3.6. SPECTRAL THEORY OF DIFFERENTIAL OPERATORS 171
3.6.4 Estimates for number of negative eigenvalues
For simplicity, it is physically sufficient for us to consider the eigenvalue problem of the
Laplace operators, given by
(3.6.21)
−∇^2 ψ+V(x)ψ=λ ψ forx∈Br,
ψ= 0 forx∈∂Br,whereBr⊂Rnis a ball with radiusr.
In physics,Vrepresents a potential function and takes negative value ina bound state,
ensuring by Theorem3.36that (3.6.21) possesses negative eigenvalues.
Here, for the potential functionV(x), we assume that
(3.6.22) V(ρx)≃ραV 0 (x) (α>− 2 ),
whereV 0 (x)is defined in the unit ballB 1 , and
(3.6.23) Ω={x∈B 1 |V 0 (x)< 0 } 6=/0.
Letθ>0 be defined by
(3.6.24) θ= inf
ψ∈L^2 (Ω,Cm)
1
||ψ||L 2∫Ω|V(x)| |ψ|^2 dx.The main result in this section is the following theorem, which provides a relation be-
tweenN,θandr, whereNis the number of negative eigenvalues of (3.6.21). Letλ 1 be the
first eigenvalue of the equation
(3.6.25)
−∆e=λe forx∈Ω,
e= 0 forx∈∂Ω,whereΩ⊂B 1 is as defined by (3.6.23).
To state the main theorem, we need to introduce a lemma, leading to the Weyl asymptotic
relation (Weyl, 1912 ).
Lemma 3.37(H. Weyl).LetλNbe the N-th eigenvalue of the m-th order elliptic operator
(3.6.26)
(− 1 )m∆me=λe for x∈Ω⊂Rn,
Dke|∂Ω= 0 for 0 ≤k≤m− 1 ,thenλNhas the asymptotical relation
(3.6.27) λN∼λ 1 N^2 m/n,
whereλ 1 is the first eigenvalue of (3.6.26).
We are now ready for the main theorem.