are twice differentiable on $S^3$ will be called regular.

If h is a regular function of degree zero, then

$$(gradh(u),u)=0.$$

Now let us write in the Formula (4) instead of $g\left(u\right)$ the function ${\displaystyle \frac{\partial h}{\partial u}}$, then add. After adding the index i, the relation (4) will lead to the equality:

$${\int }_{S^3}f\left(u\right)\Delta h\left(u\right)d\omega \left(u\right)=-{\int }_{S^3}(gradf\left(u\right),gradh\left(u\right))d\omega \left(u\right)={\int }_{S^3}h\left(u\right)\Delta f\left(u\right)d\omega \left(u\right).$$

Let us generalize the Formula (4) to the case when $f\left(u\right)$ is regular at all points, except perhaps one $u=u^{\prime }.$ The function f defined on the unit sphere $S^3$ can be continued to the entire ${\mathbb{R}}^4\backslash \left\{0\right\}$ using the formula

$$f(u,u^{\prime })=f\left({\displaystyle \frac{u-u^{\prime }}{\left|\right|u-u^{\prime }\left|\right|}}\right).$$

The function extended in this way will be singular at all points $r{u}^{\prime },r>0$.

In the volume integral (3), we remove the set $M\left(\epsilon \right)$ of points u, for which

$$r_1⩽\parallel u\parallel ⩽r_2,\left(u^{\prime },u\right)⩾\parallel u\parallel ∥u^{\prime }∥cos\epsilon ,$$

(5)

where $0<\epsilon <\pi /2$.

If $\omega \left(\epsilon \right)=S^3\cap M\left(\epsilon \right)$, then the boundary of $M\left(\epsilon \right)$ consists of sets $r_1\omega \left(\epsilon \right)$, $r_2\omega \left(\epsilon \right)$ and $N\left(\epsilon \right)$ are a subset of $M\left(\epsilon \right)$ for which the equality in the second condition (5) is satisfied. To the right side of (3), it is necessary to add the integral

$${\int }_{N\left(\epsilon \right)}f\left(u\right)g\left(u\right)v_i\left(u\right)dS\left(u\right),$$

where $v\left(u\right)$ is the unit normal to $N\left(\epsilon \right)$ directed to $M\left(\epsilon \right)$.

Lemma 2. 

The normal unit is calculated using the formula

$$v\left(u\right)={\displaystyle \frac{u^{\prime }}{∥u^{\prime }∥}}cosec\epsilon -{\displaystyle \frac{u}{\parallel u\parallel }}cot\epsilon .$$

(6)

Proof. 

Let the second condition (5) satisfy the equality

$$\sum _{i=1}^n{\displaystyle \frac{u_i^{\prime }}{\left|\right|u^{\prime }\left|\right|}}·{\displaystyle \frac{u_i}{\left|\right|u\left|\right|}}=cos\epsilon .$$

Let us differentiate the last equality with respect to $u_1:$

$${\displaystyle \frac{u_1^{\prime }}{\left|\right|u^{\prime }\left|\right|}}·{\displaystyle \frac{1}{\left|\right|u\left|\right|}}+\sum _{i=1}^4{\displaystyle \frac{u_i^{\prime }·u_i}{\left|\right|u^{\prime }\left|\right|}}·(-{\displaystyle \frac{1}{2}}{\displaystyle \frac{2u_1}{{\left|\right|u\left|\right|}^3}})={\displaystyle \frac{u_1^{\prime }}{\left|\right|u^{\prime }\left|\right|·\left|\right|u\left|\right|}}-{\displaystyle \frac{u_1}{{\left|\right|u\left|\right|}^2}}cos\epsilon .$$

In the same way it is differentiated by $u_2, u_3, u_4$. As a result, we have an expression for the corresponding gradient:

$${\displaystyle \frac{u^{\prime }}{\left|\right|u^{\prime }\left|\right|·\left|\right|u\left|\right|}}-{\displaystyle \frac{u}{{\left|\right|u\left|\right|}^2}}cos\epsilon .$$

Therefore, the required normal in $v\left(u\right)$ is collinear to the vector

$${\displaystyle \frac{u^{\prime }}{\left|\right|u^{\prime }\left|\right|}}-{\displaystyle \frac{u}{\left|\right|u\left|\right|}}cos\epsilon .$$

It is clear that the length of the vector ${\displaystyle \frac{u^{\prime }}{\left|\right|u^{\prime }\left|\right|}}-{\displaystyle \frac{u}{\left|\right|u\left|\right|}}cos\epsilon$ is equal to $sin\epsilon .$ Therefore the required normal is $v\left(u\right)$ has the form (6). □

Let $n\left(\epsilon \right)=S^3\cap N\left(\epsilon \right)$ be the boundary of $\omega \left(\epsilon \right)$ on $S^3$. The set ${\omega }^{\prime }$ of points w such that $\parallel w\parallel =1,\left(u^{\prime },w\right)=0$ is a two-dimensional spherical surface similar to $n\left(\epsilon \right)$. The area element ${\omega }^{\prime }$, the internal normal of which lies in the plane $u^{\prime }$ and $v\left(u\right)$, will be denoted by $d{\omega }^{\prime }\left(v\left(u\right)\right)$. Then on $n\left(\epsilon \right)$ we have

$$dS\left(u\right)={\epsilon }^2{r}^{2}drd{\omega }^{\prime }\left(v\left(u\right)\right),$$

where $r=\parallel u\parallel$.

Repeating our previous reasoning, we find instead of the equality (4) proves the following relation

$$\begin{array}{cc}\hfill {\int }_{S^3-\omega \left(\epsilon \right)}f\left(u\right){\displaystyle \frac{\partial g\left(u\right)}{\partial u_i}}d\omega \left(u\right)=& {\int }_{S^3-\omega \left(\epsilon \right)}g\left(u\right)\left[(3+p+q)f\left(u\right)u_i-{\displaystyle \frac{\partial f\left(u\right)}{\partial u_i}}\right]d\omega \left(u\right)\hfill \\ & +{\epsilon }^2{\int }_{n\left(\epsilon \right)}f\left(u\right)g\left(u\right)v_i\left(u\right)d{\omega }^{\prime }\left(v\left(u\right)\right).\hfill \end{array}$$

Let us introduce the function $h_1$ defined for $u\ne u^{\prime }$ by the formula

$$h_1\left(u^{\prime },u\right)=-{\displaystyle \frac{1}{{\omega }_3}}{\left[s\left(u^{\prime },u\right)\right]}^{-1},$$

(7)

where s is the spherical distance given by

$$s\left(u^{\prime },u\right)=Arccos\left[{\displaystyle \frac{\left(u^{\prime },u\right)}{\parallel u\parallel ∥u^{\prime }∥}}\right].$$

(8)

The introduced function $h_1(u,u^{\prime })$ has a singularity at $u=u^{\prime }$. A singularity of this order is called a fundamental singularity.

