Spectral numbers and manifolds with boundary

We consider a smooth submanifold $N$ with a smooth boundary in an ambient closed manifold $M$ and assign a spectral invariant $c(\alpha ,H)$ to every singular homological class $\alpha \in H_{\ast }(N)$ and a Hamiltonian $H$ defined on the cotangent bundle $T^{\ast }M$. We also derive certain properties of spectral numbers, for example we prove that spectral invariants $c_{\pm }(H,N)$ associated to the whole Floer homology $H{F}_{\ast }(H,N:M)$ of the submanifold $N$, are limits of the decreasing nested family of open sets.