Hamiltonians non-Hermitian

On the other hand, we sometimes discuss the Schrodinger equation without an absorption potential, for example, the Coulomb few-body Schrodinger equation with the Hamiltonian (3), assuming the energy to take complex values. This is analytic continuation of the quantal problem into fake, complex energies. This way, we depart from what actually occurs in nature for real energies. By regarding the nature as a special 

A technique for direct computations of the eigenvalues Er —zT/2 of H(6 = 0) with the outgoing-wave boundary condition is reviewed in detail in a chapter in Part I of this two-volume special issue of Advances in Quantum Chemistry on Unstable States in the Continuous Spectra [27]. Determination of the wavefunction of Eq. (2) with a real eigenvalue EQ using a judiciously chosen real, square-integrable basis set, followed by diagonalization of a complex Hamiltonian matrix for the whole Hq + H constructed in terms of basis functions of complex-rotated coordinates, is shown to be quite useful. 

In fact, even the usual Hamiltonian H with a real potential and the usual boundary conditions is not always Hermitian the relation 

the Hermitian property of E — T has been assumed in shifting its position, because of which the scattering amplitude has been proved to be zero irrespective of the potential V and of the energy E. 

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This raises a dilemma in treating second- and higher-order properties in coupled-cluster theory. In the EOM-CC approach, which is basically a Cl calculation for a non-Hermitian Hamiltonian H= that incorporates... 

Here, /j and rj are the l" left- and the J right-hand eigenvectors of the non-Hermitian Hamiltonian H. The operator is represented on the space spanned by the manifold created by the excitations out of a Hartree-Fock reference determinant, including the null excitation (the reference function). When we calculate the transition probability between a ground state g) and an excited state ]e), we need to evaluate and The reference function is a right-... 

This contribution considers systems which can be described with just the Hamiltonian, and do not need a dissipative term so that TZd = 0- This would be the case for an isolated system, or in phenomena where the dissipation effects can be represented by an additional operator to form a new effective non-Hermitian Hamiltonian. These will be called here Hamiltonian systems. For isolated systems with a Hermitian Hamiltonian, the normalization is constant over time and the density operator may be constructed in a simpler way. In effect, the initial operator may be expanded in its orthonormal eigenstates (density amplitudes) and eigenvalues Wn (positive populations), where n labels the states, in the form... 

Further development of this analogy leads to the non-Hermitian Hamiltonian problem describing the Bose particles. Proceeding in this way, the classical diffusion problem could be related to quantum theory of multiple scattering [115-118]. 

According to Doi [107], Zeldovich and Ovchinnikov [35], the evolution of the state of a system given by the vector (t)) obeys the Schrodinger equation with imaginary time and non-Hermitian Hamiltonian. The averaging procedure also differs from that generally-accepted in quantum mechanics. 

Despite being called a state, a resonance does not show up as an eigenstate of an Hermitian Hamiltonian. However, as it represents a particle state that is localized in space for some time and that delocalizes with a small but finite rate, a resonance is reminiscent of a stationary state, but with a decaying norm. Indeed, it can be shown that we can represent resonance states as eigenstates of a non-Hermitian Hamiltonian, whose complex eigenvalues lie in the lower half of the complex plane. 

The first volume contained nine state-of-the-art chapters on fundamental aspects, on formalism, and on a variety of applications. The various discussions employ both stationary and time-dependent frameworks, with Hermitian and non-Hermitian Hamiltonian constructions. A variety of formal and computational results address themes from quantum and statistical mechanics to the detailed analysis of time evolution of material or photon wave packets, from the difficult problem of combining advanced many-electron methods with properties of field-free and field-induced resonances to the dynamics of molecular processes and coherence effects in strong electromagnetic fields and strong laser pulses, from portrayals of novel phase space approaches of quantum reactive scattering to aspects of recent developments related to quantum information processing. 

The decay of Nal can be described in an alternative way [K.B. Mpller, N.E. Henriksen, and A.H. Zewail, J. Chem. Phys. 113, 10477 (2000)]. In the bound region of the excited-state potential energy surface, one can define a discrete set of quasi-stationary states that are (weakly) coupled to the continuum states in the dissociation channel Na + I. These quasi-stationary states are also called resonance states and they have a finite lifetime due to the coupling to the continuum. Each quasi-stationary state has a time-dependent amplitude with a time evolution that can be expressed in terms of an effective (complex, non-Hermitian) Hamiltonian. 

