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 that we know 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.

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.