Singular vectors and invariant equations for the Schrödinger algebra in <Emphasis Type="Italic">n</Emphasis> ≥ 3 space dimensions. The general case

The singular vectors in Verma modules over the Schrödinger algebra ?(n) in (n + 1)-dimensional space-time are found for the case of general representations. Using the singular vectors, hierarchies of equations invariant under Schrödinger algebras are constructed.     

Higgs Bundles,Gauge Theories and Quantum Groups

The appearance of the Bethe Ansatz equation for the Nonlinear Schrödinger equation in the equivariant integration over the moduli space of Higgs bundles is revisited. We argue that the wave functions of the corresponding two-dimensional topological U(N) gauge theory reproduce quantum wave functions of the Nonlinear Schrödinger equation in the N-particle sector. This implies the full equivalence between the above gauge theory and the N-particle sub-sector of the quantum theory of the Nonlinear Schrödinger equation. This also implies the explicit correspondence between the gauge theory and the representation theory of the degenerate double affine Hecke algebra. We propose a similar construction based on the G/G gauged WZW model leading to the representation theory of the double affine Hecke algebra.     

The Vacuum Electromagnetic Fields and the Schrödinger Equation

We consider the simple case of a nonrelativistic charged harmonic oscillator in one dimension, to investigate how to take into account the radiation reaction and vacuum fluctuation forces within the Schrödinger equation. The effects of both zero-point and thermal classical electromagnetic vacuum fields, characteristic of stochastic electrodynamics, are separately considered. Our study confirms that the zero-point electromagnetic fluctuations are dynamically related to the momentum operator p=?i ? ?/? x used in the Schrödinger equation.     

Location of Maximizers of Eigenfunctions of Fractional Schrödinger’s Equations

Eigenfunctions of the fractional Schrödinger operators in a domain D are considered, and a relation between the supremum of the potential and the distance of a maximizer of the eigenfunction from ? D is established. This, in particular, extends a recent result of Rachh and Steinerberger arXiv:1608.06604 (2017) to the fractional Schrödinger operators. We also propose a fractional version of the Barta’s inequality and also generalize a celebrated Lieb’s theorem for fractional Schrödinger operators. As applications of above results we obtain a Faber-Krahn inequality for non-local Schrödinger operators.     

Feynman formulas generated by self-adjoint extensions of the Laplace operator

The Feynman formulas give a representation of a solution of the Cauchy problem for a Schrödinger-type equation (in a special case, for a heat-type equation) using the limit of integrals of finite multiplicity over Cartesian powers of the phase space (in the special case of the configuration space). The limit thus obtained, defining an explicit representation of a one-parameter unitary group e ^it? or a similar object (in our case, this concerns the semigroup e ^t? , which is often referred to in the literature as the Schrödinger semigroup) by integral operators, is interpreted by using Feynman integrals, whereas the expression thus obtained is referred in turn as the Feynman formula. As a rule, the Chernoff theorem, which is a generalization of the well known Trotter formula, is used in the derivation of the Feynman formula.In the paper, Feynman formulas for Schrödinger semigroups e ^t? are obtained, where the role of ? is played by the operator ? _a +V which is a perturbation of the self-adjoint extension of the Laplace operator (parametrized by some a ∈ (?∞, ∞]).     

Numerical calculation of energies of some excited states in a helium atom

The energies of some excited states with the total angular momentum L=0, 1 and 2. the total spin of two electrons S=0 and 1, and the even and odd parities are precisely calculated directly from the Schrödinger equation where the mass of the helium nucleus is finite. Moreover, we find that the solutions to the equation for the excited states have some more nodes, which can be used to distinguish the states with the same spectral term.     

Bounds on the Lyapunov Exponent via Crude Estimates on the Density of States

We study the Chirikov (standard) map at large coupling λ ? 1, and prove that the Lyapounov exponent of the associated Schrödinger operator is of order log λ except for a set of energies of measure exp(?c λ ^β ) for some 1 < β < 2. We also prove a similar (sharp) lower bound on the Lyapunov exponent (outside a small exceptional set of energies) for a large family of ergodic Schrödinger operators, the prime example being the d-dimensional skew shift.     

