In particular the new ODE toolbox uses Matlab features which are not yet supported by Octave; we try to find workarounds. Interval vector and matrix operations are very fast due to extensive use of BLAS routines. However, nonlinear computations and loops may slow down the system significantly due to interpretation overhead and extensive use of the operator concept. Details and examples about timing can be found here.
Moreover, some routines treat extremely ill-conditioned problems. Details and examples are here. For download and installation under Octave see here. Many main routines work, but not everything, in particular concerning graphics. Kluwer Academic Publishers, Dordrecht, It runs under the forthcoming Matlab version Rb and is downward compatible at least to Rb. Rump at Hamburg University of Technology, Germany. This is independent of whether the commercial product is used privately or for commercial purposes. To obtain such a license contact the author Siegfried M.
Rump tuhh. However, neither an error-free processor nor an error-free program can be guaranteed. Finally, we show how one can obtain bounded wavefunctions which can approximate the Riemann—Zeta function in a more systematic though somewhat complicated fashion. We finally present a scheme based on the Degenerate Parametric Oscillator to measure the hyperbolic momentum in a physical system corresponding to the Degenerate Parametric Oscillator.
Legendre Duplication Formula
One interesting result of this study is that we are able to find a physical quantum system whose ground state wavefunction, when viewed in the appropriate basis, possesses zeros whose locations exactly match those of the Riemann—Zeta function on the critical line. It is not clear to us how our construction has relevance to the Hilbert—Polya formulation of the Riemann—Zeta hypothesis but the existence of relatively 'simple' quantum wavefunctions with randomly distributed zeros seems intriguing.
We now turn to the description of quantum mechanics on the half-line. There have been numerous expositions of the quantum mechanics of a particle moving in. Almost without exception, however, these analyses are executed in the position basis x. Here, we instead seek to make a closer correspondence between the conjugate phase-space operators , for a particle moving on the full-line , with analogous conjugate phase-space operators on the half-line.
As the latter must represent physical observables, they must be represented by Hermitian self-adjoint operators. To determine whether an operator , is Hermitian one must examine the action of the operator , and its adjoint , but also the domains , if is finite , of these operators, , and. If the actions of and are identical, i. Further it is Self-Adjoint if the actions match and so too do the domains of the operator and its adjoint,. An operator can be Hermitian but not self-adjoint and this will give rise to unphysical situations in that the action of may change the domain of the wavefunction, i.
Moreover, there is no self-adjoint extension possible for this operator. Before examining the case of , let us examine the somewhat easier case of the kinetic energy operator on the half-line,. As mentioned above we have to pay particular attention to the domain of operators on which H 0 is allowed to act.
If one chooses boundary conditions A , or B , then H 0 is Hermitian. From 10 , a more general class of boundary conditions which give an Hermitian and self-adjoint H 0 , is when. This choice of boundary condition extends the operator H 0 , into a one-parameter class of operators and this process is known as a self-adjoint extension.
In either case, the resulting operator , will be Hermitian but will not be self-adjoint. It is not obvious that we can choose alternative boundary conditions that will yield a self-adjoint extension of this non-self-adjoint operator. The method is quite simple and involves comparing the dimensions of the so-called, 'deficiency subspaces', , of the operator in question, e. These spaces are defined by. That the typical expression for the momentum operator , does not correspond to a physical operator on , seems to have only generated little attention in the literature.
On the other hand, as , the action of on , yields a displacement,. Instead we examine the operator which generates dilations of the operator:. To see if the operator is self-adjoint we use the von Neumann test and examine. Defining , this will be made more precise later on , we can recover the standard Heisenberg algebra and displacement operation, albeit in the 'exponent space': , and.
Thus by making the unitary transformation of the quantum mechanics on the half line described by the 'conjugate operators', , to the exponential representation on the full-line, , we can regain the familiar Heisenberg algebra. Dilations of now become displacements of , and one can formulate all of the standard QM on , in this hyperbolic phase-space.
We now evaluate the transition matrix elements , and resolutions of unity in the new basis in order to obtain the correct measures. First we look at the eigenstates of by taking , and where , to obtain. As , we can define the eigenstates of the coordinate conjugate to via the Fourier transform.
The quantum Mellin transform
We see that the eigenstates are not merely rescalings of the eigenstates but also include a square-root weighting. From 24 , we see that.
Thus , while from 24 , we have. As we mentioned above, one is typically familiar with the Fourier transform appearing in quantum mechanics representing the unitary change in basis.
Catalogue of GP/PARI Functions: Transcendental functions
To the best of our knowledge, the Mellin transform has never appeared in the literature playing a similar role. The Mellin transform and its inverse, of a function f is given by.
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We now return to our previous expression for the Lerch transcendent 6 , we see that. We can now ask about the quantum system where the wavefunction 45 , could appear. From 45 , we can find. From the form of 48 , we can add a constant to E , without changing the condition that 45 solves We note that. We see perfect correspondence between the wavefunction and the Zeta function zeros. Figure 1. Surprisingly a wavefunction which yields a perfect match can be found.
On the entire line, these states are defined by. However, just as the state , in the x -representation is unbounded and possesses infinite energy, the Riemann—Zeta function for is also formally unbounded and is defined there via analytic continuation. We now instead search for an intermediate wavefunction in the x -representation which does not possess any pathologies such as unbounded norm, infinite energy etc and which gives the Zeta function up to some multiplicative factor which damps far slower than above, i.
As we shall see, there are an infinite number of such examples. For the most part however, we will be unable to give simple analytic formulae for these wavefunctions.
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To find these wavefunctions we return to equation 7 ,. We see that the particular case of 59 , is obtained when we choose. An example of such a function is the one parameter family of functions. We see that the Mellin transform damps quickly with increasing t , for low values of [0,2. Figure 2. The W function however, is typically defined on the phase space x , p , where. For quantum mechanics on the half-line , we have shown above that one cannot define a self-adjoint momentum operator and thus we cannot construct an associated Wigner function.
Figure 3. Figure 4. One can see faint traces of the Riemann—Zeta zeros of the wavefunction within the Wigner function. Figure 5. Figure 6. Above we have given a mathematical derivation of a new type of unitary transformation which transforms quantum mechanics on the positive half line to a hyperbolic representation and in the process, executes a Mellin transform. The boundary condition 54 , is not typical. Instead we consider the alternative coupling Hamiltonian. We note however that 68 acts disjointly on the two half-line 'super-selection' sectors and.
From this we see that the probe's momentum is coupled to the hyperbolic momentum in each super-selection sector but the probe's position is no longer coupled to the hyperbolic position. To get a feeling for how 68 , provides us with an indirect measurement of the hyperbolic momentum of the system mode we examine the quantum Heisenberg equations of motion and their semiclassical c-number approximations,.
From this one can prove that , is a constant of the motion, while in addition. Using this constant of the motion one can arrive at coupled nonlinear dynamics for the probe mode alone,. This is a conservative system of nonlinear ordinary differential equations which corresponds precisely to the kinematics of a particle moving in two spatial dimensions x pb , p pb , under the central potential. The dynamics can be exactly solved via elliptic functions. One can find a series solution for small times to be.
Qualitatively the particle moves away from the origin initially with constant velocity as near the origin the force vanishes , but after a time , both x pb and p pb asymptote to fixed values as the particle comes to rest at the maximum of the potential V.