Schrodinger equation complex eigenvalue

The earlier sections contain a number of prototypical ("proof-of-principle"-type) examples where these ideas and methods can be used. I point out that the possibility of constructing state-specific and, hence, compact and directly interpretable wavefunctions has allowed the implementation of practical methodologies for the ab initio nonperturbative solution of the complex-eigenvalue Schrodinger equation for unstable states that are created whenflc external fields are included [10] and of the time-dependent Schrodinger equation for various types of problems [17]. 

THE FORM OF THE RESONANCE EIGENFUNCTION AND THE COMPLEX EIGENVALUE SCHRODINGER EQUATION... 

This question was answered some time ago in Ref. [101], where we showed how the corresponding complex eigenvalue Schrodinger equation (CESE) is derived via the appropriate consideration of boundary condifions. The concomitant results justify the computation of resonance sfafes in ferms of non-Hermitian, complex-energy formalism via fhe use of superposifions of square-integrable real and complex funcfions. 

For example, this form is in harmony with the superposition of energy states in Eq. (2), whose coefficients have been obtained formally by Fano [29]. Although, for the solution of particular problems involving unstable states, we have implemented, in conjunction with the methods of the SSA, the real-energy, Hermitian, Cl in the continuum formalism that characterizes Fano s theory, e.g.. Refs. [78, 82-87] and Chapter 6, in this chapter I focused on the theory and the nonperturbative method of solution of the complex eigenvalue Schrodinger equation (CESE), Eq. (27). 

The value of the functions are complex numbers, but the eigenvalues of the Schrodinger equation are real numbers. 

The 2s and 2po orbitals are real, but the 2pi and 2p orbitals are complex. Since the four orbitals have the same eigenvalue Ei, any linear combination of them also satisfies the Schrodinger equation (6.12) with eigenvalue E2. Thus, we may replace the two complex orbitals by the following linear combinations to obtain two new real orbitals... 

Solving the time-dependent Schrodinger equation for resonance states [78] one obtains a set of complex eigenvalues, which may be written in the form... 

The Ck are complex coefficients. The functions and the energies Eu are obtained as eigenfunctions and eigenvalues of the solution of the time-independent Schrodinger equation ... 

However, due to the lack of Hermiticity, the spectrum of H includes, in addition to the normal spectrum of the ordinary Hamiltonian H, the possibility of complex eigenvalues. The occurence of such then causes the general operator exp(-iHt/h) to be undefined unless one considers it in a projected subspace (42). Within such a subspace comprised of all physical solutions to the time dependent Schrodinger equation, equation (82) is perfectly well behaved. Let us consider again the time independent equation... 

The operator theorems in the last section hint at an important aspect of wave mechanical descriptions The eigenvalue of some eigenequation can be formally isolated by the process of multiplying by the complex conjugate of the wavefunction and then integrating over the physical space of the system. In the case of the eigenvalue of the Schrodinger equation, we have... 

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