Spectral analysis of higher order differential operators with unbounded coefficients

Abstract

Key words Eigenvalues, deficiency indices, absolutely continuous spectrum MSC (2010) 47E05, 34L05, 34L20Higher even order linear differential operators with unbounded coefficients are studied. For these operators the eigenvalues of the characteristic polynomials fall into distinct classes or clusters. Consequently the spectral properties, deficiency indices and spectra, of the underlying differential operators are superpositions of the contributions from the individual clusters. These results are based on a quantitative improvement of Levinson's Theorem. Our methods will also be applicable to other classes of linear differential operators.

Essential and continuous spectrum of symmetric difference equations

“…Similarly, the author together with others have discussed the deficiency indices as well as the location of absolutely continous spectrum of both self‐adjoint extenstion operators of differential operators as well as for self adjoint extenstion subspaces for difference equations. For more details see . Therefore, the results of this paper can be considered as a continuation of the investigations carried out in the above mentioned papers and in particular, as an extension of some of the results in to the discretised version.…”

“…For more details see . Therefore, the results of this paper can be considered as a continuation of the investigations carried out in the above mentioned papers and in particular, as an extension of some of the results in to the discretised version. Here, we have applied asymptotic summation to compute the deficiency indices as well as the location of both essential and continuous spectrum.…”

Deficiency indices and spectrum of fourth order difference equations with unbounded coefficients

“…Then apply the approach used in Eastham [15, Sect. 3.3] and 9. For the regularity and smoothness condition, we will need that the coefficients are thrice differentiable with …”

On essential spectra of singular linear Hamiltonian systems

“…For m = 2, l = 1, q 0 = 0, a = 0, and b = ∞, τ reduces to the Sturm-Liouville differential expressioñ τ (x)(t) = w −1 (t) − p 1 (t)x (t) + p 0 (t)x(t) , t ∈ [0, ∞). (1.6) The spectral theory for symmetric differential operators has been investigated extensively by using many methods such as asymptotic approximations of solutions, perturbation theory, and oscillation theory, and many good results have been established (cf., e.g., [2][3][4][5][9][10][11]16,24,30,32] and their references). In [3], asymptotic approximations of solutions and M (λ) functions were used to study the absolutely continuous spectra of self-adjoint operators generated by system (1.1) under certain conditions.…”