Waxman's Algorithm for non-Hermitian Hamiltonian Operators

An algorithm for finding the bound-state eigenvalues and eigenfunctions of a Hermitian Hamiltonian operator using Green's method, developed by Waxman\cite{W98},has been extended to include non-Hermitian Hamiltonian operators.

Non-Hermitian Hamiltonian operators have played an important role in many fields of physics. In nuclear physics, optical model calculations as well as Gamow shell model calculations have long been of interest in describing states in the continuum. Recently, a Gamow shell model description of weakly bound systems in neutron-rich nuclei involving configuration mixing in a single particle Berggren basis [2] has been given [3]. The Berggren basis contains bound single particle states as well as narrow resonances and non-resonant continuum. The Hamiltonian to be diagonalized in this basis is non-Hermitian. The Waxman Algorithm [1] is an iterative method based on Green's method that allows one to determine eigenstates of a Hamiltonian operator without matrix diagonalization.
Note Green's method may be applied to Hermitian as well as non-Hermitian operators. In the Waxman Algorithm approach, the coupling constant of the potential λ is determined numerically as a function of the eigenvalue, ε. ε is then varied until one obtains the value of λ used in the Hamiltonain operator. For non-Hermitian Hamitonian operators ε may be a complex number and an iterative algorithm is required to determine the complex eigenvalue corresponding to the real value of λ used in the Hamiltonian operator.
Consider the following eigenvalue problem whereT is the kinetic energy operator, λ is the real coupling constant, ε is the energy eigenvalue, andV is the potential energy operator. For non-Hermitian potentials the energy eigenvalues will in general be complex. For bound states the solution of Eq. (1) via Green's Method yields where the Green's operator,Ĝ ε , is defined aŝ and the vector |u is normalized with a reference vector r| such that With this, λ can be written as Eq. (5) can now be substituted into Eq. (2) |u =Ĝ εV |u r|ĜV |u .
For a chosen value of ε, Eq. (6) can be iterated until a convergent solution is obtained, at which point λ can be determined from Eq. (5).
If ε is chosen to be complex, λ determined from Eq. (5) will not necessarily be real. Using Re[λ] , Waxman's proof of convergence implies that there will be convergence of |λ| to the magnitude of the chosen real value of λ, λ ex , but convergence is not guaranteed to be a real solution. To converge to the real solution λ ex , i.e. where φ(λ)=0, the following method was developed.
For a matrix whose ground state is complex, an arbitrary value of ε = |ε|e iφ(ε) and a corresponding arbitrary eigenvector are chosen. |ε| is then varied incrementally until |λ| is within a small range close to the magnitude of the chosen real λ ex . FIGS. 1 and 2 show |λ| vs.
|ε| for a sample 20 x 20 non-Hermitian Hamiltonian Matrix whose lowest lying eigenvalue is complex. One can see that |λ| and |ε| are related linearly and varying φ(ε) causes a veritcal shift in |λ|.  This alternating procedure must be done in order to ensure |λ| is within a small range close to the magnitude λ ex .
Next, consider the case where the lowest lying eigenvalue is real. The iterations, however, are in the complex plane and will not always converge to the proper value of λ. In order to correct for this, the potential,V is perturbed slightly by δi * I, where δ < 1 and I is the identity matrix, such thatV + δi * I =V . ReplacingV withV in the Hamiltonian will result in a ground state with a complex eigenvalue and the algorithm as described above can be applied. FIGS. 4 and 5 show the convergence for this case.

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In the present work we have extended Waxman's algorithm to include non-Hermitian Hamiltonian operators. A convergent iterative scheme is presented to find the lowest lying eigenstate of such operators. For Hamiltonians whose ground state eigenvalues are real a simple prescription is given to guarantee convergence. Excited states may be obtained from a new start vector in which the the lower lying eigenstates are projected out of the original start vector.