On the inverse of band matrices

Band matrices and Green's matrices have been defined by E. Asplund [1]; his basic theorem shows their relationship. The definition of one-pair-matrices was given by F. R. Gantmacher and M. G. Krein [5]; they showed that the inverse of a one-pair-matrix is a tridiagonal matrix and vice versa. Introducing the concept of strict band matrices and semi-separable matrices, and making use of the above theorems, the main result on the relationship between strict band matrices and semi-separable matrices will be proved. Further a constructive method will be given to obtain the inverse of a strict band matrix.     

On the dispersion spectrum

A measure for the denseness of sequences (an) mod 1, irrational, is the dispersion constantD() introduced byH. Niederreiter. In this paper the smallest accumulation point ₁ of the set of theD() is determined and all those are explicitely given for whichD () < ₁ holds.     

The courant-fischer theorem and the spectrum of selfadjoint block band Toeplitz operators

We show that ifT(F) is a selfadjoint block Toeplitz operator generated by a trigonometric matrix polynomialF, then the spectrum ofT(F) as well as the limiting set (F) of the eigenvalues of the truncationsT _n (F) is the union of a finite collection of segments (the spectral range ofF) and at most a finite set of points for which we give an upper bound.     

On the eigenvalues of double band matrices

We consider matrices containing two diagonal bands of positive entries. We show that all eigenvalues of such matrices are of the form rζ, where r is a nonnegative real number and ζ is a pth root of unity, where p is the period of the matrix, which is computed from the distance between the bands. We also present a problem in the asymptotics of spectra in which such double band matrices are perturbed by banded matrices.     

On the spectrum of the Laplacian

In this article we prove a generalization of Weyl’s criterion for the essential spectrum of a self-adjoint operator on a Hilbert space. We then apply this criterion to the Laplacian on functions over open manifolds and get new results for its essential spectrum.     

Bose-Einstein condensation in the excited band and the energy spectrum of the Bose-Hubbard model

Based on the mean field approximation, we investigate the transition into the Bose-Einstein condensate phase in the Bose-Hubbard model with two local states and boson hopping in only the excited band. In the hard-core boson limit, we study the instability associated with this transition, which appears at excitation energies δ < |t ₀ |, where |t ₀ | is the particle hopping parameter. We discuss the conditions under which the phase transition changes from second to first order and present the corresponding phase diagrams (Θ,μ) and (|t ₀ |, μ), where Θ is the temperature and μ is the chemical potential. Separation into the normal and Bose-Einstein condensate phases is possible at a fixed average concentration of bosons. We calculate the boson Green’s function and one-particle spectral density using the random phase approximation and analyze changes in the spectrum of excitations of the “particle” or “hole” type in the region of transition from the normal to the Bose-Einstein condensate phase.     

On the quadratic Lagrange spectrum

We study the quadratic Lagrange spectrum defined by Parkkonen and Paulin by considering the approximation by elements of the orbit of a given real quadratic irrational number for the action by homographies and anti-homographies of $PSL_2(\mathbf{Z})$ on $\mathbf{R}\cup \{\infty \}$ . Our approach is based on the theory of continued fractions.     

On the lower Markov spectrum

C. Hightower found two infinite sequences of gaps in the Markov spectrum, ( _n , _n ) and ( _n , _n ) with _n and _n both Markov elements, converging to . This paper exhibits Markov elements _n ^* and _n ^* such that, for alln 1, ( _n ^* , _n ) and ( _{n n} ^* ) are gaps in the Markov spectrum. Other results include showing that, for alln 1, _n is completely isolated, while the other endpoints of the gaps are limit points in the Markov spectrum.     

On the invariant spectrum on

Motivated by the work of Abreu and Freitas 1 , we study the invariant spectrum of the Laplace operator associated to hermitian line bundles endowed with invariant metrics over .     

On the spectrum of anosov diffeomorphisms

Any Anosov diffeomorphismf induces a linear operator in the Banach space of continuous vector fields, which spectrum is contained in two annuli both being disjoint with the unit circle. The main theorem states that, if these annuli are sufficiently “narrow”, thenf has a fixed point. It is also proved that, iff acts on an infranilmanifold (e.g., on a torus), then the spectrum annuli for the algebraic automorphism corresponding tof are always more “narrow” than forf.     

On the fluctuation spectrum of plasma

The spectrum of electron phase space density fluctuations of a plasma is calculated by a novel method that parallels conventional calculations of the partition function in statistical physics. Expressions for the electric field fluctuations and the closely related form factor agree with existing results. The method clears up ambiguities about equipartition and provides a new expression for the spectrum of electrostatic phase space density fluctuations about stable non-Maxwellian equilibria.     

On the spectrum of hyponormal operators

Xia's theorem about the spectrum of hyponormal operators is reproved by a simple method and it is extended to the case of unbounded hyponormal operators with bounded imaginary part and double commutingn-tuples of hyponormal operators.