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Solvable models in quantum mechanics

Is there anyone who studied on the book "Solvable Models In Quantum Mechanics" by Albeverio? I don't succed in understanding the proof of page 116 about the eigenvalues of the Hamiltonian with point interactions. In particular I don't understand the proof of the fact that eigenvalues conrespond to the zero of the determinant of the matrix of the Hamiltonian with N point interxations: $$A(k)=\bigg[\bigg(\alpha_j-\frac{ik}{4\pi}\bigg)\delta_{jj'}-\tilde{f}(y_j-y_{j'})\bigg]_{jj'}$$ where $\tilde{f}(x)$ is $\tilde{f}(x)=\frac{e^{ik|x|}}{4\pi|x|}$ se $x\neq 0$ and $0$