The moment problem has been almost exclusively studied in the literature having (implicitly) in mind Hermitian operators (classical moment problem). With the progress of the modem projective methods of statistical mechanics and the description of relaxation phenomena via effective non-Hermitian Hamiltonians or Liouvillians, it is important to consider the moment problem also in its generalized form. In this section we consider some specific aspects of the classical moment problem, and in Section V.C we focus on peculiar aspects of the relaxation moment problem. 

The dynamics of this system is described by a non-hermitian Hamiltonian... 

However, much physical information is lost by this projection, and in addition this can be shown to produce instabilities. Otherwise one must simply use the non-Hermitian Hamiltonian matrix (61). The inclusion of sufficiently extensive basis sets will of course always reduce this problem. From a slightly different points of view, it can be said that the central purpose of the DIM approach is to reproduce the full Hamiltonian matrix—which contains important three- and four-centre contributions—by the expression (61), which involves no more than diatomic and atomic terms. This can only be done if the expansion (54) is essentially complete. Nonetheless, the DIM procedure presents a powerful approximation that has enjoyed fair success in a range of applications. 

The Chebychev method converges exponentially with the number of expansion terms n for a given step size A and is particularly advantageous and efficient when A is large. However, unlike short-time propagators such as SOD or SP, the Chebychev method is not directly applicable to time-dependent or non-Hermitian Hamiltonians. 

The above statement from [106] contains the essence of the theory and computations in terms of scaled, non-Hermitian Hamiltonians with complex coordinates that started a decade later. It is remarkable that the simple, one-page paper of Dykhne and Chaplik was not cited in the literature discussing the early developments of fhe theory of complex scaling of Hamiltonians, until 1981 [101]. 

Specifically, by diagonalizing an appropriately constructed non-Hermitian Hamiltonian matrix and by producing the final solution in the form of an expansion over configurations with complex coefficients,... 

The non-Hermitian formulation discussed in the present work is also different from a recent formulation involving non-Hermitian Hamiltonians where the mathematical axiom of Hermiticity is replaced by the condition of space-time reflection symmetry [39]. 

Recently the same problem has been reanalyzed by Dicus et al. [86], and indeed they confirmed that the survival probability deviates from exponential at long times. This model and its variants have been applied to study the effect of a distant detector (by adding an absorptive potential) [87], anomalous decay from a flat initial state [44], resonant state expansions [3], initial state reconstruction (ISR) [58], or the relevance of the non-Hermitian Hamiltonian concept (associated with a projector formalism for internal and external regions of space) in potential scattering [88]. In Ref. [88] the model was extended to a chain of delta functions to study overlapping resonances. 

Note that equation (4.62) can be considered as the expectation value of a similarity transformed (non-Hermitian) Hamiltonian. Since e tries to generate deexicitations from the reference Oif, which is impossible, eq. (4.62) is identical to eq. (4.58). Equations for the amplitudes are obtained by multiplying with an excited state. 

The second derivative terms 3 2 of the nonadiabatic coupling are mostly much smaller than the first derivative terms and hence are usually believed to be negligible. However, their omission will lead to a non-hermitian Hamiltonian due to a non-hermitian coupling... 

We treat the coupling between the ICD electron and the dication by means of the Lippmann-Schwinger equation extended to non Hermitian Hamiltonians ... 

However, this definition leads to a non-Hermitian Hamiltonian, which may not be the most optimal representation for interpretation. Therefore, one often adopts the... 

Bender, C. M. and Boettcher, S. (1998) Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry , Phys. Rev. Lett. 80, 5243. 

The left and right eigenvalues and the EOM-CC energy can be obtained from the matrix representing the non-Hermitian Hamiltonian in a suitable functional space. We limit... 

The calculation of N(E) by an absorbing boundary condition Green s function relies on the construction and inversion of a non-Hermitian Hamiltonian matrix. 

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