Let $h_2$ be a symmetric function of $u^{\prime }$ and u, regular over $S^3\times S^3$ and extended by homogeneity of degree zero over $u^{\prime }$ and u. Then, the function $h\left(u\right)=h_1\left(u\right)+h_2\left(u\right)$ satisfies the condition

$$(gradh(u),u)=0.$$

Next, assume F is also a homogeneous function of degree zero. Here, $f\left(u\right)={\displaystyle \frac{\partial F}{\partial u_i}}$. Then, using standard manipulations, we obtain

$$\begin{array}{cc}& {\int }_{S^3-\omega \left(\epsilon \right)}\left[h\left(u^{\prime },u\right)\Delta F\left(u\right)-F\left(u\right)\Delta h\left(u^{\prime },u\right)\right]d\omega \left(u\right)\hfill \\ & ={\epsilon }^2{\int }_{n\left(\epsilon \right)}\left[h\left(u^{\prime },u\right)(gradF\left(u\right),v\left(u\right))-F\left(u\right)\left(gradh\left(u^{\prime },u\right),v\left(u\right)\right)\right]d{\omega }^{\prime }\left(v\left(u\right)\right),\epsilon \to 0.\hfill \end{array}$$

Since on $n\left(\epsilon \right)$, we have $s\left(u^{\prime },u\right)=\epsilon$ and from (6) and (8)

$$grads\left(u^{\prime },u\right)=-v\left(u\right).$$

Hence, using (7) we obtain

$$\left(grad{h}_1\left(u^{\prime },u\right),v\left(u\right)\right)=-{\epsilon }^{-2}/{\omega }_3.$$

These observations and the regularity of F and $h_2$ lead us to conclude that

$$F\left(u^{\prime }\right)={\int }_{S^3}\left[h\left(u^{\prime },u\right)\Delta F\left(u\right)-F\left(u\right)\Delta h\left(u^{\prime },u\right)\right]d\omega \left(u\right).$$

Let s still be defined by the Formula (8) and, following the work of [8], we set

$$\gamma \left(s\right)=-{\displaystyle \frac{1}{{\omega }_4}}{\int }_{\pi /2}^s{cosec}^{2}t\left({\int }_{\pi }^t{sin}^2{t}^{\prime }d{t}^{\prime }\right)dt.$$

Then, the fundamental solution is determined for $u\ne u^{\prime }$ by the following formula

$$G\left(u^{\prime },u\right)=\gamma \left(s\left(u^{\prime },u\right)\right).$$

Thus,

$${\Delta }^{\prime }G\left(u^{\prime },u\right)=-{\displaystyle \frac{1}{{\omega }_4}}+\delta (u^{\prime }-u),$$

(9)

where ${\Delta }^{\prime }$ is Laplacian with respect to $u_1^{\prime }, u_2^{\prime }, u_3^{\prime }, u_4^{\prime }$. Here, $\delta (u^{\prime }, u)$ denotes the Dirac delta function on the sphere $S^3$. We also note that direct calculations make it possible to verify that

$$G\left(u^{\prime },u\right)=h_1\left(u^{\prime },u\right)+h_2\left(u^{\prime },u\right),$$

where

$$h_1(u^{\prime },u)=-{\displaystyle \frac{1}{4\pi }}·{\displaystyle \frac{1}{s(u^{\prime },u)}},$$

$$h_2(u^{\prime },u)={\displaystyle \frac{s(u^{\prime },u)}{4{\pi }^2}}·{\displaystyle \frac{coss(u^{\prime },u)}{sins(u^{\prime },u)}}-{\displaystyle \frac{1}{4\pi }}[{\displaystyle \frac{s(u^{\prime },u)}{sins(u^{\prime },u)}}·coss(u^{\prime },u)-1]·{\displaystyle \frac{1}{s(u^{\prime },u)}}.$$

Here, $h_2$ is a symmetric function of $u^{\prime }$ and u, regular over $S^3\times S^3$ and extended to degree zero homogeneity in $u^{\prime }$ and u, and $h_1$ is a function with a fundamental singularity defined by the following formula:

$$h_1(u^{\prime },u)=-{\displaystyle \frac{1}{4\pi }}{\displaystyle \frac{1}{s(u^{\prime },u)}}.$$

Note that G is defined for all u not equal to $u^{\prime }$ and is homogeneous of degree zero, and G is symmetric with respect to u and $u^{\prime }$.

The analysis of well-posed boundary value problems for the Laplace–Beltrami operator on Riemannian manifolds has developed into a rich theory encompassing both classical and modern techniques. Foundational results concerning elliptic equations on manifolds with boundaries were significantly advanced by [9], who considered elliptic and parabolic problems on manifolds where the boundary consists of components of different dimensions. This approach introduced new challenges in defining suitable function spaces and transmission conditions, which remain relevant in the current geometric analysis.

In more recent developments, spectral properties of elliptic operators under singular perturbations or lower-dimensional interfaces have drawn attention. A notable example is the study of point interactions and PT-hermiticity, as explored by [10]. Their results establish criteria for the reality of the spectrum under non-self-adjoint perturbations, which is of particular interest in non-standard boundary conditions and in settings where the Laplace–Beltrami operator is modified by singular potentials or defects on the manifold.

These contributions complement classical treatments by providing methods to handle irregular geometries or nonlocal boundary effects, which frequently arise in physical models and require a broader understanding of the concept of well-posedness beyond traditional self-adjoint frameworks.

2.2. Laplace–Beltrami Operator on a Sphere with a Cut

Let us choose a standard spherical coordinate system on $S^3$ in $R^4$ $x_1=cos\phi cos\psi cos\theta ,$ $x_2=sin\phi cos\psi cos\theta ,$ $x_3=cos\psi sin\theta ,$ $x_4=sin\psi .$ Note that the parameters $\psi$ and $\theta$ vary in the interval $(-{\displaystyle \frac{\pi }{2}},{\displaystyle \frac{\pi }{2}}),$ and the parameter $\phi$ runs over the interval $(0,2\pi ).$

On the three-dimensional sphere $S^3$, we choose a circle C defined by the relation

$$x_1^2+x_2^2={\displaystyle \frac{1}{4}}, x_3={\displaystyle \frac{1}{2}}, x_4={\displaystyle \frac{\sqrt{2}}{2}},$$

or ${\psi }_0={\displaystyle \frac{\pi }{4}}, {\theta }_0={\displaystyle \frac{\pi }{4}}, 0\le \phi \le 2\pi .$

Consider a sphere with a cut $S_c^3$ defined by the relation

$$S_c^3=S^3\backslash C.$$

Let $\delta >0.$ On a three-dimensional sphere $S^3$, consider $\delta$—a neighborhood of a circle C—defined as follows:

$${\Pi }_c\left(\delta \right)=\{(\psi ,\theta ,\phi )\in S^3:|\psi -{\displaystyle \frac{\pi }{4}}|<\delta ,|\theta -{\displaystyle \frac{\pi }{4}}|<\delta , 0\le \phi \le 2\pi \}.$$

It is convenient for us to introduce a class of functions:

$$W_{2,loc}^2\left(S_c^3\right)=\bigcup _{\delta >0}{W}_2^2(S^3\backslash {\Pi }_c\left(\delta \right)),$$

where $W_2^2(\Omega )$ is the Sobolev space on the domain $\Omega \subset S^3\subset {\mathbb{R}}^4$.

For further purposes, it is convenient to introduce the operators $K_1, K_2, K_3$ using the formulas

