Convergence of Eigenvalues and Eigenvectors for Uniformly Form-Bounded Operators

Suppose that $ A$ is an operator on a dense domain $ D(A)\subset L^2$ with compact resolvent, and with quadratic form $ q(f,g):=\langle f,Ag\rangle$ .


Let $ (r_n)_{n\in\mathbb N}$ be a sequence of quadratic forms on $ D(A)$ that are uniformly form-bounded by $ A$ , in the sense that there exists $ 0<\alpha<1$ and $ \beta>0$ independent of $ n$ such that $ $ |r_n(f,f)|\leq \alpha\cdot q(f,f)+\beta\cdot\|f\|_2,\qquad f\in D(A).$ $ Since $ \alpha<1$ the operators $ A_n$ defined by the form $ q+r_n$ all have compact resolvent.


Suppose that the $ r_n$ have a limit $ r_\infty$ , in the sense that $ $ \lim_{n\to\infty}r_n(f,f)\to r_\infty(f,f)$ $ for every $ f\in D(A)$ . Define the operator $ A_\infty$ through the form $ q+r_\infty$ . Clearly $ $ |r_\infty(f,f)|\leq \alpha\cdot q(f,f)+\beta\cdot\|f\|_2,\qquad f\in D(A),$ $ and thus $ A_\infty$ has compact resolvent as well.

Question. Does the uniform form-bound of the $ r_n$ give rise to a dominated convergence-type result for the spectrum of $ A_n$ , that is, the eigenvalues of $ A_n$ converge to that of $ A_\infty$ , and the eigenfunctions converge in $ L^2$ ?

Inaccurate zero eigenvalues for 7*7 matrix

I have symbolic entries for all the elements of a 7*7 matrix. At the symbolic level Eigenvalues gives two zeroes and five others that are extremely complicated. At the same time Det evaluates to exactly zero.

I take this to mean that, no matter what values the symbols within the matrix are assigned, I will have at least two zero eigenvalues. Exactly zero eigenvalues.

However, when I assign values to the symbols by hand, for eg., a=10; b=50, and so on, and evaluate the same codes, Eigenvalues evaluates to give two eigenvalues of the order of 10^-12. For my purpose, this magnitude cannot be treated as a zero. And I am also in need of a different eigenvalue of the same matrix which is very small, of this order or even smaller, but is not exactly zero. So I need the zeroes to show up much more accurately.

I have tried adding $ MinPrecision=50 to my code before executing the rest of it. It does not help at all.

A similar question was posted, Is there a good way to check, whether a small value produced numerically is a symbolic zero? However I could not decipher anything of use out of it.