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

Abstract

In this paper, the spectrum and the resolvent of the operator L-lambda which is generated by the differential expression l(lambda)(y) = y((m)) + Sigma([I])(gamma=1) (Sigma(gamma)(k=0) lambda(k)p(gamma)(X))y((m-gamma)) has been investigated in the space L-2(R). Here the coefficients p(gamma k)(x)=Sigma(infinity)(n-1) p gamma kn(ei alpha nx),k = 0,1,...,gamma-1; p(gamma gamma),gamma(x)= p(gamma gamma),gamma = 1,2,...,m, are constants, p(mm) not equal 0 and p(gamma k)((nu))(X), nu = 0,1,2,...,m y, 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 +infinity It has been shown that if the polynomial phi(z) = z(m) + p(11)z(m-1) + p(22)z(m-2) + ... + p(m-1),(m-1) Z + p(mm) has the simple roots omega(1),omega(2), ..., omega(m) (or one multiple root omega(0)), then the spectrum of operator L-lambda,L- is pure continuous and consists of lines Re(lambda(omega k)) = 0, k = 1,2,...,m (or of line Re(Acoo) = 0). Moreover, a countable set of spectral singularities on the continuous spectrum can exist which coincides with numbers of the form A = 0, Asp, = icen(coi cos)-', n E N, s,j= 1,2,...,m, If phi(z) = (z Nor, then the spectral singularity does not exist. The resolvent L-lambda(-1) is an integral operator in L-2(R) with the kernel of Karleman type for any lambda is an element of p(L-lambda).