# On One Controllability of the Schrödinger Equation as Coupled with the Atomic-Level Mannesmann Effect

## Abstract

**:**

## 1. Introduction

**R**

^{n}, when we firstly consider the controlled linear Schrödinger equation [4] as:

**H**[5] of the system:

^{n}$\times \mathit{R}$

^{1}provided:

_{g}$\in \mathit{G}$ commutate with the Hamiltonian

**H**of the system. In other words, the control v localized on the subdomain ω of Ω, implying via a characteristic function ${\chi}_{\omega}$ of $\omega $ a noncommutativity of U and y, can possess a physical meaning under a commutativity relation (4) with U as a part of

**H**and v as a part of y(T) at a control time T. In such a way (

**G**, +) can be considered as the group of symmetry of the Schrödinger equation in (2).

^{2}$\text{}\left(\mathsf{\Omega}\right)$ with controls in $L$

^{2}$\text{}\left(\mathsf{\Omega}\times \left(0,\text{}T\right)\right)$, provided that there exists an unique solution $y\in C(\left[0,\text{}T\right];\text{}L$

^{2}$\text{}\left(\mathsf{\Omega}\right))$ for all y

_{0}∈ $L$

^{2}$\left(\mathsf{\Omega}\right)$ only if v ∈ $L$

^{2}$\left(\omega \times \left(0,\text{}T\right)\right)$ is determined via a solution of the generalized form of the Schrödinger equation and/or via the Klein-Gordon equation, respectively.

_{1}$\in L$

^{2}$\text{}\left(\mathsf{\Omega}\right)$”.

## 2. Material and Methods (How to Choose a Control $v$ and Time T for the Schrödinger Equation)

^{n}$\times \mathit{R}$

^{1}.

**H**IS NOT EQUAL TO THE ENERGY OPERATOR

**E**VIA ANY y = y(T) OF THE SYSTEM WITH RESPECT TO THE GROUP Z4.

_{.}

**Remark**

**1.**

**Proposition**

**1.**

**KGE**) of the Schrödinger equation, via whose we find complex valued $y\left(T\right)={y}_{k}\left(T\right)$ and an appropriate ${v}_{k}.$