Path integral approach for spaces of nonconstant curvature in three dimensions

In this contribution, I show that it is possible to construct three-dimensional spaces of nonconstant curvature, i.e., three-dimensional Darboux spaces. Two-dimensional Darboux spaces have been introduced by Kalnins et al., with a path integral approach by the present author. In comparison to two dimensions, in three dimensions it is necessary to add a curvature term in the Lagrangian in order that the quantum motion can be properly defined. Once this is done, it turns out that, in the two three-dimensional Darboux spaces which are discussed in this paper, the quantum motion is similar to the two-dimensional case. In D _3d-I, we find seven coordinate systems which separate the Schrödinger equation. For the second space, D _3d-II, all coordinate systems of flat three-dimensional Euclidean space which separate the Schrödinger equation also separate the Schrödinger equation in D _3d-II. I solve the path integral on D _3d-I in the (u, v, w) system and on D _3d-II in the (u, v, w) system and in spherical coordinates.     

Weyl’s Type Estimates on the Eigenvalues of Critical Schrödinger Operators

We consider on a bounded domain \(\Omega \subset {\mathbb{R}}^N\) , the Schrödinger operator ? Δ ? V supplemented with Dirichlet boundary solutions. The potential V is either the critical inverse square potential V(x) = (N ? 2)²/4|x|^?2 or the critical borderline potential V(x) =  (1/4)dist(x, ?Ω)^?2. We present explicit asymptotic estimates on the eigenvalues of the critical Schrödinger operator in each case, based on recent results on improved Hardy–Sobolev type inequalities.     

The Geometry of the Semiclassical Wave Front Set for Schrödinger Eigenfunctions on the Torus

This paper deals with the phase space analysis for a family of Schrödinger eigenfunctions ψ _? on the flat torus ?? ⁿ = (?/2π?) ⁿ by the semiclassical Wave Front Set. We study those ψ _? such that WF_?(ψ _?) is contained in the graph of the gradient of some viscosity solutions of the Hamilton-Jacobi equation. It turns out that the semiclassical Wave Front Set of such Schrödinger eigenfunctions is stable under viscous perturbations of Mean Field Game kind. These results provide a further viewpoint, and in a wider setting, of the link between the smooth invariant tori of Liouville integrable Hamiltonian systems and the semiclassical localization of Schrödinger eigenfunctions on the torus.     

Global transformations preserving Sturm–Liouville spectral data

We show the existence of a real analytic isomorphism between the space of the impedance function ρ of the Sturm–Liouville problem ?ρ ^?2(ρ ² f′)′ +uf on (0, 1), where u is a function of ρ, ρ′, ρ″, and that of potential p of the Schrödinger equation ?y″ +py on (0, 1), keeping their boundary conditions and spectral data. This mapping is associated with the classical Liouville transformation f → ρf, and yields a global isomorphism between solutions of inverse problems for the Sturm–Liouville equations of the impedance form and those of the Schrödinger equations.     

Are wave mechanics and matrix mechanics equivalent theories?

The Eckart and Schrödinger proofs of 1926 are often described as having established the equivalence of wave mechanics and matrix mechanics as physical theories. The objective of this paper is to show that these “proofs” establish nothing of the kind. The Eckart-Schrödinger “proofs” have to do only with the formal identity of two different calculi. The question is, do the “proofs” establish the mathematical identity ofC ₁ andC ₂? Two views are possible: (1) Eckart and Schrödinger subsumed wave mechanics (C ₁) and matrix mechanics (C ₂) within a more comprehensive theory — which might be called “the operator calculus” (O). From this alone it does not follow thatC ₁ andC ₂ are formally identical. In general, the identity of two theories can never be established just by the fact that they both follow from the same premise. The other view (2) is thatO is simply a logical transformer which converts any statement ofC ₁ into a corresponding statement ofC ₂ — without adding any theoretical content of its own. That this is so could never beproved by an inductive selection of typical problems within microphysics; yet this is the actual procedure of Eckart and Schrödinger. Strictly speaking, one could consistently doubt thatC ₁ andC ₂ are ultimately identical even after sympathetically entertaining the Eckart-Schrödinger “proofs”. The really convincing argument for the equivalence asphysical theories of wave mechanics and matrix mechanics was provided by Born's statistical interpretation of theψ-function. Because here, in a frankly inductive procedure, Bornforces a physical interpretation onto bothC ₁ andC ₂ which at last makes it a matter of indifference which algorithm one chooses to express his predictions.     

