The five-dimensional Kepler Problem as an SU(2) Gauge System: Algebraic Constraint Quantization

Starting from the structural similarity between the quantum theory of gauge systems and that of the Kepler problem, an SU(2) gauge description of the five-dimensional Kepler problem is given. This non-abelian gauge system is used as a testing ground for the application of an algebraic constraint quantization scheme which can be formulated entirely in terms of observable quantities. For the quantum mechanical reduction only the quadratic Casimir of the constraint algebra, interpreted as an observable, is needed.Comment: 29 pages, Latex, no figure

Spectral properties of singular Sturm-Liouville operators with indefinite weight sgn x

We consider a singular Sturm-Liouville expression with the indefinite weight sgn x. To this expression there is naturally a self-adjoint operator in some Krein space associated. We characterize the local definitizability of this operator in a neighbourhood of $\infty$. Moreover, in this situation, the point $\infty$ is a regular critical point. We construct an operator A=(\sgn x)(-d^2/dx^2+q) with non-real spectrum accumulating to a real point. The obtained results are applied to several classes of Sturm-Liouville operators.Comment: 21 pages, LaTe

On a class of JJ-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

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

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