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Incoherence in the controlled Hamiltonian is an important limitation on the precision of coherent control in quantum information processing. Incoherence can typically be modeled as a distribution of unitary processes arising from slowly varying experimental parameters. We show how it introduces artifacts in quantum process tomography and we explain how the resulting estimate of the superoperator may not be completely positive. We then go on to attack the inverse problem of extracting an effective distribution of unitaries that characterizes the incoherence via a perturbation theory analysis of the superoperator eigenvalue spectra. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

We consider a family of Hamiltonians describing a quantum particle interacting, via zero-range forces, with a localized spin in the plane. A sub-class of Hamiltonians shows eigenvalues and/or resonances at the threshold of the continuous spectrum and generates particle dynamics not affecting the localized spin. We examine the effect of a small particle-spin interaction on the spectral structure of such unperturbed Hamiltonians. [ABSTRACT FROM AUTHOR]

In this paper, we build on our earlier proposal for the construction of a positive inner product for pseudo-Hermitian Hamiltonians and present examples to clarify the procedure. We focus on two detailed calculations where the method is used, namely, a simple (generalized 2×2 matrix) pseudo-Hermitian Hamiltonian, which can be diagonalized, and a second system where the Hamiltonian cannot be diagonalized, but can be described as a perturbation of the harmonic oscillator. When the quantum mechanical system cannot be diagonalized exactly, our construction can be carried out perturbatively and we develop the general formalism for such a perturbative calculation systematically (for real eigenvalues). [ABSTRACT FROM AUTHOR]

We study statistical models, specifically transfer matrices corresponding to a multiparameter hierarchy of braid matrices of (2n)2×(2n)2 dimensions with 2n2 free parameters (n=1,2,3,...). The simplest, 4×4 case, is treated in detail. Powerful recursion relations are constructed by explicitly giving the dependence on the spectral parameter θ of the eigenvalues of the transfer matrix at each level of coproduct sequence. A brief study of higher dimensional cases (n>=2) is presented, pointing out features of particular interest. Spin chain Hamiltonians are also briefly presented for the hierarchy. In a long final section, basic results are recapitulated with systematic analysis of their contents. Our eight vertex 4×4 case is compared to standard six vertex and eight vertex models. [ABSTRACT FROM AUTHOR]

This paper deals with an extension of the applications of the paper by Gelfand and Retakh [Funct. Anal. Appl. 25, 91 (1991)] on quasideterminant (QsD) algebraic method to eigenvalue problems in quantum mechanics. Using relevant identities on the free 1-mode bosonic algebra, we build characteristic QsDs associated with generalized spin-orbit Hamiltonians with a well defined representation which allows us to explicitly and straightforwardly compute analytical expressions of eigenenergies. Specific instances are provided on f-deformed generalized Jaynes–Cummings models and other Hamiltonian classes widely used in condensed matter physics. [ABSTRACT FROM AUTHOR]

It is possible to tell if two or more eigenvalues of a matrix are equal without calculating the eigenvalues. We use this property to detect (avoided) crossings in the spectra of Hamiltonians representable by matrices. This approach provides a pedagogical introduction to (avoided) crossings, is capable of handling realistic Hamiltonians analytically, and offers a way to visualize crossings that is sometimes superior to that provided by the spectrum directly. We illustrate the method using the Breit-Rabi Hamiltonian to describe the hyperfine-Zeeman structure of the ground-state hydrogen atom in a uniform magnetic field. [ABSTRACT FROM AUTHOR]

We prove the integrability of the general quantum Hamiltonian systems governed by an arbitrary non-skew-symmetric, so(3)-valued, nondynamical classical r-matrix with spectral parameters. We consider the most interesting example of these quantum integrable systems, namely, the so(3) “generalized Gaudin systems” in detail. In the case of an arbitrary r-matrix which is “diagonal” in the sl(2) basis we calculate the spectrum and the eigenvalues of the corresponding Hamiltonians using the algebraic Bethe ansatz technique. [ABSTRACT FROM AUTHOR]

