THE SPECTRAL PROBLEM OF THE LAPLACE OPERATOR ON A CIRCLE
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This paper investigates the spectral problem of the Laplace operator defined on a circle. The problem is formulated on a one-dimensional compact smooth manifold under periodic boundary conditions. It is shown that the Laplace operator is a self-adjoint and positive operator with a purely discrete spectrum. The eigenvalues are explicitly determined and form an unbounded increasing sequence, while the corresponding eigenfunctions are represented by trigonometric functions. These eigenfunctions constitute a complete orthonormal basis in the Hilbert space establishing a direct connection between the spectral decomposition of the Laplace operator and classical Fourier analysis. In addition, the asymptotic behavior of the eigenvalues is analyzed and shown to be consistent with general spectral laws for compact manifolds. The results obtained provide a clear and rigorous framework for understanding fundamental spectral properties and have important applications in mathematical physics, particularly in quantum mechanics and heat conduction models.
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