115 research outputs found

### On a class of $J$-self-adjoint operators with empty resolvent set

In the present paper we investigate the set $\Sigma_J$ of all
$J$-self-adjoint extensions of a symmetric operator $S$ with deficiency indices
$$ which commutes with a non-trivial fundamental symmetry $J$ of a Krein
space $(\mathfrak{H}, [\cdot,\cdot])$, SJ=JS. Our aim is to describe different
types of $J$-self-adjoint extensions of $S$. One of our main results is the
equivalence between the presence of $J$-self-adjoint extensions of $S$ with
empty resolvent set and the commutation of $S$ with a Clifford algebra
${\mathcal C}l_2(J,R)$, where $R$ is an additional fundamental symmetry with
$JR=-RJ$. This enables one to construct the collection of operators
$C_{\chi,\omega}$ realizing the property of stable $C$-symmetry for extensions
$A\in\Sigma_J$ directly in terms of ${\mathcal C}l_2(J,R)$ and to parameterize
the corresponding subset of extensions with stable $C$-symmetry. Such a
situation occurs naturally in many applications, here we discuss the case of an
indefinite Sturm-Liouville operator on the real line and a one dimensional
Dirac operator with point interaction

### On Domains of PT Symmetric Operators Related to -y''(x) + (-1)^n x^{2n}y(x)

In the recent years a generalization of Hermiticity was investigated using a
complex deformation H=p^2 +x^2(ix)^\epsilon of the harmonic oscillator
Hamiltonian, where \epsilon is a real parameter. These complex Hamiltonians,
possessing PT symmetry (the product of parity and time reversal), can have real
spectrum. We will consider the most simple case: \epsilon even. In this paper
we describe all self-adjoint (Hermitian) and at the same time PT symmetric
operators associated to H=p^2 +x^2(ix)^\epsilon. Surprisingly it turns out that
there are a large class of self-adjoint operators associated to H=p^2
+x^2(ix)^\epsilon which are not PT symmetric

### Eigenvalue estimates for singular left-definite Sturm-Liouville operators

The spectral properties of a singular left-definite Sturm-Liouville operator
$JA$ are investigated and described via the properties of the corresponding
right-definite selfadjoint counterpart $A$ which is obtained by substituting
the indefinite weight function by its absolute value. The spectrum of the
$J$-selfadjoint operator $JA$ is real and it follows that an interval
$(a,b)\subset\mathbb R^+$ is a gap in the essential spectrum of $A$ if and only
if both intervals $(-b,-a)$ and $(a,b)$ are gaps in the essential spectrum of
the $J$-selfadjoint operator $JA$. As one of the main results it is shown that
the number of eigenvalues of $JA$ in $(-b,-a) \cup (a,b)$ differs at most by
three of the number of eigenvalues of $A$ in the gap $(a,b)$; as a byproduct
results on the accumulation of eigenvalues of singular left-definite
Sturm-Liouville operators are obtained. Furthermore, left-definite problems
with symmetric and periodic coefficients are treated, and several examples are
included to illustrate the general results.Comment: to appear in J. Spectral Theor

### Variational principles for self-adjoint operator functions arising from second-order systems

Variational principles are proved for self-adjoint operator functions arising
from variational evolution equations of the form $\langle\ddot{z}(t),y \rangle + \mathfrak{d}[\dot{z} (t), y] + \mathfrak{a}_0
[z(t),y] = 0.$ Here $\mathfrak{a}_0$ and $\mathfrak{d}$ are densely defined,
symmetric and positive sesquilinear forms on a Hilbert space $H$. We associate
with the variational evolution equation an equivalent Cauchy problem
corresponding to a block operator matrix $\mathcal{A}$, the forms $\mathfrak{t}(\lambda)[x,y] := \lambda^2\langle x,y\rangle +
\lambda\mathfrak{d}[x,y] + \mathfrak{a}_0[x,y],$ where $\lambda\in \mathbb C$
and $x,y$ are in the domain of the form $\mathfrak{a}_0$, and a corresponding
operator family $T(\lambda)$. Using form methods we define a generalized
Rayleigh functional and characterize the eigenvalues above the essential
spectrum of $\mathcal{A}$ by a min-max and a max-min variational principle. The
obtained results are illustrated with a damped beam equation.Comment: to appear in Operators and Matrice

### Analyticity and Riesz basis property of semigroups associated to damped vibrations

Second order equations of the form $z'' + A_0 z + D z'=0$ in an abstract
Hilbert space are considered. Such equations are often used as a model for
transverse motions of thin beams in the presence of damping. We derive various
properties of the operator matrix $A$ associated with the second order problem
above. We develop sufficient conditions for analyticity of the associated
semigroup and for the existence of a Riesz basis consisting of eigenvectors and
associated vectors of $A$ in the phase space

### Spectral bounds for singular indefinite Sturm-Liouville operators with $L^1$--potentials

The spectrum of the singular indefinite Sturm-Liouville operator
$A=\text{\rm sgn}(\cdot)\bigl(-\tfrac{d^2}{dx^2}+q\bigr)$ with a real
potential $q\in L^1(\mathbb R)$ covers the whole real line and, in addition,
non-real eigenvalues may appear if the potential $q$ 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 $|\lambda|\leq |q|_{L^1}^2$ on the absolute values of the non-real
eigenvalues $\lambda$ of $A$ is obtained. Furthermore, separate bounds on the
imaginary parts and absolute values of these eigenvalues are proved in terms of
the $L^1$-norm of the negative part of $q$.Comment: to appear in Proc. Amer. Math. So

### Bounds on the non-real spectrum of differential operators with indefinite weights

Ordinary and partial differential operators with an indefinite weight
function can be viewed as bounded perturbations of non-negative operators in
Krein spaces. Under the assumption that 0 and $\infty$ are not singular
critical points of the unperturbed operator it is shown that a bounded additive
perturbation leads to an operator whose non-real spectrum is contained in a
compact set and with definite type real spectrum outside this set. The main
results are quantitative estimates for this set, which are applied to
Sturm-Liouville and second order elliptic partial differential operators with
indefinite weights on unbounded domains.Comment: 27 page

### Numerical Range and Quadratic Numerical Range for Damped Systems

We prove new enclosures for the spectrum of non-selfadjoint operator matrices
associated with second order linear differential equations $\ddot{z}(t) + D
\dot{z} (t) + A_0 z(t) = 0$ in a Hilbert space. Our main tool is the quadratic
numerical range for which we establish the spectral inclusion property under
weak assumptions on the operators involved; in particular, the damping operator
only needs to be accretive and may have the same strength as $A_0$. By means of
the quadratic numerical range, we establish tight spectral estimates in terms
of the unbounded operator coefficients $A_0$ and $D$ which improve earlier
results for sectorial and selfadjoint $D$; in contrast to numerical range
bounds, our enclosures may even provide bounded imaginary part of the spectrum
or a spectral free vertical strip. An application to small transverse
oscillations of a horizontal pipe carrying a steady-state flow of an ideal
incompressible fluid illustrates that our new bounds are explicit.Comment: 27 page

- â€¦