- For $\mathit{H}\ne E$ we cannot consider the same mass m, but ${m}_{\mathit{H}}$ and ${m}_{E}$ at the control time T. Namely with respect to $\left(\mathsf{\Omega}\times \left(0,\text{}T\right)\right)$ and $\left(\omega \times \left(0,\text{}T\right)\right)$, or vice versa.
- It is analogously transferred via the usual form of the Klein-Gordon equation$$\left(\hslash {\partial}_{\mu}{\partial}_{\mu}+{m}^{2}\right)\text{}\varphi \left(x,t\right)=0\text{}in\text{}{\mathit{R}}^{\mathit{n}}\times {\mathit{R}}^{\mathbf{1}}$$Into an euclidean equation of motion$$\left(\hslash {\partial}_{\mu}{\partial}_{\mu}-{M}^{2}\right)\varphi \left(x,t\right)=0\text{}in\text{}{\mathit{R}}^{\mathbf{3}}\times {\mathit{R}}^{\mathbf{1}},\text{}\mu =0,\text{}1,\text{}2,\text{}3\phantom{\rule{0ex}{0ex}}\mathrm{with}\text{}m\to iM$$
- By subtitution $y\to \varphi \left(x,t\right)$ from (10) in (22) we get, using the TP-Complex, a complex valued mass in a form$${M}^{2}=\frac{{\hslash}^{2}{z}^{2}\left({z}^{2}-2z+1\right)}{{b}^{2}{t}^{2}\left({z}^{2}-2z-\mathit{\sigma}+1\right)\left({z}^{2}-2z+\mathit{\sigma}\right)}+\phantom{\rule{0ex}{0ex}}+\frac{i{\hslash}^{2}\mathrm{z}({z}^{5}\left(2\mathrm{b}+1\right)-2{z}^{4}\left(5b+3\right)-2{z}^{3}\left(b\left(2\sigma -9\right)+2\left(\sigma -3\right)+2{z}^{2}\left(b\left(6\sigma -7\right)+6\sigma -5\right)+z\left(2b\left({\sigma}^{2}-5\sigma +2\right)+3\left({\sigma}^{2}-3\sigma +1)\right)-2\sigma \left(\sigma -1\right)\left(b+1\right)\right)\right)}{{b}^{2}{t}^{2}{\left(-{z}^{2}+2z+\sigma \right)}^{\frac{3}{2}}\text{}{\left({z}^{2}-2z-\sigma +1\right)}^{\frac{3}{2}}}\phantom{\rule{0ex}{0ex}}\mathit{\sigma}:=\sigma \left[\mathit{K}\mathit{G}\mathit{E}\right]$$This reminds [12], where the complex valued mass for an unstable particle is written as$${M}^{2}\to {M}^{2}-iM\Gamma ,\text{}\Gamma \tau =\hslash .$$The quantity $\Gamma $ means the width of the particle decaying as $1\to n$ into n particles and $\tau $ is a corresponding lifetime of the particle.
- I is easy to see from the last relation, that via putting $\tau =T$, the imaginary part of (23) must vanish and a system is “stabilized”. The corresponding mass reads consequently:$$M=\frac{2{\left(z-1\right)}^{2}\sqrt{\left({z}^{4}-4{z}^{3}+{z}^{2}\left(5-2\mathit{\sigma}\right)+2{z}^{1}\left(2\mathit{\sigma}-1\right)+{\mathit{\sigma}}^{2}-\mathit{\sigma}\text{}\right)}+sign\left({z}^{5}-6{z}^{4}-4{z}^{3}\left(\mathit{\sigma}-3\right)+2{z}^{2}\left(6\mathit{\sigma}-5\right)+3z\left({\mathit{\sigma}}^{2}-3\mathit{\sigma}+1\right)-2{\mathit{\sigma}}^{2}+2\mathit{\sigma}\right)\u2502\hslash z/t\u2502}{{z}^{5}-6{z}^{4}-4{z}^{3}\left(\sigma -3\right)+2{z}^{2}\left(6\sigma -5\right)+3z\left({\sigma}^{2}-3\sigma +1\right)-2{\sigma}^{2}+2\sigma}\phantom{\rule{0ex}{0ex}}\mathit{\sigma}:=\sigma \left[\mathit{K}\mathit{G}\mathit{E}\right]$$
- Analogously like in (12), we let a possible co-prime generator pattern for Z4 vanish in the last relation, i.e.,$$-4{z}^{3}+2{z}^{1}\left(2\mathit{\sigma}-1\right)=0\text{}\mathrm{for}\text{}t=T\phantom{\rule{0ex}{0ex}}\mathit{\sigma}=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\left(2{z}^{2}+1\right),$$From which it follows that neither Z4 nor ($\mathit{G},+$) can be groups of symmetry of the
**KGE**. - In order to make a totally avoided but partially permitted coupling b between the groups Z4 and ($\mathit{G},+$) in (23) more transparent, we consider Z4 = {0, 1, 2, 3} only for the index $\mu $ taking values from this set with respect to coordinate-related azimuthal quantum number l in the Schrödinger equation, so that we allow $\sigma \to \mathit{\sigma}$ in (10) obtaining after the substitution from (26) the field$$\varphi \left(x,t\right)\propto \mathrm{exp}(i\text{}asin\left(\surd 2\sqrt{\left(4z+1\right)}/2\right)$$
- Finally we get the b-coupled control ${\mathit{v}}_{\mathit{b}}\equiv {v}_{k}$ applicable in the (10) at$$\varphi \left(x,t\right)\to {y}_{k}\left(T\right)\text{}for\text{}z+\sigma =0,\text{}\mathrm{for}\text{}\sigma \text{}\mathrm{chosen}\text{}\mathrm{from}\text{}{\sigma}_{2,3}\text{}\mathrm{with}\text{}t=T\text{}\mathrm{in}\text{}\left(18\right)$$Yielding explicitly$$k\propto \frac{(64\left(p-6\right)co{s}^{4}\left(i\text{}ln\left({y}_{k}\left(T\right)\right)-\left(p+2\right)\right)}{(2p\left(192co{s}^{4}\left(i\text{}ln\left({y}_{k}\left(T\right)\right)+1\right)\right)},\text{}p\in \omega \phantom{\rule{0ex}{0ex}}y\to 0,\text{}k=\frac{\left(p-6\right)}{6p}\text{}and\text{}z=\frac{1}{2}$$
**Consequence****1.**All zeros of the state y lie at$z=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$for Im$z=0,\text{}\forall \text{}p\text{}\ge 7$. This can imply that for all primes, i.e., including 2, 3, 5, y can transfer to the Riemann zeta function, emerging an appropriate non-zero imaginary part and leading thus in a quantum chaos, which we wanted to avoid for the control parameter$k>0$. - Putting further consequently for the non-zero $y\left(T\right)=u+i{\mathit{v}}_{\mathit{k}}$ in (19) with$$\varphi \left(x,t\right)\to {\mathit{v}}_{\mathit{k}}\text{}\mathrm{with}\text{}z+\sigma =0,\text{}\sigma \to l=0,\text{}1,\text{}2,\text{}3,\dots $$We get, for $u=\sqrt{\left(1-{\mathit{v}}_{\mathit{k}}{}^{\mathbf{2}}\right)}$ with ${\mathit{v}}_{\mathit{k}}:=exp\left(i\text{}asin\text{}\left(\frac{\sqrt{2}\sqrt{1-4l}}{2}\right)\right),$$$k=\frac{-4\mathrm{cos}{\left(\pi \text{}sign\left({v}_{k}\right)\sqrt{\left(-sign\left({v}_{k}{}^{2}-1\right)\right)}-\pi \text{}sign\left({v}_{k}\right)-2\mathrm{asin}\left({v}_{k}\right)\right)-\left(p+2\right)\mathrm{sin}\left(\pi \text{}sign\text{}({v}_{k}\text{}\right)\sqrt{\left(-sign\left({v}_{k}{}^{2}-1\right)\right)}-\pi \text{}sign\left({v}_{k}\right)-2\mathrm{asin}\left({v}_{k}\right))}^{2}+4}{2p{\left(2\mathrm{cos}\left(\text{}\pi \text{}sign\left({v}_{k}\right)\sqrt{\left(-sign\left({v}_{k}{}^{2}-1\right)\right)}-\pi \text{}sign\text{}\left({v}_{k}\right)-2\mathrm{asin}\left({v}_{k}\right)\right)-\mathrm{sin}(\pi \text{}sign\text{}\left({v}_{k}\right)\sqrt{\left(-sign\left({v}_{k}{}^{2}-1\right)\right)}-\pi \text{}sign\text{}\left({v}_{k}\right)-2\mathrm{asin}\left({v}_{k}\right)\right)}^{2}+2}$$

