Feynman formulas for particles with position-dependent mass

In this thesis we introduce the S-theory. We apply the S-theory to variational problems of immersed Riemann surfaces, Elasticity theory and Spin theory. 
\nFrom classical mechanics it is known that geometric problems can be substantially simplified by symmetry considerations. The Noether theorem describes in this case how to find conservation laws from the invariance property of the observed energy. At first, the motivation was to give the space of immersions M = {f : M → R3}, for an oriented Riemannian manifold M of dimension less than or equal to three, an Euclidean invariant description. For geometric energies, i.e. energies which are invariant under Euclidean transformations, one would be able to derive conservation laws for the respective variational problems. It is a well known fact that immersed space curves are uniquely determined by their curvature functions (κ1,κ2,τ) up to Euclidean transformations. In the case of immersed surfaces and immersed 3-manifolds the situation becomes more complicated and leads to a non conformal deformation theory (S-Theorie). 
\nThe S-Theory builds the foundations of this thesis. Starting from a Riemannian manifold M and a reference metric ⟨ , ⟩ one can model any other Riemmanian metric g through a positive definite and self adjoint operator S via g = ⟨S,S⟩. 
\nThe operator S is an isometry, i.e. g = S∗⟨,⟩. From this point of view we compute the Levi- Civita connection with respect to g out of Levi-Civita connection of the reference metric and the operator S. 
\nFurther we introduce Spin bundles over two and three dimensional oriented Riemannian manifolds. Thereby Spin bundles, in contrast to most of the literature, have additionally a quaternionic structure. Many formulas become clearer and more accessible. We show that any Spin bundle Σ over (M, ⟨ , ⟩) has a unique Spin connection and compute via the S-Theory the deformed Spin connection with respect to the metric g. Eventually we consider the induced Spin bundle of an immersion f : M → R3 and compute the corresponding Spin connection and find a new interpretation of the Gauss-Codazzi equation. We introduce the Dirac operator and compute its deformation with respect to the metric g. 
\nWe are now able to give the space of immersed surfaces M = {f : M → R3} the desired geometric description. We compute the normal space of M and formulate the Noether theorem. As an application we compute conservation laws for the Willmore functional. 
\nFinally we introduce the intrinsic version of the famous distance-squared energy on a n-dimensional Riemannian manifold. We introduce the corresponding stress tensor and show that it’s closeness is a characterization for critical points of the energy. Thereby the S-theory provides a fundamental concept for the understanding of the stress tensor. In the case of a Riemann surface, the S-theory provides a canonical harmonic rotation 1-form for critical points of the distance-squared energy.

Remark on the decay for damped string and beam equations

A new exactly solvable two-state model with an asymmetric pulse shape and a constant detuning is presented. It is an asymmetrized generalization of the Rosen-Zener model. The final populations of the states are expressed in terms of elementary functions and the phase of the gamma function. In contrast to the temporary asymmetric pulses studied so far, the transition probability in this model can vanish for a number of combinations of the free parameters.

We derive an exact solution of an explicitly time-dependent multichannel model of quantum mechanical nonadiabatic transitions. Our model corresponds to the case of a single linear diabatic energy level interacting with a band of an arbitrary N states, for which the diabatic energies decay with time according to the Coulomb law. We show that the time-dependent Schrödinger equation for this system can be solved in terms of Meijer functions whose asymptotics at a large time can be compactly written in terms of elementary functions that depend on the roots of an Nth order characteristic polynomial. Our model can be considered a generalization of the Demkov–Osherov model. In comparison to the latter, our model allows one to explore the role of curvature of the band levels and diabatic avoided crossings.

On lifting of operators to Hilbert spaces induced by positive selfadjoint operators

A two-dimensional ray model is introduced to realize the non-Markovian speedup evolution of a center massive particle gravitationally coupled to a controllable environment (multilayer arrangement of the massive particles). By controlling the environment, for instance by choosing a judicious mass of the environmental particles or by changing the separation distance of each massive particle, two dynamical crossover behaviors from Markovian to non-Markovian and from no-speedup to speedup are achieved due to the gravitational interactions between the system particle and environmental particles. It is obvious that the critical mass of the environmental particles or the critical separation distance for these two dynamical crossover behaviors restrict each other directly. The larger the value of the mass of the environmental particles is, the smaller the value of the critical separation distance should be requested. In addition, it should be emphasized that the non-Markovian dynamics is the principal physical reason for the speedup evolution of the system massive particle. Particularly, the non-Markovianity of the dynamics process of the system massive particle in the even ray case has better correspondence with the quantum speed limit time than that in the singular ray case.

It is known that the nonlinear nonhomogeneous backward Cauchy problem ut(t) + Au(t) = f (t, u(t)), 0 ⩽ t < τ with u(τ) = φ, where A is a densely defined positive self-adjoint unbounded operator on a Hilbert space, is ill-posed in the sense that small perturbations in the final value can lead to large deviations in the solution. We show, under suitable conditions on φ and f, that a solution of the above problem satisfies an integral equation involving the spectral representation of A, which is also ill-posed. Spectral truncation is used to obtain regularized approximations for the solution of the integral equation, and error analysis is carried out with exact and noisy final value φ. Also stability estimates are derived under appropriate parameter choice strategies. This work extends and generalizes many of the results available in the literature, including the work by Tuan (2010) for linear homogeneous final value problem and the work by Jana and Nair (2016b) for linear nonhomogeneous final value problem.

