2,013 research outputs found
Spectral bounds for singular indefinite Sturm-Liouville operators with --potentials
The spectrum of the singular indefinite Sturm-Liouville operator
with a real
potential covers the whole real line and, in addition,
non-real eigenvalues may appear if the potential assumes negative values. A
quantitative analysis of the non-real eigenvalues is a challenging problem, and
so far only partial results in this direction were obtained. In this paper the
bound on the absolute values of the non-real
eigenvalues of is obtained. Furthermore, separate bounds on the
imaginary parts and absolute values of these eigenvalues are proved in terms of
the -norm of the negative part of .Comment: to appear in Proc. Amer. Math. So
PT-Symmetric Hamiltonians as couplings of dual pairs
In the seminal paper (Bender & Boettcher, 1998) a new view of quantum mechanics was proposed. This new view differs from the old one in that the restriction on the Hamiltonian to be Hermitian is relaxed: now the Hamiltonian is PT -symmetric. Here P is parity and T is time reversal. Since 1998, PT -symmetric Hamiltonians have been analyzed intensively by many authors. In Mostafazadeh (2002) PT -symmetry was embedded into the more general mathematical framework of pseudo-Hermiticity or, what is the same, self-adjoint operators in Kre˘ın spaces, see (Langer & Tretter, 2004; Azizov & Trunk, 2012; Hassi & Kuzhel, 2013; Leben & Trunk, 2019). For a general introduction to PT -symmetric quantum mechanics we refer to the overview paper of Mostafazadeh (2010) and to the books of Moiseyev (2011) and Bender (2019)
Relative oscillation theory and essential spectra of Sturm--Liouville operators
We develop relative oscillation theory for general Sturm-Liouville
differential expressions of the form and prove perturbation results and invariance of essential
spectra in terms of the real coefficients , , . The novelty here is
that we also allow perturbations of the weight function in which case the
unperturbed and the perturbed operator act in different Hilbert spaces.Comment: 15 page
Lower bounds for self-adjoint Sturm-Liouville operators
The numerical range and the quadratic numerical range is used to study the spectrum of a class of block operator matrices. We show that the approximate point spectrum is contained in the closure of the quadratic numerical range. In particular, the spectral enclosures yield a spectral gap. It is shown that these spectral bounds are tighter than classical numerical range bounds
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