Spectral interpretation of Primes

[zetafunction]

the idea is

is there a Linear operator L so $$L | \phi _n > =p_n |\phi_n >$$

with p_n being the nth prime and L a linear operator , is it possible to have an spectral interpretation for prime numbers ?

 

[ramsey2879]

the idea is

is there a Linear operator L so $$L | \phi _n > =p_n |\phi_n >$$

with p_n being the nth prime and L a linear operator , is it possible to have an spectral interpretation for prime numbers ?

I don't think your formula makes sense, but I am not sure. It looks like you have L and p(n)each divide $$\phi_n$$ but should that really be so since $$\phi_n$$ should be less than p(n)?
PS There are over 20,000 articles in science direct that combined the words spectral and primes. Some mention a spectral analysis of the prime intevals etc., but I don't have access to the specific articles.