63 research outputs found
Hydrogen atom in space with a compactified extra dimension and potential defined by Gauss' law
We investigate the consequences of one extra spatial dimension for the
stability and energy spectrum of the non-relativistic hydrogen atom with a
potential defined by Gauss' law, i.e. proportional to . The additional
spatial dimension is considered to be either infinite or curled-up in a circle
of radius . In both cases, the energy spectrum is bounded from below for
charges smaller than the same critical value and unbounded from below
otherwise. As a consequence of compactification, negative energy eigenstates
appear: if is smaller than a quarter of the Bohr radius, the corresponding
Hamiltonian possesses an infinite number of bound states with minimal energy
extending at least to the ground state of the hydrogen atom.Comment: 10 page
Spectral analysis of non-self-adjoint Jacobi operator associated with Jacobian elliptic functions
We perform the spectral analysis of a family of Jacobi operators
depending on a complex parameter . If the spectrum of
is discrete and formulas for eigenvalues and eigenvectors are
established in terms of elliptic integrals and Jacobian elliptic functions. If
, , the essential spectrum of covers
the entire complex plane. In addition, a formula for the Weyl -function as
well as the asymptotic expansions of solutions of the difference equation
corresponding to are obtained. Finally, the completeness of
eigenvectors and Rodriguez-like formulas for orthogonal polynomials, studied
previously by Carlitz, are proved.Comment: published version, 2 figures added; 21 pages, 3 figure
Root system of singular perturbations of the harmonic oscillator type operators
We analyze perturbations of the harmonic oscillator type operators in a
Hilbert space H, i.e. of the self-adjoint operator with simple positive
eigenvalues satisfying . Perturbations
are considered in the sense of quadratic forms. Under a local subordination
assumption, the eigenvalues of the perturbed operator become eventually simple
and the root system forms a Riesz basis.Comment: 16 pages; extended Section 5; published versio
Spectra of definite type in waveguide models
We develop an abstract method to identify spectral points of definite type in
the spectrum of the operator . The method is
applicable in particular for non-self-adjoint waveguide type operators with
symmetries. Using the remarkable properties of the spectral points of definite
type, we obtain new results on realness of weakly coupled bound states and of
low lying essential spectrum in the -symmetric
waveguide. Moreover, we show that the pseudospectrum has a normal tame behavior
near the low lying essential spectrum and exclude the accumulation of non-real
eigenvalues to this part of the essential spectrum. The advantage of our
approach is particularly visible when the resolvent of the unperturbed operator
cannot be explicitly expressed and most of the mentioned spectral conclusions
are extremely hard to prove using direct methods.Comment: 15 pages, 4 figures, submitte
Eigenvalues of one-dimensional non-self-adjoint Dirac operators and applications
We analyze eigenvalues emerging from thresholds of the essential spectrum of
one-dimensional Dirac operators perturbed by complex and non-symmetric
potentials. In the general non-self-adjoint setting we establish the existence
and asymptotics of weakly coupled eigenvalues and Lieb-Thirring inequalities.
As physical applications we investigate the damped wave equation and armchair
graphene nanoribbons.Comment: 16 page
Approximations of spectra of Schr\ odinger operators with complex potentials on
We study spectral approximations of Schr\ odinger operators with complex potentials on , or exterior domains , by domain truncation. Our weak assumptions cover wide classes of potentials for which has discrete spectrum, of approximating domains , and of boundary conditions on such as mixed Dirichlet/Robin type. In particular, need not be bounded from below and may be singular. We prove generalized norm resolvent convergence and spectral exactness, i.e. approximation of all eigenvalues of by those of the truncated operators without spectral pollution. Moreover, we estimate the eigenvalue convergence rate and prove convergence of pseudospectra. Our results are illustrated by numerical computations for several examples, such as complex harmonic and cubic oscillators for . The talk is based on a joint work with S. B\ ogli and C. Tretter
Differential operators admitting various rates of spectral projection growth
We consider families of non-self-adjoint perturbations of self-adjoint
harmonic and anharmonic oscillators. The norms of spectral projections of these
operators are found to grow at intermediate rates from arbitrarily slowly to
exponentially rapidly.Comment: 35 pages, significant revisions, accepted version (published version
may differ
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