Abstract. We review certain results for amplitudes of spectral flowed operators in string theory on AdS3. We present the modified Knizhnik-Zamolodchikov and null vector equations to be satisfied by correlators including w = 1 operators. We then discuss the three point function of two w = 1 and one w = 0 operators in the x-basis, and perform a consistency check on the definition of the w = 1 ope...

For differential operators which are invariant under the action of an abelian group Bloch theory is the tool of choice to analyze spectral properties. By shedding some new non-commutative light on this we motivate the introduction of a noncommutative Bloch theory for elliptic operators on Hilbert C-modules. It relates properties of C-algebras to spectral properties of module operators such as b...

We discuss the band-gap structure and the integrated density of states for periodic elliptic operators in the Hilbert space L2(R m), for m ≥ 2. We specifically consider situations where high contrast in the coefficients leads to weak coupling between the period cells. Weak coupling of periodic systems frequently produces spectral gaps or spectral concentration. Our examples include Schrödinger ...

For differential operators which are invariant under the action of an abelian group Bloch theory is the tool of choice to analyze spectral properties. By shedding some new non-commutative light on this we motivate the introduction of a noncommutative Bloch theory for elliptic operators on Hilbert C-modules. It relates properties of C-algebras to spectral properties of module operators such as b...

Abstract. We review certain results for amplitudes of spectral flowed operators in string theory on AdS3. We present the modified Knizhnik-Zamolodchikov and null vector equations to be satisfied by correlators including w = 1 operators. We then discuss the three point function of two w = 1 and one w = 0 operators in the x-basis, and perform a consistency check on the definition of the w = 1 ope...

We discuss the division method for subspectra which appears to be one of the key approaches in the study of spectral properties of self-adjoint differential vector-operators, that is operators generated as a direct sum of self-adjoint extensions on an Everitt-Markus-Zettl multi-interval system. In the current work we show how the division method may be applied to obtain the ordered spectral rep...

It is by now well known that the wave functions of rational solutions to the KP hierarchy which can be achieved as limits of the pure n-soliton solutions satisfy an eigenvalue equation for ordinary differential operators in the spectral parameter. This property is known as “bispectrality” and has proved to be both interesting and useful. In this note, it is shown that all pure soliton solutions...

Abstract. We construct non-random bounded discrete half-line Schrödinger operators which have purely singular continuous spectral measures with fractional Hausdorff dimension (in some interval of energies). To do this we use suitable sparse potentials. Our results also apply to whole line operators, as well as to certain random operators. In the latter case we prove and compute an exact dimensi...

New unique characterization results for the potential V (x) in connection with Schrödinger operators on R and on the half-line [0,∞) are proven in terms of appropriate Krein spectral shift functions. Particular results obtained include a generalization of a well-known uniqueness theorem of Borg and Marchenko for Schrödinger operators on the half-line with purely discrete spectra to arbitrary sp...

This text is a survey of recent results obtained by the author and collaborators on different problems for non-self-adjoint operators. The topics are: Kramers-Fokker-Planck type operators, spectral asymptotics in two dimensions and Weyl asymptotics for the eigenvalues of non-self-adjoint operators with small random perturbations. In the introduction we also review the notion of pseudo-spectrum ...