Problem (by  Y. Wei)

Please prove that in the complex Hilbert space L²(R³), the operator

H=K-ε/r

is not positive semidefinite, i.e.,  

there exists a function f₀(x,y,z)  in L²(R³)  , such that the inner product

<f₀,Hf₀>     <   0.

Here ε is a positive number, (especially ε=1/138), r=√(x²+y²+z²),  and operator K is defined as

Kf (x,y,z)=(F^-1kFf)(x,y,z).

F is Fourier transform

Ff  (k₁,k₂,k₃)=(2π)^-3/2∫ f(x,y,z)exp(-ik₁x-ik₂y-ik₃z)dxdydz

 F^-1the inversion and k=√(k₁²+k₂²+k₃²).

Yuchuan Wei, (Ph D of physics)

7. 26, 2024

yuchuanwei@gmail.com 

非诚勿扰，奖金1万元

Problem.docx