Consider this differential operator $$\mathcal{H}(\phi(\mathbf{x})) = -\triangle + V(\mathbf{x})H_\epsilon (\phi(\mathbf{x}))$$ where $\mathbf{x} \in \mathbb{R}^2$, $\phi : \mathbb{R}^2 \rightarrow \mathbb{R}^2$, and $H_\epsilon$ is a relaxed version of the Heaviside step function.

I am interested in studying spectra of $\mathcal{H}$ as $\phi$ varies but am not sure about the definitions. Suppose $(v,\lambda)$ is an eigenpair of $\mathcal{H}$, I want to know $\nabla v$ and $\nabla \lambda$.

For example, I compute $\nabla \lambda$ as $$\mathcal{H}v = \lambda v$$ $$\implies \nabla (\mathcal{H}v) = \nabla (\lambda v)$$ $$\implies \nabla \mathcal{H} v + \mathcal{H} \nabla v = \nabla \lambda v + \lambda \nabla v$$ $$\implies \nabla \lambda = v^T \nabla \mathcal{H} v$$ $$\implies \nabla \lambda = v^T V(\mathbf{x}) \delta_\epsilon(\phi(\mathbf{x})) v$$

which is weird since $\nabla \lambda$ is a number and I don't know that I computed $\nabla \mathcal{H}$ properly. It's the same way Terry does the computation in the single variable case (cf http://terrytao.wordpress.com/2008/10/28/when-are-eigenvalues-stable/ ).

Any advice? I hope to use $\phi$ in a level set method for domain decomposition.

Edit: Restrict everything to the discrete case (ie $\mathcal{H}$ is a matrix, like in the link above) and I still don't know how to do this. I guess the concise way to state my question is "Given a matrix $\mathcal{H}(\phi)$ does anyone know a formula for the first variation of the eigenvalues or eigenvectors of $\mathcal{H}$?"

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In Terry Tao's computation, he's differentiating a <strong>matrix</strong> with coefficients depending on a variable t. You're quite more ambitious here, since you plan to differentiate an unbounded operator depending on two variables. I guess it would help you to read some chapters of Kato's book on the perturbation theory of operators to get a feeling for the difficulties involved – Piero D'Ancona Sep 20 '11 at 11:50
Yes. To be honest all I really need is the answer when $\mathcal{H}$ is a matrix as I am coding this. Do you have any advice in the matrix case? – dranxo Sep 20 '11 at 21:12