Spurious eigensolutions in the boundary integral equation (BIE) or boundary element method (BEM) for doublyconnected domain problems, the eccentric and annular membranes, are studied analytically and numerically in this paper. For the mathematical analysis, we employ the null-field integral equation, the degenerate kernels and the Fourier series to prove the existence of spurious eigensolutions in the continuous system. Examples of eccentric case, annular membrane and general shape are solved by using the null-field equation approach and BEM, respectively. Based on the numerical experiment, computer-assisted proof for the existence of spurious eigenvalues in companion with the trivial outer boundary data is given. The spurious eigenvalue is found to be the true eigenvalue of a circular membrane with the inner radius. The SVD structure for the four influence matrices is examined. Also, the trivial outer boundary densities are found in case of spurious eigenvalues and are shown in the bar chart.

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