$$\begin{array}{cc}\hfill & \left(K_{1}z\right)\left(\phi \right)=\underset{\delta \to 0}{lim}{\int }_{{\displaystyle \frac{\pi }{4}}-\delta }^{{\displaystyle \frac{\pi }{4}}+\delta }({\displaystyle \frac{\partial \widehat{z}\left(\frac{\pi }{4}-\delta ,\theta ,\phi \right)}{\partial \psi }}+\widehat{z}\left({\displaystyle \frac{\pi }{4}}-\delta ,\theta ,\phi \right)-\hfill \\ \hfill & -{\displaystyle \frac{\partial \widehat{z}\left(\frac{\pi }{4}+\delta ,\theta ,\phi \right)}{\partial \psi }}-\widehat{z}\left({\displaystyle \frac{\pi }{4}}+\delta ,\theta ,\phi \right))cos\theta d\theta +\hfill \\ \hfill & +\sqrt{2}\underset{\delta \to 0}{lim}{\int }_{{\displaystyle \frac{\pi }{4}}-\delta }^{{\displaystyle \frac{\pi }{4}}+\delta }\left({\displaystyle \frac{\partial \widehat{z}\left(\psi ,\frac{\pi }{4}-\delta ,\phi \right)}{\partial \theta }}-{\displaystyle \frac{\partial \widehat{z}\left(\psi ,\frac{\pi }{4}-\delta ,\phi \right)}{\partial \theta }}\right)d\psi ,\hfill \\ \hfill & \left(K_{2}z\right)\left(\phi \right)=\underset{\delta \to 0}{lim}{\int }_{{\displaystyle \frac{\pi }{4}}-\delta }^{{\displaystyle \frac{\pi }{4}}+\delta }\left(\widehat{z}\left({\displaystyle \frac{\pi }{4}}+\delta ,\theta ,\phi \right)-\widehat{z}\left({\displaystyle \frac{\pi }{4}}-\delta ,\theta ,\phi \right)\right)cos\theta d\theta ,\hfill \\ \hfill & \left(K_{3}z\right)\left(\phi \right)=\underset{\delta \to 0}{lim}{\int }_{{\displaystyle \frac{\pi }{4}}-\delta }^{{\displaystyle \frac{\pi }{4}}+\delta }\left(\widehat{z}\left(\psi ,{\displaystyle \frac{\pi }{4}}+\delta ,\phi \right)-\widehat{z}\left(\psi ,{\displaystyle \frac{\pi }{4}}+\delta ,\phi \right)\right)d\psi .\hfill \end{array}$$

Linear operators $K_i$ for $i=1,2,3$ transform functions z defined on $S^3$ into functions $l=K_{i}z$ defined on the circle C.

It is necessary to define the following function class

$$W_{2,K}^2\left(S_c^3\right)=\{z\in W_{2,loc}^2\left(S_c^3\right):K_{i}z\in L_2(0,2\pi ),i=1,2,3,{\int }_{S^3}zdV=0\}.$$

Take an arbitrary element z from the class $W_{2,K}^2\left(S_c^3\right)$. Let us denote by $T\left(x\right)={\int }_{S^3\backslash {\Pi }_c\left(\delta \right)}G(x,y){\Delta }_{S_y^3}z\left(y\right)dV\left(y\right)$.

Let us transform T using the expression of the Laplace–Beltrami operator

$$T\left({\psi }^{\prime },{\theta }^{\prime },{\phi }^{\prime }\right)={\int }_{S^3\backslash {\Pi }_c\left(\delta \right)}G({\psi }^{\prime },{\theta }^{\prime },{\phi }^{\prime };\psi ,\theta ,\phi )·({\displaystyle \frac{1}{cos\psi }}{\displaystyle \frac{\partial }{\partial \psi }}(cos\psi {\displaystyle \frac{\partial \widehat{z}}{\partial \psi }})+$$

$$\left.+{\displaystyle \frac{1}{cos\theta {cos}^2\psi }}{\displaystyle \frac{\partial }{\partial \theta }}\left(cos\theta {\displaystyle \frac{\partial \widehat{z}}{\partial \theta }}\right)+{\displaystyle \frac{1}{{cos}^2\psi {cos}^2\theta }}{\displaystyle \frac{{\partial }^2\widehat{z}}{\partial {\phi }^2}}\right)3{cos}^2\psi cos\theta d\psi d\theta d\phi =$$

$$=3{\int }_{S^3\backslash {\Pi }_c\left(\delta \right)}G\left({\psi }^{\prime },\theta ,{\phi }^{\prime };\psi ,\theta ,\phi \right)cos\psi cos\theta {\displaystyle \frac{\partial }{\partial \psi }}\left(cos\psi {\displaystyle \frac{\partial \widehat{z}}{\partial \psi }}\right)d\psi d\theta d\phi +$$

$$+3{\int }_{S^3\backslash {\Pi }_c\left(\delta \right)}G\left({\psi }^{\prime },{\theta }^{\prime },{\phi }^{\prime };\psi ,\theta ,\phi \right)·{\displaystyle \frac{\partial }{\partial \theta }}\left(cos\theta {\displaystyle \frac{\partial \widehat{z}}{\partial \theta }}\right)d\psi d\theta d\phi +$$

$$+3{\int }_{S^3\backslash {\Pi }_c\left(\delta \right)}G\left(\psi ;\theta ,{\phi }^{\prime };\psi ,\theta ,\phi \right){\displaystyle \frac{1}{cos\theta }}{\displaystyle \frac{{\partial }^2\widehat{z}}{\partial {\phi }^2}}d\psi d\theta d\phi =I_{\psi }+I_{\theta }+I_{\phi }.$$

Let us separately transform the following integral:

$$I_{\theta }=3{\int }_0^{2\pi }d\phi \left({\int }_{{\displaystyle \frac{\pi }{4}}-\delta }^{{\displaystyle \frac{\pi }{4}}+\delta }\left(\left({\int }_{-{\displaystyle \frac{\pi }{2}}}^{{\displaystyle \frac{\pi }{4}}-\delta }+{\int }_{{\displaystyle \frac{\pi }{4}}+\delta }^{{\displaystyle \frac{\pi }{2}}}\right)G{\displaystyle \frac{\partial }{\partial \theta }}\left(cos\theta {\displaystyle \frac{\partial \widehat{z}}{\partial \theta }}\right)d\theta \right)d\psi \right)+$$

$$+3{\int }_0^{2\pi }d\phi \left(\left({\int }_{-{\displaystyle \frac{\pi }{2}}}^{{\displaystyle \frac{\pi }{4}}-\delta }+{\int }_{{\displaystyle \frac{\pi }{4}}+\delta }^{{\displaystyle \frac{\pi }{2}}}\right)\left({\int }_{-{\displaystyle \frac{\pi }{2}}}^{{\displaystyle \frac{\pi }{2}}}G{\displaystyle \frac{\partial }{\partial \theta }}\left(cos\theta {\displaystyle \frac{\partial \widehat{z}}{\partial \theta }}\right)d\theta \right)d\psi \right)=$$

$$=3{\int }_0^{2\pi }d\phi \left({\left.{\int }_{{\displaystyle \frac{\pi }{4}}-\delta }^{{\displaystyle \frac{\pi }{4}}+\delta }\left(cos\theta {\displaystyle \frac{\partial \widehat{z}}{\partial \theta }}G-\widehat{z}cos\theta {\displaystyle \frac{\partial G}{\partial \theta }}\right)\right|}_{\theta ={\displaystyle \frac{\pi }{4}}+\delta }^{\theta ={\displaystyle \frac{\pi }{4}}-\delta }d\psi \right)+$$

$$+3{\int }_0^{2\pi }d\phi \left({\left.\left({\int }_{-{\displaystyle \frac{\pi }{2}}}^{{\displaystyle \frac{\pi }{4}}-\delta }+{\int }_{{\displaystyle \frac{\pi }{4}}+\delta }^{{\displaystyle \frac{\pi }{2}}}\right)\left(cos\theta {\displaystyle \frac{\partial \widehat{z}}{\partial \theta }}G-\widehat{z}cos\theta {\displaystyle \frac{\partial G}{\partial \theta }}\right)\right|}_{\theta =-{\displaystyle \frac{\pi }{2}}}^{\theta ={\displaystyle \frac{\pi }{2}}}d\psi \right)+$$

$$+3\int {\displaystyle \frac{1}{cos\theta {cos}^2\psi }}\widehat{z}{\displaystyle \frac{\partial }{\partial \theta }}\left(cos\theta {\displaystyle \frac{\partial G}{\partial \theta }}\right){cos}^2\psi cos\theta d\psi d\theta d\phi .$$

