Hamiltonian level

We will first discuss the simple transformation of a 2 X 2 matrix to complex symmetric form. The model is quite general. We will further establish a simple condition for degeneracy with Segre characteristic r = 2 (see Refs. [6,9,10]). We will consider a specific metric A, which in general may not be positive definite. We will distinguish between two principal cases (1) All 22 12 21 0 and (2) An = — A22 = 1 A12 = A21 = 0. 

Before introducing any particular models, we realize that the secular problem below, c.f. the Hamiltonian models used in Ref. [9], can be written 

As already mentioned in the introduction, we will allow non-Hermitian extensions of quantum mechanics and hence it is natural to choose the first of the three forms in equation (3). Note that the formula in the middle is mostly used when working with positive definite metrics and self-adjoint Hamiltonians. 

A general theorem in linear algebra, see Ref. [10] for more details, says that any finite matrix representation can be transformed to complex symmetric form. For this reason, we can without loss of generality set up a complex symmetric model for our generalized dynamical picture based entirely on the parameters //u, H22, and v with 

As usual we will try to diagonalize the matrix H via a suitable similarity transformation, finding that it is not always possible. One obtains with r = 2 

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Recall that the rank of a homology group (as distinct from the rank of a fundamental group) is the number of independent infinite-order generators. Above, we have formulated the question Can any three-dimensional closed manifold be a Hamiltonian level surface of an integrable system The answer is given by the following theorem. 

The Topology of Hamiltonian-Level Surfaces of an Integrable System and of the Corresponding One-Dimensional Graphs... 

Consider the polyad = 6 of the Hamiltonian ( Al.2.7). This polyad contains the set of levels conventionally assigned as [6, 0, ], [5, 1],. . ., [0, 6], If a Hamiltonian such as ( Al.2.7) described the spectrum, the polyad would have a pattern of levels with monotonically varymg spacing, like that shown in figure Al.2.8. 

We now add die field back into the Hamiltonian, and examine the simplest case of a two-level system coupled to coherent, monochromatic radiation. This material is included in many textbooks (e.g. [6, 7, 8, 9, 10 and 11]). The system is described by a Hamiltonian having only two eigenstates, i and with energies = and Define coq = - co. The most general wavefunction for this system may be written as... 

The strategy for representing this differential equation geometrically is to expand both H and p in tenns of the tln-ee Pauli spin matrices, 02 and and then view the coefficients of these matrices as time-dependent vectors in three-dimensional space. We begin by writing die the two-level system Hamiltonian in the following general fomi. 

Nuclear spin relaxation is caused by fluctuating interactions involving nuclear spins. We write the corresponding Hamiltonians (which act as perturbations to the static or time-averaged Hamiltonian, detemiming the energy level structure) in tenns of a scalar contraction of spherical tensors ... 

The interaction of the electron spin s magnetic dipole moment with the magnetic dipole moments of nearby nuclear spins provides another contribution to the state energies and the number of energy levels, between which transitions may occur. This gives rise to the hyperfme structure in the EPR spectrum. The so-called hyperfme interaction (HFI) is described by the Hamiltonian... 

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