We consider a self-adjoint, purely absolutely continuous operator M. Let P be a rank one operator Pu=⟨χ,u⟩χ such that for β0 Hβ0≔M+β0P has a simple eigenvalue E0 embedded in its absolutely continuous spectrum, with corresponding eigenvector ψ. Let Hω be a rank one perturbation of the operator Hβ0, namely, Hω =M+(β0+ω)P. Under suitable conditions, the operator Hω has no point spectrum in a neighborhood of Eω, for ω small. Here, we study the evolution of the state χ under the Hamiltonian Hω, in particular, we obtain explicit estimates for its sojourn time τω(ψ)= ʃ∞-∞ ∣ ⟨ψ,e-iHωtψ⟩∣² dt. By perturbation theory, we prove that τω(ψ) is is finite for ω≠0, and that for ω small it is of order ω-2. Finally, by using an analytic deformation technique, we estimate the sojourn time for the Friedrichs model in ℝn. [ABSTRACT FROM AUTHOR]

When an eigenvector of a semibounded operator is positive, we show that a remarkably simple argument allows to obtain upper and lower bounds for its associated eigenvalue. This theorem is a substantial generalization of Barta-type inequalities and can be applied to non-necessarily purely quadratic Hamiltonians. An application for a magnetic Hamiltonian is given and the case of a discrete Schrödinger operator is also discussed. It is shown how this approach leads to some explicit bounds on the ground-state energy of a system made of an arbitrary number of attractive Coulombian particles. [ABSTRACT FROM AUTHOR]

We consider the non-Hermitian eigenvalue problems -u″(z)±(iz)mu(z)=λu(z), m>=3, under every rapid decay boundary condition that is symmetric with respect to the imaginary axis in the complex z plane. We prove that the eigenvalues λ are all real and positive. [ABSTRACT FROM AUTHOR]

In this paper we study the integrability of natural Hamiltonian systems with a homogeneous polynomial potential. The strongest necessary conditions for their integrability in the Liouville sense have been obtained by a study of the differential Galois group of variational equations along straight line solutions. These particular solutions can be viewed as points of a projective space of dimension smaller by one than the number of degrees of freedom. We call them Darboux points. We analyze in detail the case of two degrees of freedom. We show that, except for a radial potential, the number of Darboux points is finite and it is not greater than the degree of the potential. Moreover, we analyze cases when the number of Darboux points is smaller than maximal. For two degrees of freedom the above-mentioned necessary condition for integrability can be expressed in terms of one nontrivial eigenvalue of the Hessian of potential calculated at a Darboux point. We prove that for a given potential these nontrivial eigenvalues calculated for all Darboux points cannot be arbitrary because they satisfy a certain relation which we give in an explicit form. We use this fact to strengthen maximally the necessary conditions for integrability and we show that in a generic case, for a given degree of the potential, there is only a finite number of potentials which satisfy these conditions. We also describe the nongeneric cases. As an example we give a full list of potentials of degree four satisfying these conditions. Then, investigating the differential Galois group of higher order variational equations, we prove that, except for one discrete family, among these potentials only those which are already known to be integrable are integrable. We check that a finite number of potentials from the exceptional discrete family are not integrable, and we conjecture that all of them are not integrable. [ABSTRACT FROM AUTHOR]

Some basic properties of the energy level sets for parameter-dependent systems are analyzed. A class of Hamiltonians depending linearly on two parameters is considered as a working example. It is shown that an approximate computation of the boundary of the level set can be useful to derive analytical representations of the eigenvalues of the system. The expressions deduced for the eigenenergies have the correct analytic behavior in the whole range of all physical parameters. In the simplest case, a nonvariational upper bound to the ground state energy is derived that satisfies both virial and Hellmann-Feynman theorems, as well as firstorder perturbation theory. Furthermore, some approximations are obtained, within the framework of the sealing variational method, that are numerical upper and lower bounds to the exact eigenvalues. Applications are illustrated with a family of anharmonic oscillators. [ABSTRACT FROM AUTHOR]

We derive a nearly optimal upper bound on the running time in the adiabatic theorem for a switching family of Hamiltonians. We assume the switching Hamiltonian is in the Gevrey class Gα as a function of time, and we show that the error in adiabatic approximation remains small for running times of order g-2 |ln g |6α. Here g denotes the minimal spectral gap between the eigenvalue(s) of interest and the rest of the spectrum of the instantaneous Hamiltonian. [ABSTRACT FROM AUTHOR]