Schrödinger’s Equation with Gauge Coupling Derived from a Continuity Equation

A quantization procedure without Hamiltonian is reported which starts from a statistical ensemble of particles of mass m and an associated continuity equation. The basic variables of this theory are a probability density ρ, and a scalar field S which defines a probability current j=ρ ? S/m. A first equation for ρ and S is given by the continuity equation. We further assume that this system may be described by a linear differential equation for a complex-valued state variable χ. Using these assumptions and the simplest possible Ansatz χ(ρ,S), for the relation between χ and ρ,S, Schrödinger’s equation for a particle of mass m in a mechanical potential V(q,t) is deduced. For simplicity the calculations are performed for a single spatial dimension (variable q). Using a second Ansatz χ(ρ,S,q,t), which allows for an explicit q,t-dependence of χ, one obtains a generalized Schrödinger equation with an unusual external influence described by a time-dependent Planck constant. All other modifications of Schrödinger’ equation obtained within this Ansatz may be eliminated by means of a gauge transformation. Thus, this second Ansatz may be considered as a generalized gauging procedure. Finally, making a third Ansatz, which allows for a non-unique external q,t-dependence of χ, one obtains Schrödinger’s equation with electrodynamic potentials A,φ in the familiar gauge coupling form. This derivation shows a deep connection between non-uniqueness, quantum mechanics and the form of the gauge coupling. A possible source of the non-uniqueness is pointed out.     

Inverse Spectral Problems for Schrödinger Operators

In this article we find some explicit formulas for the semi-classical wave invariants at the bottom of the well of a Schrödinger operator. As an application of these new formulas for the wave invariants, we improve the inverse spectral results proved by Guillemin and Uribe in [GU]. They proved that under some symmetry assumptions on the potential V(x), the Taylor expansion of V(x) near a non-degenerate global minimum can be recovered from the knowledge of the low-lying eigenvalues of the associated Schrödinger operator in \({\mathbb R^n}\) . We prove similar inverse spectral results using fewer symmetry assumptions. We also show that in dimension 1, no symmetry assumption is needed to recover the Taylor coefficients of V(x).     

The Schrödinger Equation,the Zero-Point Electromagnetic Radiation,and the Photoelectric Effect

A Schrödinger type equation for a mathematical probability amplitude Ψ(x,t) is derived from the generalized phase space Liouville equation valid for the motion of a microscopic particle, with mass M and charge e, moving in a potential V(x). The particle phase space probability density is denoted Q(x,p,t), and the entire system is immersed in the “vacuum” zero-point electromagnetic radiation. We show, in the first part of the paper, that the generalized Liouville equation is reduced to a simpler Liouville equation in the equilibrium limit where the small radiative corrections cancel each other approximately. This leads us to a simpler Liouville equation that will facilitate the calculations in the second part of the paper. Within this second part, we address ourselves to the following task: Since the Schrödinger equation depends on \(\hbar \), and the zero-point electromagnetic spectral distribution, given by \(\rho _{0}{(\omega )} = \hbar \omega ^{3}/2 \pi ^{2} c^{3}\), also depends on \(\hbar \), it is interesting to verify the possible dynamical connection between ρ₀(ω) and the Schrödinger equation. We shall prove that the Planck’s constant, present in the momentum operator of the Schrödinger equation, is deeply related with the ubiquitous zero-point electromagnetic radiation with spectral distribution ρ₀(ω). For simplicity, we do not use the hypothesis of the existence of the L. de Broglie matter-waves. The implications of our study for the standard interpretation of the photoelectric effect are discussed by considering the main characteristics of the phenomenon. We also mention, briefly, the effects of the zero-point radiation in the tunneling phenomenon and the Compton’s effect.     