If $\delta \to +0,$ then we have the limit relation

$$\underset{\delta \to 0}{lim}{I}_{\theta }={\displaystyle \frac{3\sqrt{2}}{2}}{\int }_0^{2\pi }d\phi \underset{\delta \to 0}{lim}{\int }_{{\displaystyle \frac{\pi }{4}}-\delta }^{{\displaystyle \frac{\pi }{4}}+\delta }\left({\displaystyle \frac{\partial \widehat{z}\left(\psi ,\frac{\pi }{4}-\delta ,\phi \right)}{\partial \theta }}G\left({\psi }^{\prime },{\theta }^{\prime },{\phi }^{\prime };\psi ,{\displaystyle \frac{\pi }{4}}-\delta ,\phi \right)-\right.$$

$$\left.-{\displaystyle \frac{\partial \widehat{z}\left(\psi ,\frac{\pi }{4}-\delta ,\phi \right)}{\partial \theta }}G\left({\psi }^{\prime };{\theta }^{\prime },{\phi }^{\prime };\psi ,{\displaystyle \frac{\pi }{4}}+\delta ,\phi \right)\right)d\psi -$$

$$-{\displaystyle \frac{3\sqrt{2}}{2}}{\int }_0^{2\pi }d\phi \underset{\delta \to 0}{lim}{\int }_{{\displaystyle \frac{\pi }{4}}-\delta }^{{\displaystyle \frac{\pi }{4}}+\delta }\left(\widehat{z}\left(\psi ,{\displaystyle \frac{\pi }{4}}+\delta ,\phi \right){\displaystyle \frac{\partial G\left({\psi }^{\prime },{\theta }^{\prime },{\phi }^{\prime };\psi ,\frac{\pi }{4}+\delta ,\phi \right)}{\partial \theta }}-\right.$$

$$\left.-\widehat{z}\left(\psi ,{\displaystyle \frac{\pi }{4}}+\delta ,\phi \right){\displaystyle \frac{\partial G\left({\psi }^{\prime },{\theta }^{\prime },{\phi }^{\prime };\psi ,\frac{\pi }{4}+\delta ,\phi \right)}{\partial \theta }}\right)d\psi +$$

$$+3\int {\displaystyle \frac{1}{cos\theta {cos}^2\psi }}\widehat{z}{\displaystyle \frac{\partial }{\partial \theta }}\left(cos\theta {\displaystyle \frac{\partial G}{\partial \theta }}\right){cos}^2\psi cos\theta d\psi d\theta d\phi =$$

$$={\displaystyle \frac{3\sqrt{2}}{2}}{\int }_0^{2\pi }G\left({\psi }^{\prime },{\theta }^{\prime },{\phi }^{\prime };{\displaystyle \frac{\pi }{4}},{\displaystyle \frac{\pi }{4}},\phi \right)d\phi \underset{\delta \to 0}{lim}{\int }_{{\displaystyle \frac{\pi }{4}}-0}^{{\displaystyle \frac{\pi }{4}}+\delta }\left({\displaystyle \frac{\partial \widehat{z}\left(\psi ,\frac{\pi }{4}-\delta ,\phi \right)}{\partial \theta }}-{\displaystyle \frac{\partial \widehat{z}\left(\psi ,\frac{\pi }{4}-\delta ,\phi \right)}{\partial \theta }}\right)d\psi -$$

$$-{\displaystyle \frac{3\sqrt{2}}{2}}{\int }_0^{2\pi }{\displaystyle \frac{\partial G\left({\psi }^{\prime },{\theta }^{\prime },{\phi }^{\prime };\frac{\pi }{4},\frac{\pi }{4},\phi \right)}{\partial \theta }}d\phi \underset{\delta \to 0}{lim}{\int }_{{\displaystyle \frac{\pi }{4}}-\delta }^{{\displaystyle \frac{\pi }{4}}+\delta }\left(\widehat{z}\left(\psi ,{\displaystyle \frac{\pi }{4}}+\delta ,\phi \right)-\widehat{z}\left(\psi ,{\displaystyle \frac{\pi }{4}}+\delta ,\phi \right)\right)d\psi +$$

$$+3\int {\displaystyle \frac{1}{cos\theta {cos}^2\psi }}\widehat{z}{\displaystyle \frac{\partial }{\partial \theta }}\left(cos\theta {\displaystyle \frac{\partial G}{\partial \theta }}\right){cos}^2\psi cos\theta d\psi d\theta d\phi .$$

Similarly, we transform the value of the integral $I_{\psi }$:

$$I_{\psi }=3{\int }_0^{2\pi }d\phi \left({\int }_{{\displaystyle \frac{\pi }{4}}-\delta }^{{\displaystyle \frac{\pi }{4}}+\delta }\left(\left({\int }_{-{\displaystyle \frac{\pi }{2}}}^{{\displaystyle \frac{\pi }{4}}-\delta }+{\int }_{{\displaystyle \frac{\pi }{4}}+\delta }^{{\displaystyle \frac{\pi }{2}}}\right){\displaystyle \frac{\partial }{\partial \psi }}\left(cos\psi {\displaystyle \frac{\partial \widehat{z}}{\partial \psi }}\right)cos\psi cos\theta Gd\psi \right)d\theta +\right.$$

$$+3{\int }_0^{2\pi }d\phi \left(\left({\int }_{-{\displaystyle \frac{\pi }{2}}}^{{\displaystyle \frac{\pi }{4}}-\delta }+{\int }_{{\displaystyle \frac{\pi }{4}}+\delta }^{{\displaystyle \frac{\pi }{2}}}\right)\left({\int }_{-{\displaystyle \frac{\pi }{2}}}^{{\displaystyle \frac{\pi }{2}}}{\displaystyle \frac{\partial }{\partial \psi }}\left(cos\psi {\displaystyle \frac{\partial \widehat{z}}{\partial \psi }}\right)cos\psi cos\theta Gd\psi \right)d\theta =\right.$$

$$=3{\int }_0^{2\pi }d\phi \left({\left.{\int }_{{\displaystyle \frac{\pi }{4}}-\delta }^{{\displaystyle \frac{\pi }{4}}+\delta }\left({cos}^2\psi cos\theta {\displaystyle \frac{\partial \widehat{z}}{\partial \psi }}G-zcos\psi {\displaystyle \frac{\partial }{\partial \psi }}(cos\psi cos\theta G)\right)\right|}_{\psi ={\displaystyle \frac{\pi }{4}}+\delta }^{\psi ={\displaystyle \frac{\pi }{4}}-\delta }d\theta \right)+$$

$$+3{\int }_0^{2\pi }d\phi \left(\left({\int }_{-{\displaystyle \frac{\pi }{2}}}^{{\displaystyle \frac{\pi }{4}}-\delta }+{\int }_{{\displaystyle \frac{\pi }{4}}+\delta }^{{\displaystyle \frac{\pi }{2}}}\right)\left({\left.{\int }_{-{\displaystyle \frac{\pi }{2}}}^{{\displaystyle \frac{\pi }{2}}}\left({cos}^2\psi cos\theta {\displaystyle \frac{\partial \widehat{z}}{\partial \psi }}G-zcos\psi {\displaystyle \frac{\partial }{\partial \psi }}(cos\psi cos\theta G)\right)\right|}_{\psi =-{\displaystyle \frac{\pi }{2}}}^{\psi ={\displaystyle \frac{\pi }{2}}}d\theta \right)+\right.$$

$$+3{\int }_{S^3\backslash {\Pi }_c\left(\delta \right)}z{\displaystyle \frac{1}{cos\psi }}{\displaystyle \frac{\partial }{\partial \psi }}\left(cos\psi {\displaystyle \frac{\partial G}{\partial \psi }}\right){cos}^2\psi cos\theta d\psi d\theta d\phi =$$

