The Hamiltonian model of quantum error correction code in the literature is often constructed with the help of its stabilizer formalism. But there have been many known examples of nonadditive codes which are beyond the standard quantum error correction theory using the stabilizer formalism. In this paper, we suggest the other type of Hamiltonian formalism for quantum error correction code witho...

Given a one-dimensional weighted Dirac operator we can define a spectral measure by virtue of singular Weyl–Titchmarsh–Kodaira theory. Using the theory of de Branges spaces we show that the spectral measure uniquely determines the Dirac operator up to a gauge transformation. Our result applies in particular to radial Dirac operators and extends the classical results for Dirac operators with one...

Since 1970s, Lie algebras have played more important and extensive roles in nonlinear partial differential equations and theoretical physics than they did before. One of the most interesting examples is the birth of the theory of Hamiltonian operators in middle 1970s, which was a work of Gel’fand, Dikii and Dorfman (cf. [GDi1-2], [GDo]). The existence of certain Hamiltonian operators associated...

We study the Schrödinger operator −∆ − αδ(x − Γ) in L 2 (R 3) with a δ interaction supported by an infinite non-planar surface Γ which is smooth, admits a global normal parameterization, and is asymptotically planar in a suitable sense. We show that for a large enough α > 0 this operator has a non-empty discrete spectrum and derive an asymptotic expansion of the eigenvalues in terms of a " two-...

We consider compact connected six dimensional symplectic mani-folds with Hamiltonian SU(2) or SO(3) actions with cyclic principal stabilizers. We classify such manifolds up to equivariant symplectomorphisms.

Many shape analysis methods treat the geometry of an object as a metric space captured by the Laplace-Beltrami operator. In this thesis we present an adaptation of a classical operator from quantum mechanics to shape analysis where we suggest to integrate a scalar function through a unified elliptical Hamiltonian operator. We study the addition of a potential function to the Laplacian as a gene...

For a given standard Hamiltonian H with arbitrary complex scalar and vector potentials in one-dimension, we construct an invertible antilinear operator τ such that H is τ -anti-pseudo-Hermitian, i.e., H = τHτ−1. We use this result to give the explicit form of a linear Hermitian invertible operator with respect to which any standard PT symmetric Hamiltonian with a real degree of freedom is pseud...

In this article, we prove the similarity and, in the focusing case, the J-unitary equivalence of the free Hamiltonian and the restriction of the full Hamiltonian to the maximal invariant subspace on which its spectrum is real for the matrix Zakharov–Shabat system under suitable conditions on the potentials. This restriction of the full Hamiltonian is shown to be a scalar-type spectral operator ...

The long standing problem of the ordering ambiguity in the definition of the Hamilton operator for a point particle in curved space is naturally resolved by using the powerful geometric calculus based on Clifford Algebra. The momentum operator is defined to be the vector derivative (the gradient) multiplied by −i; it can be expanded in terms of basis vectors γμ as p = −iγ∂μ. The product of two ...

In arXiv:0710.5653v1 M. Znojil claims that he has found and corrected an error in my paper: [Phys. Lett. B 650, 208 (2007), arXiv:0706.1872v2] and that it is possible to escape its main conclusion, namely that the unitarity of the time-evolution and observability of the Hamiltonian imply time-independence of the metric operator. In this note I give a very short calculation showing that the anal...