A study of fractional Schrödinger equation composed of Jumarie fractional derivative

In this paper we have derived the fractional-order Schrödinger equation composed of Jumarie fractional derivative. The solution of this fractional-order Schrödinger equation is obtained in terms of Mittag–Leffler function with complex arguments, and fractional trigonometric functions. A few important properties of the fractional Schrödinger equation are then described for the case of particles in one-dimensional infinite potential well. One of the motivations for using fractional calculus in physical systems is that the space and time variables, which we often deal with, exhibit coarse-grained phenomena. This means infinitesimal quantities cannot be arbitrarily taken to zero – rather they are non-zero with a minimum spread. This type of non-zero spread arises in the microscopic to mesoscopic levels of system dynamics, which means that, if we denote x as the point in space and t as the point in time, then limit of the differentials dx (and dt) cannot be taken as zero. To take the concept of coarse graining into account, use the infinitesimal quantities as (Δx) ^α (and (Δt) ^α ) with 0 < α < 1; called as ‘fractional differentials’. For arbitrarily small Δx and Δt (tending towards zero), these ‘fractional’ differentials are greater than Δx (and Δt), i.e. (Δx) ^α > Δx and (Δt) ^α > Δt. This way of defining the fractional differentials helps us to use fractional derivatives in the study of dynamic systems.     

Estimation of transition probabilities in a hydrogen atom under the influence of a short electric pulse based on the path integral

Schemes for estimation of the path integral on the basis of the saddle-point method are tested. Estimation of the excitation probabilities of 1s–2s, 2p transitions in a hydrogen atom under the influence of a short electric pulse is chosen as a test problem. The obtained results are compared with the previous calculations based on the solution of the nonstationary Schrödinger equation by the finite element method.     

Justification of the Lattice Equation for a Nonlinear Elliptic Problem with a Periodic Potential

We justify the use of the lattice equation (the discrete nonlinear Schrödinger equation) for the tight-binding approximation of stationary localized solutions in the context of a continuous nonlinear elliptic problem with a periodic potential. We rely on properties of the Floquet band-gap spectrum and the Fourier–Bloch decomposition for a linear Schrödinger operator with a periodic potential. Solutions of the nonlinear elliptic problem are represented in terms of Wannier functions and the problem is reduced, using elliptic theory, to a set of nonlinear algebraic equations solvable with the Implicit Function Theorem. Our analysis is developed for a class of piecewise-constant periodic potentials with disjoint spectral bands, which reduce, in a singular limit, to a periodic sequence of infinite walls of a non-zero width. The discrete nonlinear Schrödinger equation is applied to classify localized solutions of the Gross–Pitaevskii equation with a periodic potential.     

Sharp Two–Sided Heat Kernel Estimates for Critical Schrödinger Operators on Bounded Domains

On a smooth bounded domain \(\Omega \subset {\bf {\rm R}}^N\) we consider the Schrödinger operators ? Δ ? V, with V being either the critical borderline potential V(x) =  (N ? 2)²/4 |x|^?2 or V(x) =  (1/4) dist(x, ?Ω)^?2, under Dirichlet boundary conditions. In this work we obtain sharp two-sided estimates on the corresponding heat kernels. To this end we transform the Schrödinger operators into suitable degenerate operators, for which we prove a new parabolic Harnack inequality up to the boundary. To derive the Harnack inequality we have established a series of new inequalities such as improved Hardy, logarithmic Hardy Sobolev, Hardy-Moser and weighted Poincaré. As a byproduct of our technique we are able to answer positively to a conjecture of E. B. Davies.