The function , where is Lerch's transcendent, satisfies the following two-dimensional formally self- adjoint second-order hyperbolic differential equation: where . The corresponding differential expression determines a densely defined symmetric operator (the minimal operator) on the Hilbert space , where . We obtain a description of the domains of definition of some symmetric extensions of the minimal operator. We show that formal solutions of the eigenvalue problem for these symmetric extensions are represented by functional series whose structure resembles that of the Fourier series of . We discuss sufficient conditions for these formal solutions to be eigenfunctions of the resulting symmetric differential operators. We also demonstrate a close relationship between the spectral properties of these symmetric differential operators and the distribution of the zeros of some special analytic functions analogous to the Riemann zeta function. Bibliography: 15 titles.

We study orthogonal and symmetric operators in non-Archimedean Hilbert spaces in the connection with p-adic quantization. This quantization describes measurements with finite precision. Symmetric (bounded) operators in the p-adic Hilbert spaces represent physical observables. We study spectral properties of one of the most important quantum operators, namely, the operator of the position (which is represented in the p-adic Hilbert L2-space with respect to the p-adic Gaussian measure). Orthogonal isometric isomorphisms of p-adic Hilbert spaces preserve precisions of measurements. We study properties of orthogonal operators. It is proved that each orthogonal operator in the non-Archimedean Hilbert space is continuous. However, there exist discontinuous operators with the dense domain of definition which preserve the inner product. There also exist nonisometric orthogonal operators. We describe some classes of orthogonal isometric operators and we study some general questions of the theory of non-Archimedean Hilbert spaces (in particular, general connections between topology, norm and inner product).

As we already pointed out in the introduction, the understanding of time observables in quantum mechanics has been hindered by the early appreciation that there could be no such “observable” in the standard, von Neumann formalism. A statement of such a view is that provided by Pauli [[1]], whereby a self-adjoint positive operator cannot have a self-adjoint conjugate operator.

Kernel-based Fisher discriminant analysis on the Riemannian manifold for nuclear atypia scoring of breast cancer

Multi-alkali photocathode in super generation image intensifier is different from previous multi-alkali photocathode between the production processes,so the photoelectric emission characteristics are different from previous multi-alkali photocathode.In this paper,through the measurement of multi-alkali photocathode spectral reflectivity and transmissivity,according to the law of conservation of energy,cathode spectral absorption rate was obtained.Spectral absorption rate indicates that,only when the photon greater than 1.333 eV,cathode absorption rates began to increase quickly.The cathode spectral absorption shows that cathode will not absorb any photons if light incident photon is less than the absorption limit,i.e.933 nm wave absorption limit.In the cathode surface electron affinity further reduced circumstances,cathode photoemission long wave theory threshold is determined by long-wave absorption limit.In the electronic transition process after absorption of a photon,transition energies increase less than the absorption of the incident photon energy,i.e.the presence of an energy loss.The higher of a photon is,the higher electronic transition level is,the more loss is.At the same time,the higher of a photon is;the higher-energy level transition electron,the lower electronic transition probability is.Photocathode quantum efficiency is determined by the absorption rate,the transition probability and transition level,energy loss of diffusion process and other factors,thus photocathode quantum efficiency is in the presence of long wave threshold and shortwave threshold also.Photocathode quantum efficiency in 587 nm reaches the maximum value,after that decreases with the photon increases,when 3.6 eV,the quantum efficiency is reduced to zero.Cathode in photon of 3.6 eV the absorption coefficient is still high,but due to the electronic transition probability is low,while the electron diffusion process of loss is big,thus in spite of cathode on lower wavelength has high absorption coefficient,but the quantum efficiency is still low.Therefore on shortwave,cathode absorbed shortwave photon is converted into a photoconductive,lattice vibration and other forms of energy;the photoelectric emission utilization rate is very low.

Let H be an abstract separable Hilbert space. We will consider the Hilbert space H1 whose elements are functionsf(x) with domain H and we will also consider the set of self-adjoint operators Q(x) in H of the form Q(x)=A+B(x). In this formula A≥E, B(x)≥0, and the operator B(x) is bounded for all x. An operator L0 is defined on the set of finite, infinitely differentiable (in the strong sense) functions y(x) eH1 according to the formula: L0y=−y″ + Q(x)y (−∞<x<∞). It is proved that the closure of the operator L0 is a self-adjoint operator in H1 under the given assumptions.

Chapter 9 deals with trace class perturbations of parts of the spectrum of self-adjoint operators. In the first section, the decomposition of a self-adjoint operator and its spectrum into a pure point part, a singularly continuous part, and an absolutely continuous part is obtained. Let A be a self-adjoint operator, and D a self-adjoint trace class operator acting on the same Hilbert space, and set B=A+D. Krein’s spectral shift associated with such a pair {B,A} is a function on the real line which involves an integral representation formula for the trace of the difference of functions of A and B. The spectral shift and its properties and the proof of Krein’s trace formula are central themes of this chapter. We follow M.G. Krein’s original approach to the spectral shift which was based on perturbation determinants. A short introduction into perturbation determinants is given in a separate section. Another main topic of Chap. 9 is the spectral theory of rank one perturbations B=A+α〈⋅,u〉u of the operator A, where α is a real number, and u is a unit vector. We prove the Aronszajn–Donoghue theorem. It characterizes the various parts of the spectrum of the self-adjoint operator B in terms of boundary values on ℝ of the holomorphic function 〈R z (A)u,u〉.

Relative form boundedness and compactness for a second-order differential operator