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This was probably done somewhere 100 times, but I can't find a reference.

Assume that we have a bounded star-shaped domain $\Omega\subset \mathbb{R}^n$ with piece-wise smooth boundary and a general second elliptic operator $P$ with smooth coefficients. I can shrink my domain to a point and get this way a family of domains $\Omega_t$ parametrised by the interval [0,1] such that $\Omega_0$ is just a point $x\in \Omega$ and $\Omega_1 = \Omega$. Let $P_t$ be the restriction of $P$ to $\Omega_t$, $t\in(0,1]$ with Dirichlet or Neumann boundary conditions. I would like to understand if it is possible to obtain assymptotics of the eigenvalues of $P_t$. A formal expansion seems to suggest that $$ \lambda_t \sim \frac{\lambda}{t^2}, $$ where $\lambda$ is an eigenvalue of an operator $\tilde P$ obtained as the Fourier transform of the principal symbol of $P$ at the point $p$ (essentially just freezing the coefficients and dropping low order terms) with Dirichlet (or Neumann) boundary conditions.

Could someone give me a reference or a justification why this is true or not, and how to possibly prove it rigorously? Thank you!

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    $\begingroup$ For a general operator, you could have a problem if it is not self adjoint. For self adjoint operator, You have Weyl's Lemma which tells you that thanks to the min-max, there exists a positive constant $C$ such that for all $n$, the $n$-th eigenvalue of your operator is bounded above and below by the $n$-th eigenvalue of the laplacian: $$C\lambda_n(\Delta)\leq \lambda_n(P)\leq \frac1C \lambda_n(\Delta) $$ . That gives you the order of magntitude. Once you have this, you can freeze/make asymptotics. $\endgroup$
    – username
    Commented Jan 15, 2022 at 16:26

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