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

We consider the boundary value problem for the deflection of a finite beam on an elastic foundation subject to vertical loading. We construct a one-to-one correspondence $\Gamma$ from the set of equivalent well-posed two-point boundary conditions to $\mathrm{gl}(4,\mathbb{C})$. Using $\Gamma$, we derive eigenconditions for the integral operator $\mathcal{K}_\mathbf{M}$ for each well-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 well understood. Using our eigenconditions, we show that, for each nonzero real $\lambda \not \in \mathrm{Spec}\,\mathcal{K}_{l,\alpha,k}$, there exists a real well-posed boundary condition $\mathbf{M}$ such that $\lambda \in \mathrm{Spec}\,\mathcal{K}_\mathbf{M}$. This in particular shows that the integral operators $\mathcal{K}_\mathbf{M}$ arising from well-posed boundary conditions, may not be positive nor contractive in general, as opposed to $\mathcal{K}_{l,\alpha,k}$.
Comment: Changes: 1. $\mathbf{Y}_\lambda(x)$ are defined for $\lambda$ and $x$ such that $\det\mathbf{X}_\lambda(x) \neq 0$. 2. The boundary condition for infinitely long beam in Introduction. 3. Some sign corrections in (B.9). 4. Inclusion of the new item [4] in References