The non-normality of Wilson-type lattice Dirac operators has important consequences — the application of the usual concepts from the textbook (hermitian) quantum mechanics should be reconsidered. This includes an appropriate definition of observables and the refinement of computational tools. We show that the truncated singular value expansion is the optimal approximation to the inverse operato...

For a weakly pseudo-Hermitian linear operator, we give a spectral condition that ensures its pseudo-Hermiticity. This condition is always satisfied whenever the operator acts in a finite-dimensional Hilbert space. Hence weak pseudo-Hermiticity and pseudo-Hermiticity are equivalent in finite-dimensions. This equivalence extends to a much larger class of operators. Quantum systems whose Hamiltoni...

The quantum deformation of the oscillator algebra and its implications on the phase operator are studied from a view point of an index theorem by using an explicit matrix representation. For a positive deformation parameter q or q = exp(2πiθ) with an irrational θ, one obtains an index condition dim ker a − dim ker a † = 1 which allows only a non-hermitian phase operator with dim ker e iϕ − dim ...

We show that a (non-Hermitian) Hamiltonian H admitting a complete biorthonormal set of eigenvectors is pseudo-Hermitian if and only if it has an anti-linear symmetry, i.e., a symmetry generated by an anti-linear operator. This implies that the eigenvalues of H are real or come in complex conjugate pairs if and only if H possesses such a symmetry. In particular, the reality of the spectrum of H ...

where ρ ≥ 1 is a mass parameter corresponding to a bare fermion mass mb = m(ρ − 1), and Q is the hermitian Wilson operator with a negative mass parameter −m. Note that γ5 sign(Q) is unitary. We can also define a hermitian overlap operator Dh = γ5Du. Both operators are normal, i.e. they commute with their adjoints. We will approximate the matrix sign function using the Zolotarev partial fraction...

In this talk we present the results published recently in Ref. [1], where we showed how to introduce a quark chemical potential in the overlap Dirac operator. The resulting operator satisfies a Ginsparg-Wilson relation and has exact zero modes. It is no longer γ5-Hermitian, but its nonreal eigenvalues still occur in pairs. We compute the spectral density of the operator on the lattice and show ...

We investigate variational methods for finding approximate solutions to the Fokker-Planck equation, especially in cases lacking detailed balance. These schemes fall into two classes: those in which a Hermitian operator is constructed from the (non-Hermitian) Fokker-Planck operator, and those which are based on soluble approximations to this operator. The various approaches are first tested on a...

We show how to introduce a quark chemical potential in the overlap Dirac operator. The resulting operator satisfies a Ginsparg-Wilson relation and has exact zero modes. It is no longer gamma5 Hermitian, but its nonreal eigenvalues still occur in pairs. We compute the spectral density of the operator on the lattice and show that, for small eigenvalues, the data agree with analytical predictions ...

A possible method to investigate non-Hermitian Hamiltonians is suggested through finding a Hermitian operator η+ and defining the annihilation and creation operators to be η+ -pseudo-Hermitian adjoint to each other. The operator η+ represents the η+ -pseudo-Hermiticity of Hamiltonians. As an example, a non-Hermitian and non-PT-symmetric Hamiltonian with imaginary linear coordinate and linear mo...

In quantum mechanics it is well known that observables are represented by hermitian matrices (or operators). Uncertainty relations are represented as some kinds of trace inequalities satisfied by two observables and one density matrix (or operator). Now we try to release the hermitian restriction on observables. This is only a mathematical interest. In this case we give several non-hermitian ex...