Relative form boundedness and compactness for a second-order differential operator

If A and L are self-adjoint operators on a Hilbert space H such that A is nonnegative and L⩾ εI for some ε>0 we study conditions under which A is form bounded or form compact with respect to L and contrast these concepts with the stronger properties that A be relatively operator bounded or compact with respect to L. In particular several definitions of form compactness are shown to be equivalent. The principal application of the theory is to self-adjoint second order operators. In this case conditions that A be form bounded or form compact with respect to L are shown in some cases to be necessary and sufficient. Examples include the energy operator of the hydrogen atom.