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 $\mathrm{\Gamma }$ from the set of equivalent well-posed two-point boundary conditions to $\mathrm{g}\mathrm{l}(4,\mathbb{C})$. Using $\mathrm{\Gamma }$, we derive eigenconditions for the integral operator ${\mathcal{K}}_{\mathbf{M}}$ for each well-posed two-point boundary condition represented by $$\mathbf{M}$ \in $\mathrm{g}\mathrm{l}$(4,8,$\mathbb{C}$)$. Special features of our eigenconditions include; (1) they isolate the effect of the boundary condition $\mathbf{M}$ on $\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}{\mathcal{K}}_{\mathbf{M}}$, (2) they connect $\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}{\mathcal{K}}_{\mathbf{M}}$ to $\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}{\mathcal{K}}_{l,\alpha ,k}$ whose structure has been well understood. Using our eigenconditions, we show that, for each nonzero real $\lambda \in ̸\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}{\mathcal{K}}_{l,\alpha ,k}$, there exists a real well-posed boundary condition $\mathbf{M}$ such that $\lambda \in $\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}$,$\mathcal{K}$_$\mathbf{M}$$. 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 ${\mathcal{K}}_{l,\alpha ,k}$.