Albert Einstein, " Wolfgang E. Pauli, " This homepage addresses the following three Millenium problems (resp. links to corresponding homepages): : as 0 is an element of L(2) this means that the inf-term above is at most equal 1; therefore the theorem states that this value can be arbitrarily close approximated.A separable Hilbert scale can be built from the solutions of the eigenvalue equation K(x) = l*x, where K denotes a symmetric and compact operator: LemmaIn case the domain of such a compact operator is the L(2) Hilbert space the corresponding eigenfunctions build the basis of this Hilbert space. The concept of "wave package" enables also continuous spectra. Therefore, such "wave packages" require a domain extension (e.g. L(2) --> H(-1/2) in order to ensure convergent inner products and related norms. "Wave packages" are also called "eigen-differentials" (H. Weyl), playing a key role in quantum mechanics in the context of the spectral representation of hermitian operators (D. Hilbert, J. von Neumann, P. A. M. Dirac).: for no more than countable values l(i) the equation K(x)=l*x possesses non-trivial solutions x(i), where lim(l(i)) is equal zero.Schrödinger's "purely quantum wave" visionThis is about half-odd integers, rather than integers to be applied to wave-mechanical vibrations which correspond to the motion of particles of a gas resp. the eigenvalues and eigenfunctions of the harmonic quantum oscillator still governed by the Heisenberg uncertainty inequality. The alternatively proposed H(1/2) energy space enables Schrödinger's vision (see (ScE) Statistical Thermodynamic, The n-Particle Problem (7.23) ff): let w denotes the angular frequency, h the (h-bar) Planck constant and e := w*h/2): then Schrödinger's "half-odd integer vision" is about the following replacement: n=0: E(0) = e --> E(1/2) = 1 * e n>0: E(1) = 1* w*h --> E(3/2) = 2 * e E(2) = 2* w*h --> E(5/2) = 3 * e ... E(n) = n* w*h , n>0 --> E ((n+1)/2) = (n+1) * e , n=0,1.2.... As a consequence the corresponding eigenvalue and eigenfunction solutions of the number operator (i.e. the product of generation and annihilation operators) start with index n=1, not already with n=0.... The generalized Hermite polynomials satisfy different differential equations for even and odd polynomials. In (KrA) the spectral analysis for those generalized (even and odd) Hermite polynomials is provided. For the special case of the Schrödinger differential equation the spectrum of its related Schrödinger equation operator L is discrete, consisting of the odd integers. The corresponding eigenfunctions form a complete orthogonal set in the weighted-L(2) space. Our alternatively proposed Schrödinger (Calderón) equation operator differs from the standard operator L by its combination with the Hilbert-transform operator H and an extension of the L(2) space. This enables a spectral representation of the alternatively proposed Schrödinger equation operator with vanishing (!) constant Fourier term being replaced by a continuous spectrum summand (modelling the "ground state zero" eigenfunction/ eigendifferential) "governing" the corresponding complementary space of L(2).In (DaD) the analysis for the Schrödinger equation by wavelets is provided. The Berry conjecture is about the Riemann Zeta function as a model for the quantum chaos (BeM). A "space lab" view comment ...:)We can think (hear and watch) the Yoda quote "may the FORCE be with us" and mathematics can model this FORCE/POWER/ENERGY in a way that all corresponding physical (law) models are consistent; ... the bad (or good?) news is, that's it and that's all! ... or we are back to Kant's conception of physical matter in the Opus posthumum, where Kant postulated the existence of an ether which fills the whole space and time with its moving forces (Wong, W-C. G).
The proposed framework also provides an answer to Derbyshine's question, ("Prime Obsession") The answer, in a nutshell:
Regarding the proposed alternative quantization approach we also refer to the Berry-Keating conjecture. This is about an unknown quantization . This is in contrast to canonical quantization, which leads to the Heisenberg uncertainty principle and the natural numbers as spectrum of the harmonic quantum oscillator. The Hamiltonian needs to be self-adjoint so that the quantization can be a realization of the Hilbert-Polya conjecture.H
The Navier-Stokes equations describe the motion of fluids. The http://www.navier-stokes-equations.com The "standard" weak Hopf solution is not well posed (which is therefore also the case for the corresponding classication solution(s) due to not appropriately defined domains of the underlying velocity and pressure operators. The proposed solution also overcomes the "Serin gap" issue, as a consequence of the bounded non-linear term wih respect to the appropriate energy norm.
The YME are concerned with quantum field theory. Its related Millenium problem is about an appropriate mathematical model to govern the current "mass gap" of the YME, which is the difference in energy between the vacuum and the next lowest energy field. We propose to apply the same solution concept to solve the "mass gap" issue of the YME. This provides a truly infinitesimal geometry (H. Weyl), enabling the concept of Riemann that force is a pseudo force only, which results from distortions of the geometrical structure. The baseline is a common Hilbert space framework (for all (nearby action) differential equations) - providing the mathematical concept of a geometrical structure (while Riemann's manifold concept provides only a metric space and related affine connections) - replacing "force type" specific gauge fields and its combination model(s) for the electromagnetic, the strong and the weak nuclear power "forces" - building an integrated (no longer "force" dependent dynamical matter-field interaction laws) universal field model (including the gravity "force") As a consequence there is no "mass" and therefore no (YME-) "mass gap" anymore, but there is an appropriate vacuum (Hilbert) energy space, which is governed by the Heisenberg uncertainty principle. http://www.quantum-gravitation.de/
| ||||||||||||||