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We consider the boundary value problem for the deflection of a finite beam onan elastic foundation subject to vertical loading. We construct a one-to-onecorrespondence $\Gamma$ from the set of equivalent well-posed two-pointboundary conditions to $\mathrm{gl}(4,\mathbb{C})$. Using $\Gamma$, we deriveeigenconditions for the integral operator $\mathcal{K}_\mathbf{M}$ for eachwell-posed two-point boundary condition represented by $\mathbf{M} \in\mathrm{gl}(4,8,\mathbb{C})$. Special features of our eigenconditions include;(1) they isolate the effect of the boundary condition $\mathbf{M}$ on$\mathrm{Spec}\,\mathcal{K}_\mathbf{M}$, (2) they connect$\mathrm{Spec}\,\mathcal{K}_\mathbf{M}$ to$\mathrm{Spec}\,\mathcal{K}_{l,\alpha,k}$ whose structure has been wellunderstood. Using our eigenconditions, we show that, for each nonzero real$\lambda \not \in \mathrm{Spec}\,\mathcal{K}_{l,\alpha,k}$, there exists a realwell-posed boundary condition $\mathbf{M}$ such that $\lambda \in\mathrm{Spec}\,\mathcal{K}_\mathbf{M}$. This in particular shows that theintegral operators $\mathcal{K}_\mathbf{M}$ arising from well-posed boundaryconditions, may not be positive nor contractive in general, as opposed to$\mathcal{K}_{l,\alpha,k}$.

Spectral analysis for the class of integral operators arising from  well-posed boundary value problems of finite beam deflection on elastic  foundation: characteristic equation