$$=3{\int }_0^{2\pi }d\phi \left({\left.{\int }_{{\displaystyle \frac{\pi }{4}}-\delta }^{{\displaystyle \frac{\pi }{4}}+\delta }\left({cos}^2\psi {\displaystyle \frac{\partial z}{\partial \psi }}G-z{cos}^2\psi {\displaystyle \frac{\partial G}{\partial \psi }}+zcos\psi sin\psi G\right)\right|}_{\psi ={\displaystyle \frac{\pi }{4}}+\delta }^{\psi ={\displaystyle \frac{\pi }{4}}-\delta }cos\theta d\theta \right)+$$

$$+3{\int }_0^{2\pi }d\phi \left({\left.\left({\int }_{-{\displaystyle \frac{\pi }{2}}}^{{\displaystyle \frac{\pi }{4}}-\delta }+{\int }_{{\displaystyle \frac{\pi }{4}}+\delta }^{{\displaystyle \frac{\pi }{2}}}\right)\left({cos}^2\psi {\displaystyle \frac{\partial z}{\partial \psi }}G-z{cos}^2\psi {\displaystyle \frac{\partial G}{\partial \psi }}+zcos\psi sin\psi G\right)\right|}_{\psi =-{\displaystyle \frac{\pi }{2}}}^{\psi ={\displaystyle \frac{\pi }{2}}}cos\theta d\theta \right)+$$

$$+3{\int }_{S^3\backslash {\Pi }_c\left(\delta \right)}z{\displaystyle \frac{1}{cos\psi }}{\displaystyle \frac{\partial }{\partial \psi }}\left(cos\psi {\displaystyle \frac{\partial G}{\partial \psi }}\right){cos}^2\psi cos\theta d\psi d\theta d\phi .$$

We also have the limiting relation:

$$\underset{\delta \to 0}{lim}{I}_{\psi }={\displaystyle \frac{3}{2}}{\int }_0^{2\pi }d\phi \underset{\delta \to 0}{lim}{\int }_{{\displaystyle \frac{\pi }{4}}-\delta }^{{\displaystyle \frac{\pi }{4}}+\delta }\left({\displaystyle \frac{\partial \widehat{z}\left(\frac{\pi }{4}-\delta ,\theta ,\phi \right)}{\partial \psi }}G\left({\psi }^{\prime },{\theta }^{\prime },{\phi }^{\prime };{\displaystyle \frac{\pi }{4}}-\delta ,\theta ,\phi \right)-\right.$$

$$\left.-\widehat{z}\left({\displaystyle \frac{\pi }{4}}-\delta ,\theta ,\phi \right){\displaystyle \frac{\partial G\left({\psi }^{\prime },{\theta }^{\prime },{\phi }^{\prime };\frac{\pi }{4}-\delta ,\theta ,\phi \right)}{\partial \psi }}+z\left({\displaystyle \frac{\pi }{4}}-\delta ,\theta ,\phi \right)G\left({\psi }^{\prime },{\theta }^{\prime },{\phi }^{\prime };{\displaystyle \frac{\pi }{4}}-\delta ,\theta ,\phi \right)\right)cos\theta d\theta -$$

$$-{\displaystyle \frac{3}{2}}{\int }_0^{2\pi }d\phi \underset{\delta \to 0}{lim}{\int }_{{\displaystyle \frac{\pi }{4}}-\delta }^{{\displaystyle \frac{\pi }{4}}+\delta }\left({\displaystyle \frac{\partial \widehat{z}\left(\frac{\pi }{4}+\delta ,\theta ,\phi \right)}{\partial \psi }}G\left({\psi }^{\prime },{\theta }^{\prime },{\phi }^{\prime };{\displaystyle \frac{\pi }{4}}+\delta ,\theta ,\phi \right)-\right.$$

$$\left.-\widehat{z}\left({\displaystyle \frac{\pi }{4}}+\delta ,\theta ,\phi \right){\displaystyle \frac{\partial G\left({\psi }^{\prime },{\theta }^{\prime },{\phi }^{\prime };\frac{\pi }{4}+\delta ,\theta ,\phi \right)}{\partial \psi }}+z\left({\displaystyle \frac{\pi }{4}}+\delta ,\theta ,\phi \right)G\left({\psi }^{\prime },{\theta }^{\prime },{\phi }^{\prime };{\displaystyle \frac{\pi }{4}}+\delta ,\theta ,\phi \right)\right)cos\theta d\theta +$$

$$+3{\int }_{S^3\backslash {\Pi }_c\left(\delta \right)}z{\displaystyle \frac{1}{cos\psi }}{\displaystyle \frac{\partial }{\partial \psi }}\left(cos\psi {\displaystyle \frac{\partial G}{\partial \psi }}\right){cos}^2\psi cos\theta d\psi d\theta d\phi =$$

$$={\displaystyle \frac{3}{2}}{\int }_0^{2\pi }G\left({\psi }^{\prime },{\theta }^{\prime },{\phi }^{\prime };{\displaystyle \frac{\pi }{4}},{\displaystyle \frac{\pi }{4}},\phi \right)d\phi \underset{\delta \to 0}{lim}{\int }_{{\displaystyle \frac{\pi }{4}}-\delta }^{{\displaystyle \frac{\pi }{4}}+\delta }\left({\displaystyle \frac{\partial \widehat{z}\left(\frac{\pi }{4}-\delta ,\theta ,\phi \right)}{\partial \psi }}+\right.$$

$$\left.+\widehat{z}\left({\displaystyle \frac{\pi }{4}}-\delta ,\theta ,\phi \right)-{\displaystyle \frac{\partial \widehat{z}\left(\frac{\pi }{4}+\delta ,\theta ,\phi \right)}{\partial \psi }}-z\left({\displaystyle \frac{\pi }{4}}+\delta ,\theta ,\phi \right)\right)cos\theta d\theta +$$

$$+{\displaystyle \frac{3}{2}}{\int }_0^{2\pi }{\displaystyle \frac{\partial G\left({\psi }^{\prime },{\theta }^{\prime },{\phi }^{\prime };\frac{\pi }{4},\frac{\pi }{4},\phi \right)}{\partial \psi }}d\phi \underset{\delta \to 0}{lim}{\int }_{{\displaystyle \frac{\pi }{4}}-0}^{{\displaystyle \frac{\pi }{4}}+\delta }\left(\widehat{z}\left({\displaystyle \frac{\pi }{4}}+\delta ,\theta ,\phi \right)-\widehat{z}\left({\displaystyle \frac{\pi }{4}}-\delta ,\theta ,\phi \right)\right)cos\theta d\theta +$$

$$+3{\int }_{S^3\backslash {\Pi }_c\left(\delta \right)}z{\displaystyle \frac{1}{cos\psi }}{\displaystyle \frac{\partial }{\partial \psi }}\left(cos\psi {\displaystyle \frac{\partial G}{\partial \psi }}\right){cos}^2\psi cos\theta d\psi d\theta d\phi .$$

