Perturbations by quadratic forms and invariance of essential spectra

On Lp -spectra and essential spectra of second-order elliptic operators

“…Theorem 3 is a direct generalization of the main result in [21] and of Example 1 from [13]. In the latter paper only self-adjoint operators are studied, while in the former one there is a requirement of ultracontractivity and the conditions on the coef®cients of the operator are much more restrictive.…”

“…This problem was addressed by M. S. Birman [7], M. Reed and B. Simon [23], M. Schechter [24], D. E. Edmunds and W. D. Evans [11], R. Hempel [13], E. M. Ouhabaz [21], and in [22] in a more general setting.…”

“…Using (13) we estimate the left-hand side of (12) as follows: lkuk n n tr À1 u; ru n À 1 > «tu n = 2 l À ckuk n n > «jk=u n = 2 k 2 2 «jkuk n n l À c À «jkuk n n > «jku n = 2 k 2 W 1; 2 > «jC d ku n = 2 k 2 2q = n ; provided l > c «j : l 0 . In the last step we used Sobolev's inequality with the constant C d , taking into account the fact that 1=p À 1=q < 2=d.…”

Trace estimates and invariance of the essential spectrum

“…Various sufficient conditions exist under which the resolvent difference (H + 1) −1 − (H + 1) −1 is compact and, subsequently, H andH have the same essential spectrum. See [6,7,8,5] and references therein. These conditions typically involve some decay of the differencẽ a ij − a ij of the respective coefficients near infinity.…”