# Reference request Eigenspace decomposition Hodge Laplacian on forms on manifolds with boundary

I know that on a connected, compact, oriented Riemannian manifold without boundary the Hodge Laplacian $$\Delta_k=(d+\delta)^2$$ (acting on $$k$$-forms) admits an orthogonal eigenspace decomomposition of $$L^2\Lambda^k(M)$$, consisting of smooth eigenforms, where all eigenvalues have finite multiplicity, are non-negative and only accumulate at infinity. A reference for this is for instance:

The Laplacian on a Riemannian manifold: An Introduction to Analysis on Manifolds (1997), Cambridge University Press. By Steven Rosenberg [Theorem $$1.30$$]

I was wondering whether the same holds true for compact manifolds with boundary, provided we impose some appropriate boundary conditions. Rosenberg deals first with the case of $$0$$-forms before proving the general result. He mentions after the $$\Lambda^0(M)$$ case, that we may allow boundary if we impose Dirichlet or Neumann boundary conditions. I also found references for this case. However he does not say anything about a similar result for general $$k$$-forms on manifolds with boundary.

So my question is, do we also have an orthogonal eigenspace decomposition of $$L^2\Lambda^k(M)$$ in the case of compact manifolds with boundary (under some appropriate boundary condtions, e.g. vanishing normal component)?

So far I was not able to find any reference for such a result. If anyone is aware of a paper or book dealing with this problem I would highly appretiate it.