It is necessary to transform the integral $I_{\phi }.$

$$I_{\phi }={\int }_{{\displaystyle \frac{\pi }{4}}-\delta }^{{\displaystyle \frac{\pi }{4}}+\delta }d\psi \left({\int }_{-{\displaystyle \frac{\pi }{2}}}^{{\displaystyle \frac{\pi }{4}}-\delta }+{\int }_{{\displaystyle \frac{\pi }{4}}+\delta }^{{\displaystyle \frac{\pi }{2}}}\right){\displaystyle \frac{d\theta }{cos\theta }}{\int }_0^{2\pi }{\displaystyle \frac{{\partial }^2\widehat{z}}{\partial {\phi }^2}}Gd\phi =$$

$$={\int }_{{\displaystyle \frac{\pi }{4}}-\delta }^{{\displaystyle \frac{\pi }{4}}+\delta }d\psi \left({\int }_{-{\displaystyle \frac{\pi }{2}}}^{{\displaystyle \frac{\pi }{4}}-\delta }+{\int }_{{\displaystyle \frac{\pi }{4}}+\delta }^{{\displaystyle \frac{\pi }{2}}}\right){\displaystyle \frac{d\theta }{cos\theta }}{\left({\displaystyle \frac{\partial \widehat{z}}{\partial \phi }}G\right|}_{\phi =0}^{\phi =2\pi }-{\int }_0^{2\pi }{\displaystyle \frac{\partial \widehat{z}}{\partial \phi }}{\displaystyle \frac{\partial G}{\partial \phi }}d\phi )=$$

$$={\left.{\int }_{{\displaystyle \frac{\pi }{4}}-\delta }^{{\displaystyle \frac{\pi }{4}}+\delta }d\psi \left({\int }_{-{\displaystyle \frac{\pi }{2}}}^{{\displaystyle \frac{\pi }{4}}-\delta }+{\int }_{{\displaystyle \frac{\pi }{4}}+\delta }^{{\displaystyle \frac{\pi }{2}}}\right){\displaystyle \frac{d\theta }{cos\theta }}\left({\displaystyle \frac{\partial \widehat{z}}{\partial \phi }}G-\widehat{z}{\displaystyle \frac{\partial G}{\partial \phi }}\right)\right|}_{\phi =0}^{\phi =2\pi }+$$

$$+3{\int }_{S^3\backslash {\Pi }_c\left(\delta \right)}z{\displaystyle \frac{1}{{cos}^2\psi {cos}^2\theta }}{\displaystyle \frac{{\partial }^{2}G}{\partial {\phi }^2}}{cos}^2\psi cos\theta d\psi d\theta d\phi .$$

Let us write down the limit for the integral:

$$\underset{\delta \to 0}{lim}{I}_{\phi }=3\underset{\delta \to 0}{lim}{\int }_{S^3\backslash {\Pi }_c\left(\delta \right)}{\displaystyle \frac{1}{{cos}^2\psi {cos}^2\theta }}{\displaystyle \frac{{\partial }^{2}G\left({\psi }^{\prime },\theta ,{\phi }^{\prime };\psi ,\theta \phi \right)}{\partial {\phi }^2}}{cos}^2\psi cos\theta d\psi d\theta d\phi .$$

Thus, we obtain the value $T\left({\psi }^{\prime },{\theta }^{\prime },{\phi }^{\prime }\right)$:

$$\begin{array}{cc}\hfill & T\left({\psi }^{\prime },{\theta }^{\prime },{\phi }^{\prime }\right)={\int }_{S^3\backslash C}\widehat{z}\left({\psi }^{\prime },{\theta }^{\prime },{\phi }^{\prime };\psi ,\theta ,\phi \right)\Delta G\left({\psi }^{\prime },{\theta }^{\prime },{\phi }^{\prime };\psi ,\theta ,\phi \right)dV+\hfill \\ \hfill & +{\displaystyle \frac{3}{2}}{\int }_0^{2\pi }G\left({\psi }^{\prime },{\theta }^{\prime },{\phi }^{\prime };{\displaystyle \frac{\pi }{4}},{\displaystyle \frac{\pi }{4}},\phi \right)\left(K_{1}z\right)\left(\phi \right)d\phi +\hfill \\ \hfill & +{\displaystyle \frac{3}{2}}{\int }_0^{2\pi }{\displaystyle \frac{\partial G\left({\psi }^{\prime },{\theta }^{\prime },{\phi }^{\prime };\frac{\pi }{4},\frac{\pi }{4},\phi \right)}{\partial \psi }}\left(K_{2}z\right)\left(\phi \right)d\phi -\hfill \\ \hfill & -{\displaystyle \frac{3\sqrt{2}}{2}}{\int }_0^{2\pi }{\displaystyle \frac{\partial G\left({\psi }^{\prime },{\theta }^{\prime },{\phi }^{\prime };\frac{\pi }{4},\frac{\pi }{4},\phi \right)}{\partial \theta }}\left(K_{3}z\right)\left(\phi \right)d\phi .\hfill \end{array}$$

(10)

This implies a statement that describes the structure of the elements for the introduced space.

Lemma 2. 

For any function $z\in W_{2,K}^2\left(S_c^3\right)$, the representation holds

$$\begin{array}{cc}\hfill & z\left({\psi }^{\prime },{\theta }^{\prime },{\phi }^{\prime },\psi ,\theta ,\phi \right)=T\left({\psi }^{\prime },{\theta }^{\prime },{\phi }^{\prime }\right)-{\displaystyle \frac{3}{2}}{\int }_0^{2\pi }G\left({\psi }^{\prime },{\theta }^{\prime },{\phi }^{\prime },{\displaystyle \frac{\pi }{4}},{\displaystyle \frac{\pi }{4}},\phi \right)\left(K_{1}z\right)\left(\phi \right)d\phi -\hfill \\ \hfill & -{\displaystyle \frac{3}{2}}{\int }_0^{2\pi }{\displaystyle \frac{\partial G\left({\psi }^{\prime },{\theta }^{\prime },{\phi }^{\prime };\frac{\pi }{4},\frac{\pi }{4},\phi \right)}{\partial \psi }}\left(K_{2}z\right)\left(\phi \right)d\phi +\hfill \\ \hfill & +{\displaystyle \frac{3\sqrt{2}}{2}}{\int }_0^{2\pi }{\displaystyle \frac{\partial G\left({\psi }^{\prime },{\theta }^{\prime },{\phi }^{\prime };\frac{\pi }{4},\frac{\pi }{4},\phi \right)}{\partial \theta }}\left(K_{3}z\right)\left(\phi \right)d\phi ,\hfill \end{array}$$

(11)

where $T\in W_2^2\left(S^3\right).$ The representation (11) is unique.

Since the representation (11) is unique, we can introduce the operator J, which assigns to each element $z\in W_{2K}^2$ the element $T\in W_2^2\left(S^3\right)$, that is, $T=Jz$. Using the operator $J,$ we define the operator $B_{max}$ by the formulas

$$B_{max}z=B_{0}Jz,$$

where $B_0$ is the Laplace–Beltrami operator on $S^3$.

Proof. 

Substituting the relation (9) into the equality (10):

$$T=z\left({\psi }^{\prime },{\theta }^{\prime },{\phi }^{\prime };\psi ,\theta ,\phi \right)-{\displaystyle \frac{1}{{\omega }_4}}{\int }_{S^3}zdV+{\displaystyle \frac{3}{2}}{\int }_0^{2\pi }G\left({\psi }^{\prime },{\theta }^{\prime },{\phi }^{\prime };{\displaystyle \frac{\pi }{4}},{\displaystyle \frac{\pi }{4}},\phi \right)\left(K_{1}z\right)\left(\phi \right)d\phi +$$

$$+{\displaystyle \frac{3}{2}}{\int }_0^{2\pi }{\displaystyle \frac{\partial G\left({\psi }^{\prime },{\theta }^{\prime },{\phi }^{\prime };\frac{\pi }{4},\frac{\pi }{4},\phi \right)}{\partial \psi }}\left(K_{2}z\right)\left(\phi \right)d\phi -$$

$$-{\displaystyle \frac{3\sqrt{2}}{2}}{\int }_0^{2\pi }{\displaystyle \frac{\partial G\left({\psi }^{\prime },{\theta }^{\prime },{\phi }^{\prime };\frac{\pi }{4},\frac{\pi }{4},\phi \right)}{\partial \theta }}\left(K_{3}z\right)\left(\phi \right)d\phi .$$

Since $z\in W_{2k}$, that ${\int }_{S^3}zdV=0$. □

The following statement plays an important role in further constructions.

