Estimates of the Negative Spectrum

This short chapter plays a crucial role in the whole book bridging Parts I, II and Part V. Namely here we combine the results of Parts II, III devoted to local semiclassical spectral asymptotics with the variational estimates of Rozenblioum type (upper estimates for $${\mathsf {N}}^-(A)$$ ) in order to derive upper and lower estimates for $${\mathsf {N}}^-(A)={\mathsf {N}}(-\infty , A)$$ where $${\mathsf {N}}^-(A)$$ is the number of negative eigenvalues of the self-adjoint operator A (counting their multiplicities) provided $$(-\infty , 0)$$ does not intersect with the essential spectrum of A and $${\mathsf {N}}^-(A)=\infty $$ otherwise.