This thesis investigates Lieb-Thirring and Cwikel-Lieb-Rozenblum (CLR) type inequalities for Schrödinger operators in dimensions one and two, with a focus on overcoming the weak coupling problem associated with the failure of the CLR inequality in these dimensions. To this end we start by investigating three two-dimensional Schrödinger operators with extra repulsive factors in the form of: a repulsive Hardy potential, a restriction to antisymmetric functions, and an Aharonov-Bohm magnetic field.

Under the assumption that the electric potential is radially non-increasing we establish semi-classical bounds on the number of negative eigenvalues for each of these operators. Our proof lies in generalising a one-dimensional bound of Calogero and Cohn to operator-valued potentials.

For the same operators we then derive a family of weighted CLR type inequalities by applying the Birman-Schwinger principle. An interpolation argument leads us to ascertain weak forms, which we show to be saturated in the strong coupling limit by a class of long-range potentials. In the Aharonov-Bohm case this enables us to deduce the optimal dependence of the constants on the flux of the field.

Finally in one dimension, we examine Schrödinger operators with an additional fixed attractive potential. Following the method of factorisation we derive Lieb-Thirring type bounds which measure the distance between the eigenvalues of the original and the perturbed operator. Applying this to the Pölsch-Teller potential leads to an improvement of the Lieb-Thirring inequality for partial sums of eigenvalues.