Theorem 2. 

For any $f\in L_2\left(S^3\right)$ and arbitrary ${\eta }_1,{\eta }_2,{\eta }_3\in L_2\left(C\right)$ the problem

$$\begin{array}{c}{B}_{max}z=f\left(x\right),x\in S^3\backslash C,\\ \left(K_{i}z\right){{|}_C={\eta }_i|}_C,i=1,2,3,\end{array}$$

has a unique solution in the class $W_{2K}^2(S^3\backslash C)$.

Let us introduce the operators ${\gamma }_1,{\gamma }_2,{\gamma }_3$ in the form of the following integral operators for $\forall f\in L_2\left(S^3\right)$

$$\left({\gamma }_{i}f\right)\left(\phi \right)=3{\int }_{-{\displaystyle \frac{\pi }{2}}}^{{\displaystyle \frac{\pi }{2}}}{\int }_{-{\displaystyle \frac{\pi }{2}}}^{{\displaystyle \frac{\pi }{2}}}{\alpha }_{i}f{cos}^2\psi cos\theta d\psi d\theta ,i=1, 2, 3,$$

(12)

where ${\alpha }_i(\psi ,\theta )$ is an arbitrary element of $L_2\left({(-{\displaystyle \frac{\pi }{2}},{\displaystyle \frac{\pi }{2}})}^2\right)$.

In this case, the main result of this paper follows from Theorem 1.

Remark 2. 

The characterization of $ker{B}_{max}$ given in Lemma is independent of the specific choice of the removed curve C, provided that C is a smooth closed geodesic on $S^3$. In particular, the same conclusion holds for any such C, even though the explicit form of the restriction ${G|}_C$ will depend on its position in $S^3$. This invariance reflects the underlying symmetry of the Laplace–Beltrami operator on the sphere and will be used in the proof of Theorem 1.

Lemma 2. 

Let $S^3\backslash C$ denote the three-dimensional sphere with a smooth closed curve C removed. Let $G(·,·)$ be the Green’s function of the Laplace–Beltrami operator on $S^3$ with respect to a fixed point outside C. Then, the kernel of the associated boundary integral operator on C is generated by the restriction of G to C, that is,

$$ker{B}_{max}=\mathrm{span}\left\{{G|}_C\right\}.$$

Proof. 

The boundary operator $B_{max}$ is defined via the limiting values of the single-layer potential associated with the Laplace–Beltrami operator. By construction, any function u in $ker{B}_{max}$ corresponds to a harmonic function on $S^3\backslash C$ whose trace on C satisfies the homogeneous boundary condition encoded in $B_{max}$. The representation of u via Green’s function shows that its trace must be proportional to ${G|}_C$. Conversely, ${G|}_C$ clearly satisfies the boundary condition, and hence belongs to the kernel. Therefore, the kernel is exactly the one-dimensional space spanned by ${G|}_C$. □

Theorem 2. 

Let the function ${\alpha }_i(\psi ,\theta )$ belong to the space $L_2\left({(-{\displaystyle \frac{\pi }{2}},{\displaystyle \frac{\pi }{2}})}^2\right)$ for $i=1,2,3$, and operators ${\gamma }_i$ be determined by the formulas (12). Then, for any $f\in L_2\left(S^3\right)$ the following problem

$$\begin{array}{cc}& B_{max}z=f\left(x\right),x\in S^3\backslash C,\hfill \\ & \left(K_{i}z\right)\left(\phi \right)={\gamma }_i\left(B_{max}z\right),i=1,2,3,\hfill \end{array}$$

has a unique solution in the space $W_{2K}^2\left(S^3\backslash C\right).$

The proof of Theorems 1 and 2 is carried out in exactly the same way as Theorem 2, which was proven in the work [5].

Example 2. 

Let ${\alpha }_i(\psi ,\theta ),$ be a sufficiently smooth function that satisfies the equality

$$3{\int }_{-{\displaystyle \frac{\pi }{2}}}^{{\displaystyle \frac{\pi }{2}}}{\int }_{-{\displaystyle \frac{\pi }{2}}}^{{\displaystyle \frac{\pi }{2}}}\left({\displaystyle \frac{\partial }{\partial \psi }}\left(cos\psi {\displaystyle \frac{\partial {\alpha }_i}{\partial \psi }}\right)cos\psi cos\theta +{\displaystyle \frac{\partial }{\partial \theta }}\left(cos\theta {\displaystyle \frac{\partial {\alpha }_i}{\partial \theta }}\right)\right)d\psi d\theta =0.$$

(13)

Let us write down the boundary value problem

$$3{\int }_{-{\displaystyle \frac{\pi }{2}}}^{{\displaystyle \frac{\pi }{2}}}{\int }_{-{\displaystyle \frac{\pi }{2}}}^{{\displaystyle \frac{\pi }{2}}}{\alpha }_i(\psi ,\theta )\Delta Jz{cos}^2\psi cos\theta d\psi d\theta =$$

$$={\left.3{\int }_{-{\displaystyle \frac{\pi }{2}}}^{{\displaystyle \frac{\pi }{2}}}\left({\alpha }_i{cos}^2\psi cos\theta {\displaystyle \frac{\partial Jz}{\partial \psi }}-Jzcos\psi {\displaystyle \frac{\partial }{\partial \psi }}\left({\alpha }_{i}cos\psi cos\theta \right)\right)\right|}_{\psi =-{\displaystyle \frac{\pi }{2}}}^{\psi ={\displaystyle \frac{\pi }{2}}}d\theta +$$

$$+3{\int }_{-{\displaystyle \frac{\pi }{2}}}^{{\displaystyle \frac{\pi }{2}}}{\int }_{-{\displaystyle \frac{\pi }{2}}}^{{\displaystyle \frac{\pi }{2}}}Jz{\displaystyle \frac{\partial }{\partial \psi }}\left(cos\psi {\displaystyle \frac{\partial {\alpha }_i}{\partial \psi }}\right)cos\psi cos\theta d\psi d\theta +{\left.3{\int }_{-{\displaystyle \frac{\pi }{2}}}^{{\displaystyle \frac{\pi }{2}}}\left({\alpha }_{i}cos\theta {\displaystyle \frac{\partial Jz}{\partial \theta }}-Jzcos\theta {\displaystyle \frac{\partial {\alpha }_i}{\partial \theta }}\right)\right|}_{\theta =-{\displaystyle \frac{\pi }{2}}}^{\theta ={\displaystyle \frac{\pi }{2}}}d\psi +$$

$$+3{\int }_{-{\displaystyle \frac{\pi }{2}}}^{{\displaystyle \frac{\pi }{2}}}{\int }_{-{\displaystyle \frac{\pi }{2}}}^{{\displaystyle \frac{\pi }{2}}}Jz{\displaystyle \frac{\partial }{\partial \theta }}\left(cos\theta {\displaystyle \frac{\partial {\alpha }_i}{\partial \theta }}\right)d\psi d\theta +3{\int }_{-{\displaystyle \frac{\pi }{2}}}^{{\displaystyle \frac{\pi }{2}}}{\int }_{-{\displaystyle \frac{\pi }{2}}}^{{\displaystyle \frac{\pi }{2}}}{\alpha }_i{\displaystyle \frac{1}{cos\theta }}{\displaystyle \frac{{\partial }^{2}Jz}{\partial {\phi }^2}}d\psi d\theta .$$

