Hermitian function

Abstract: For the positive semidefinite system of linear equations of a block two-by-two structure, by making use of the Hermitian/skew-Hermitian splitting iteration technique we establish a class of preconditioned Hermitian/skew-Hermitian splitting iteration methods. Theoretical analysis shows that the new method converges unconditionally to the unique solution of the linear system. Moreover, the optimal choice of the involved iteration parameter and the corresponding asymptotic convergence rate are computed exactly. Numerical examples further confirm the correctness of the theory and the effectiveness of the method.

Abstract: The performance of the second-order methods for excitation energies CC2 and ADC(2) is investigated and compared with the more approximate CIS and CIS(D) methods as well as with the coupled-cluster models CCSD, CCSDR(3) and CC3. As a by-product of this investigation the first implementation of analytic excited state gradients for ADC(2) and CIS(D∞) is reported. It is found that for equilibrium structures and vibrational frequencies the second-order models CIS(D), ADC(2) and CC2 give often results close to those obtained with CCSD. The main advantage of CCSD lies in its robustness with respect to strong correlation effects. For adiabatic excitation energies CC2 is found to give from all second-order methods for excitation energies (including CCSD) the smallest mean absolute errors. ADC(2) and CIS(D∞) are found to give almost identical results. An advantage of ADC(2) compared to CC2 is that the excitation energies are obtained as eigenvalues of a Hermitian secular matrix, while in coupled-cluster response the excitation energies are obtained as eigenvalues of a non-Hermitian Jacobi matrix. It is shown that, as a consequence of the lack of Hermitian symmetry, the latter methods will in general not give a physically correct description of conical intersections between states of the same symmetry. This problem does not appear in ADC(2).

Abstract: We show that a quantum system possessing an exact antilinear symmetry, in particular PT-symmetry, is equivalent to a quantum system having a Hermitian Hamiltonian. We construct the unitary operator relating an arbitrary non-Hermitian Hamiltonian with exact PT-symmetry to a Hermitian Hamiltonian. We apply our general results to PT-symmetry in finite dimensions and give the explicit form of the above-mentioned unitary operator and Hermitian Hamiltonian in two dimensions. Our findings lead to the conjecture that non-Hermitian CPT-symmetric field theories are equivalent to certain nonlocal Hermitian field theories.

Abstract: Non-Hermitian theory is a theoretical framework used to describe open systems. It offers a powerful tool in the characterization of both the intrinsic degrees of freedom of a system and the interactions with the external environment. The non-Hermitian framework consists of mathematical structures that are fundamentally different from those of Hermitian theories. These structures not only underpin novel approaches for precisely tailoring non-Hermitian systems for applications but also give rise to topologies not found in Hermitian systems. In this Review, we provide an overview of non-Hermitian topology by establishing its relationship with the behaviours of complex eigenvalues and biorthogonal eigenvectors. Special attention is given to exceptional points — branch-point singularities on the complex eigenvalue manifolds that exhibit nontrivial topological properties. We also discuss recent developments in non-Hermitian band topology, such as the non-Hermitian skin effect and non-Hermitian topological classifications. Non-Hermitian theory consists of mathematical structures that are used to describe open systems, which can give rise to non-Hermitian topology not found in Hermitian systems. This Review provides an overview of non-Hermitian band topology and discusses recent developments, such as the non-Hermitian skin effect and non-Hermitian topological classifications.