On the spectrum of the pencil of high order differential operators with almost periodic coefficients

In this paper, the spectrum and the resolvent of the operator L λ which is generated by the differential expression ℓ λ ( y ) = y ( m ) + ∑ γ = 1 m ( ∑ k = 0 γ λ k p γ k ( x ) ) y ( m − γ ) has been investigated in the space L 2 ( R ) . Here the coefficients p γ k ( x ) = ∑ n = 1 ∞ p γ k n e i α n x , k = 0 , 1 , … , γ − 1 ; p γ γ ( x ) = p γ γ , γ = 1 , 2 , … , m , are constants, p m m ≠ 0 and p γ k ( ν ) ( x ) , ν = 0 , 1 , 2 , … , m − γ , are Bohr almost-periodic functions whose Fourier series are absolutely convergent. The sequence of Fourier exponents of coefficients (these are positive) has a unique limit point at +∞. It has been shown that if the polynomial ϕ ( z ) = z m + p 11 z m − 1 + p 22 z m − 2 + ⋯ + p m − 1 , m − 1 z + p m m has the simple roots ω 1 , ω 2 , … , ω m (or one multiple root ω 0 ), then the spectrum of operator L λ is pure continuous and consists of lines Re ( λ ω k ) = 0 , k = 1 , 2 , … , m (or of line Re ( λ ω 0 ) = 0 ). Moreover, a countable set of spectral singularities on the continuous spectrum can exist which coincides with numbers of the form λ = 0 , λ s j n = i α n ( ω j − ω s ) − 1 , n ∈ N , s , j = 1 , 2 , … , m , j ≠ s . If ϕ ( z ) = ( z − ω 0 ) m , then the spectral singularity does not exist. The resolvent L λ − 1 is an integral operator in L 2 ( R ) with the kernel of Karleman type for any λ ∈ ρ ( L λ ) .