We investigate a lattice version of the Yang-Lee model which is characterized by a non-Hermitian quantum spin chain Hamiltonian. We propose a new way to implement PT -symmetry on the lattice, which serves to guarantee the reality of the spectrum in certain regions of values of the coupling constants. In that region of unbroken PT -symmetry we construct a Dyson map, a metric operator and find th...

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 ...

The study of MDS self-dual codes has attracted lots of attention in recent years. There are many papers on determining existence of q−ary MDS self-dual codes for various lengths. There are not existence of q−ary MDS self-dual codes of some lengths, even these lengths < q. We generalize MDS Euclidean self-dual codes to near MDS Euclidean self-dual codes and near MDS isodual codes. And we obtain ...

We give two characterization theorems for pseudo-Hermitian (possibly nondiagonalizable) Hamiltonians with a discrete spectrum that admit a block-diagonalization with finite-dimensional diagonal blocks. In particular, we prove that for such an operator H the following statements are equivalent. 1. H is pseudo-Hermitian; 2. The spectrum of H consists of real and/or complex-conjugate pairs of eige...

Hermitian codes belong to a powerful class of algebraic geometry codes, which allow to correct many independent errors. We show that Hermitian codes can correct many bursts of errors as well. Decoding of a Hermitian (N,K) code over q2 can be reduced to decoding of interleaved extended Reed-Solomon codes. Using this fact, we propose an efficient unique decoding algorithm correcting up to (N −K)/...

Signatures of quadratic forms over formally real fields have been generalized in [BP2] to hermitian forms over central simple algebras with involution over such fields. This was achieved by means of an application of Morita theory and a reduction to the quadratic form case. A priori, signatures of hermitian forms can only be defined up to sign, i.e., a canonical definition of signature is not p...

For an Hermitian matrix the QR transform is diagonally similar to two steps of the LR transforms. Even for non-Hermitian matrices the QR transform may be written in rational form.

We study analogues of classical inequalities for the eigenvalues of sums of Hermitian matrices for the cone of admissible elements in the pseudo-Hermitian case. In particular, we obtain analogues of the Lidskii-Wielandt inequalities.

Incomplete LDL∗ factorizations sometimes produce an inde nite preconditioner even when the input matrix is Hermitian positive de nite. The two most popular iterative solvers for Hermitian systems, MINRES and CG, cannot use such preconditioners; they require a positive de nite preconditioner. We present two new Krylov-subspace solvers, a variant of MINRES and a variant of CG, both of which can b...