For a non-Hermitian Hamiltonian H possessing a real spectrum, we introduce a canonical orthonormal basis in which a previously introduced unitary mapping of H to a Hermitian Hamiltonian h takes a simple form. We use this basis to construct the observables O α of the quantum mechanics based on H. In particular, we introduce pseudo-Hermitian position and momentum operators and a pseudo-Hermitian quantization scheme that relates the latter to the ordinary classical position and momentum observables. These allow us to address the problem of determining the conserved probability density and the underlying classical system for pseudo-Hermitian and in particular P T -symmetric quantum systems. As a concrete example we construct the Hermitian Hamiltonian h, the physical observables O α , the localized states, and the conserved probability density for the non-Hermitian P T -symmetric square well. We achieve this by employing an appropriate perturbation scheme. For this system, we conduct a comprehensive study of both the kinematical and dynamical effects of the non-Hermiticity of the Hamiltonian on various physical quantities. In particular, we show that these effects are quantum mechanical in nature and diminish in the classical limit. Our results provide an objective assessment of the physical aspects of P T -symmetric quantum mechanics and clarify its relationship with both the conventional quantum mechanics and the classical mechanics.

We present a generalization of the perturbative construction of the metric operator for non-Hermitian Hamiltonians with more than one perturbation parameter. We use this method to study the non-Hermitian scattering Hamiltonian:where ζ ± and α are respectively complex and real parameters and δ(x) is the Dirac delta function. For regions in the space of coupling constants ζ ± where H is quasi-Hermitian and there are no complex bound states or spectral singularities, we construct a (positive-definite) metric operator η and the corresponding equivalent Hermitian Hamiltonian h. η turns out to be a (perturbatively) bounded operator for the cases that the imaginary part of the coupling constants have opposite sign, ℑ(ζ + ) = −ℑ(ζ − ). This in particular contains the PT -symmetric case: ζ + = ζ * − . We also calculate the energy expectation values for certain Gaussian wave packets to study the nonlocal nature of h or equivalently the non-Hermitian nature of H. We show that these physical quantities are not directly sensitive to the presence of PT -symmetry.A Hamiltonian operator H is called PT -symmetric if it has parity-time reversal symmetry, i.e., [H, PT ] = 0. Since the publication of the pioneering work of Bender and Boettecher [1] non-Hermitian PT -symmetric Hamiltonians have received much attention. This has led to the discovery of a number of interesting theoretical [2,3,4,5,6,7] as well as experimental [8] implications of PT -symmetric Hamiltonians. For extensive reviews see [9,10] and references therein.Perhaps the most prominent feature of a non-Hermitian PT -symmetric Hamiltonian H is that its spectrum is symmetric about the real axis of the complex plane. In particular, if H has a discrete spectrum, either it is purely real or the nonreal eigenvalues come in complex-conjugate pairs [1,7]. It turns out that this is a characteristic property of a wider class of non-Hermitian Hamiltonian operators called the pseudo-Hermitian operators [11,12,13]. A Hamiltonian H is said to be pseudo-Hermitian if its adjoint H † satisfies

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We study the backstepping stabilization of higher order linear and nonlinear Schrödinger equations on a finite interval, where the boundary feedback acts from the left Dirichlet boundary condition. The plant is stabilized with a prescribed rate of decay. The construction of the backstepping kernel is based on a challenging successive approximation analysis. This contrasts with the case of second order pdes. Second, we consider the case where the full state of the system cannot be measured at all times but some partial information such as measurements of a boundary trace are available. For this problem, we simultaneously construct an observer and the associated backstepping controller which is capable of stabilizing the original plant. Wellposedness and regularity results are provided for all pde models. Although the linear part of the model is similar to the KdV equation, the power type nonlinearity brings additional difficulties. We give two examples of boundary conditions and partial measurements. We also present numerical algorithms and simulations verifying our theoretical results to the fullest extent. Our numerical approach is novel in the sense that we solve the target systems first and obtain the solution to the feedback system by using the bounded invertibility of the backstepping transformation.

In this paper, we prove the output feedback stabilization for the linearized Korteweg-de Vries (KdV) equation posed on a finite domain in the case the full state of the system cannot be measured. We assume that there is a sensor at the left end point of the domain capable of measuring the first and second order boundary traces of the solution. This allows us to design a suitable observer system whose states can be used for constructing boundary feedbacks acting at the right endpoint so that both the observer and the original plant become exponentially stable. Stabilization of the original system is proved in the L 2 -sense, while the convergence of the observer system to the original plant is also proved in higher order Sobolev norms. The standard backstepping approach used to construct a left endpoint controller fails and presents mathematical challenges when building right endpoint controllers due to the overdetermined nature of the related kernel models. In order to deal with this difficulty we use the method of [18] which is based on using modified target systems involving extra trace terms. In addition, we show that the number of controllers and boundary measurements can be reduced to one, with the cost of a slightly lower exponential rate of decay. We provide numerical simulations illustrating the efficacy of our controllers. of the following problem (so called "target system"):in Ω × R + , w(0, t) = w(L, t) = w x (L, t) = 0 in R + , w(x, 0) = w 0 (x) ≡ u 0 − x 0 k(x, y)u 0 (y)dy in Ω.