Taking into account the conditions (13), the following relation follows:

$${\int }_{-{\displaystyle \frac{\pi }{2}}}^{{\displaystyle \frac{\pi }{2}}}{\int }_{-{\displaystyle \frac{\pi }{2}}}^{{\displaystyle \frac{\pi }{2}}}{\alpha }_i(\psi ,\theta )\Delta Jz{cos}^2\psi cos\theta d\psi d\theta ={\int }_{-{\displaystyle \frac{\pi }{2}}}^{{\displaystyle \frac{\pi }{2}}}{\int }_{-{\displaystyle \frac{\pi }{2}}}^{{\displaystyle \frac{\pi }{2}}}{\alpha }_i{\displaystyle \frac{1}{cos\theta }}{\displaystyle \frac{{\partial }^{2}Jz}{\partial {\phi }^2}}d\psi d\theta .$$

Thus, Theorem 2 implies the correct solvability of the following boundary value problem:

$$\begin{array}{cc}& B_{max}z=f\left(x\right),x\in S^3\backslash C,\hfill \\ & \left(K_{i}z\right)\left(\phi \right)={\int }_{-{\displaystyle \frac{\pi }{2}}}^{{\displaystyle \frac{\pi }{2}}}{\int }_{-{\displaystyle \frac{\pi }{2}}}^{{\displaystyle \frac{\pi }{2}}}{\alpha }_i{\displaystyle \frac{1}{cos\theta }}{\displaystyle \frac{{\partial }^{2}Jz}{\partial {\phi }^2}}d\psi d\theta .\hfill \end{array}$$

In the conclusion of this paper, we write out a theorem for the boundary value problem.

Theorem 2. 

For $\forall {\alpha }_i\in W_2^2\left({(-{\displaystyle \frac{\pi }{2}},{\displaystyle \frac{\pi }{2}})}^2\right)$, the equality holds

$${\int }_{-{\displaystyle \frac{\pi }{2}}}^{{\displaystyle \frac{\pi }{2}}}{\int }_{-{\displaystyle \frac{\pi }{2}}}^{{\displaystyle \frac{\pi }{2}}}\left({\displaystyle \frac{\partial }{\partial \psi }}\left(cos\psi {\displaystyle \frac{\partial {\alpha }_i}{\partial \psi }}\right)cos\psi cos\theta +{\displaystyle \frac{\partial }{\partial \theta }}\left(cos\theta {\displaystyle \frac{\partial {\alpha }_i}{\partial \theta }}\right)\right)d\psi d\theta =0.$$

Then, for $\forall f\in L_2\left(S^3\right)$ the right-hand problem

$$\begin{array}{cc}& B_{max}z=f\left(x\right),x\in S^3\backslash C,\hfill \\ & \left(K_{i}z\right)\left(\phi \right)={\int }_{-{\displaystyle \frac{\pi }{2}}}^{{\displaystyle \frac{\pi }{2}}}{\int }_{-{\displaystyle \frac{\pi }{2}}}^{{\displaystyle \frac{\pi }{2}}}{\alpha }_i{\displaystyle \frac{1}{cos\theta }}{\displaystyle \frac{{\partial }^{2}Jz}{\partial {\phi }^2}}d\psi d\theta .\hfill \end{array}$$

has a unique solution.

2.3. Spectral Properties and Symmetry Aspects

Let

$$M=S^3\backslash C$$

denote the three-dimensional sphere embedded in ${\mathbb{R}}^4$ with a smooth circular cut C. The Laplace–Beltrami operator ${\Delta }_{S^3}$ acts on complex-valued functions $u:M\to \mathbb{C}$ as

$${\Delta }_{S^3}u={\displaystyle \frac{1}{\sqrt{\left|g\right|}}}{\partial }_i\left(\sqrt{\left|g\right|}{g}^{ij}{\partial }_{j}u\right),$$

(14)

where g is the induced metric on $S^3$.

Due to the cut C, the manifold M is disconnected along C, requiring additional boundary conditions to specify the behavior of u across the cut. We define the domain of the operator as

$$\begin{array}{cc}\hfill D\left({\Delta }_{S^3}\right)=\{& u\in H^2(M\backslash C)|{\left[u\right]}_C=\alpha u{{|}_{C^{-}}+\beta u|}_{C^{+}}=0,\hfill \\ \hfill & {\left[{\displaystyle \frac{\partial u}{\partial n}}\right]}_C=\mu {\displaystyle \frac{\partial u}{\partial n}}{|}_{C^{-}}+\nu {\displaystyle \frac{\partial u}{\partial n}}{|}_{C^{+}}=0\},\hfill \end{array}$$

(15)

where $\alpha ,\beta ,\mu ,\nu \in \mathbb{C}$ are complex coupling parameters, and $C^{-},C^{+}$ denote the two sides of the cut.

Unless the parameters satisfy certain conjugation relations (e.g., $\alpha =\overline{\beta }$, $\mu =\overline{\nu }$), the operator ${\Delta }_{S^3}$ is non-self-adjoint. The well-posedness of the boundary value problem

$${\Delta }_{S^3}u=f,u\in D\left({\Delta }_{S^3}\right),$$

(16)

is established by verifying that ${\Delta }_{S^3}$ is closed and possesses a bounded inverse on its range under the given jump conditions.

The formulation parallels approaches in [11] where boundary conditions are treated within the framework of quadratic forms and symmetry considerations.

The spectrum $\sigma \left({\Delta }_{S^3}\right)$ depends sensitively on the jump conditions. In general, we expect a mixture of discrete and continuous spectral components: (1) For localized eigenfunctions away from C, eigenvalues ${\left\{{\lambda }_n\right\}}_{n=1}^{\infty }$ accumulate at $+\infty$, resembling the unperturbed Laplace–Beltrami spectrum on $S^3$ but shifted by boundary effects. (2) Certain coupling parameters allow wave-like modes along the cut, contributing a continuous part to the spectrum.

Example 2. 

For the boundary conditions

$${\left[u\right]}_C=0,{\left[{\displaystyle \frac{\partial u}{\partial n}}\right]}_C{=\kappa u|}_C,\kappa \in \mathbb{R},$$

(17)

separation of variables in hyperspherical coordinates $(\chi ,\theta ,\phi )$ yields eigenfunctions of the form

$$u_{lmn}(\chi ,\theta ,\phi )=R_{ln}\left(\chi \right)Y_{lm}(\theta ,\phi ),$$

(18)

where $Y_{lm}$ are spherical harmonics on $S^2$ and $R_{ln}\left(\chi \right)$ satisfy a Sturm–Liouville problem with modified boundary conditions at C. The eigenvalues ${\lambda }_{ln}\left(\kappa \right)$ shift continuously with κ, illustrating the spectral sensitivity to the defect.

Although the circular cut breaks the full $\mathrm{SO}\left(4\right)$ symmetry of $S^3$, it preserves axial symmetry under the subgroup $\mathrm{SO}\left(2\right)\times \mathrm{SO}\left(2\right)$ corresponding to rotations that leave C invariant. This residual symmetry constrains the structure of eigenfunctions and is closely related to a separation of variables.

Similar symmetry-driven analyses appear in works [11,12,13].

These works show that even partial symmetries can be leveraged to classify solutions and analyze spectral behavior, underscoring the relevance of our problem to the symmetry readership.

3. Conclusions

This paper studies delta-like perturbations of the Laplace–Beltrami operator on a three-dimensional sphere. The paper considers the three-dimensional sphere ${\mathbb{R}}^4$ as a Riemannian manifold. Under setting correct problems, the previously known fundamental solution for the Laplace–Beltrami operator is used. A correct problem is posed for the Laplace–Beltrami operator on a three-dimensional sphere in a four-dimensional Euclidean space with a cut, which represents a circle. An example is given that illustrates the structure of a correct problem for the Laplace–Beltrami operator on a three-dimensional sphere from which a circle has been removed. In the future, it is necessary to study the spectral properties of operators corresponding to the stated correctly solvable problems for the Laplace–Beltrami operator on a sphere with a cut along a manifold of small dimensions.