Consider Hilbert spaces $V$,$H$; a closed quadratic form $a$ with domain $V$; and its associated operator $A$ on $H$. (If necessary, the form can be assumed to be coercive.) For the sake of simplicity, assume the embedding of $V$ into $H$ to be compact (even trace class, if necessary), so that $A$ has purely point (real) spectrum.


Assume the operator $A$ to have a Weyl-type spectral asymptotics.


Now, take a new quadratic form $b$ which is in some sense small: form bounded or form compact, for example, so that $a+b$ still is a closed quadratic form. Are any conditions on $b$ available that would ensure the operator associated with $a+b$ to have Weyl-type spectral asymptotics, too?


Thank you in advance.



Weyl asymptotics vs. form perturbations
Let me elaborate a little bit on my idea. Denote by $\lambda\_j$ the eigenvalues of the operator $A$. Then Weyl-asymptotics means that
$$
 N(E) = \\#(j: \lambda\_j < E) < C \cdot E^\alpha.
$$
Consider now the operator $(A)^{-1}$ with eigenvalues $\lambda^{-1}$ ... and consider its Schatten $p$ norm
$$
 \|A^{-1} \|\_p = \sum \_{j} \lambda\_j^{- p} = p \int\_0^{\infty} \frac{N(E) dE}{E^{p-1}}.
$$
This is finite for $p >\alpha + 1$, so it suffices to investigate stability of the $p$ norm of the resolvent, which is standard.