## 3. Results (The Order of Primary Cyclic Groups and Other Optimization Data with Their Important Application)

_{j}.

**D**of an occurrence of primes 2, 3, 5 over the control zone $\omega $, provided that such domain should be “tunnelled” as $\sigma \to l$, breaking the symmetry of

**D**for all $p\in \left(5,\text{}{p}_{j}\right).$ It follows immediately that, if

**D**represents (quantum-mechanically) some potential wall that a particle must tunnel in order to take place in some l – configuration, then a suppression of the explicit occurrence of (not perturbed) potential energy U in the model (1) is legal.

**D**as the representation of the potential wall, so that (through its violent tunnelling in a time $\tau $) we only realize a “goal”, i.e., the A5-symmetry of

**D**. Obviously, the governing equation of such process should be uniquely (22) in the Z4-symmetric, uncontrolled form”, i.e., with an inadmissibly perturbed solution $\varphi \left(x,\text{}t\right)={y}_{1}$ and nonzero source term $g\left(\tau \right):$

**D**. The corresponding structural damaging can be “dressed” in the by Z4-caused disintegration of the internal surface of semi-product, like it is demonstrated in the Figure 1. (Here the Mannesmann effect and its evolution were “replaced” by a removing of the material from the centre of the cylindrical billet by premature mechanical drilling.) It is reached a direct connection between A5 and Z4 by a “bypassing” the primes $p=7,\text{}11,\text{}13\in \omega $ for $k<0$, losing the control zone ω and connecting (2, 3, 5) with (17, 31, 47, 67) in the ω-avoiding way. As we see from a ring-shaped, four-times periodically “bitten” metal in the internal surface of the “pierced semi-product” in the Figure 1, the cyclic symmetry of this process is evident in some destructive composition of the LET and tunneling in the state ${y}_{1}$ at $k=0$ (avoiding both $k<0$ and ${k}_{j}$ > 0 in such a way). We deduce on this basis that contrary to the Linear Energy Transfer at ${k}_{j}$> 0, the controlled tunnelling occurs, consuming some amount of potential energy U for it and, “quasi-macroscopically” taken, with some plasticity suppression at an irregular plastic deformation during the Mannesmann process at $k<0$.

## 4. Discussions

- (a)
**Control parameter**${k}_{j}$**> 0**for regularly distributed primes ${p}_{j}$ = 17, 31, 47, 67 determining the order of the primary cyclic groups of symmetry of the model solutions in l-configurations of electrons (in atoms), corresponding specially with the LET-therapy of cancer (by C, Ne, Si, Ar–LET beams). This fact induces generally the existence of the controllable processes, the solutions of whose are given by a superposition of two, classically quite disjoint states of four chemical elements (we call such induced processes generally as the Schrödinger cat’s type problems).- (b)
**Control parameter**$k<0$ for the densely-as compared with ${p}_{j}$-distribution of the unique, cyclically repeated difference 4 in their serial numbers—distributed primes $p=7,\text{}11,\text{}13$ characterizing the control zone $\omega $ with the domain**D**existing over it. This domain is quantum-mechanically connected with a potential wall separating the groups A5 and Z4 mutually. “Quasi-macroscopically” viewed, the domain**D**is connected with a pierced metal semi-product. The goal of the tunneling (which is in a k-sense an opposite or complementary process to the LET) by an electron is a reaching of l-configuration of electrons within for the LET-beam usable atoms. The potential energy U is partially consumed for this process. Correspondingly, if “quasi-macroscopically” taken, the plasticity of the irregularly deformed and by means of the piercing plug “tunneled” metal semi-product is partially suppressed.- (c)
**Uncontrolled instance at**$k=0$. It is the individual case, when an existence of some target state y_{1}could be considered (however, here the physically substantiated existence of the control time T is excluded and cannot be even ad hoc defined) in the quantum-mechanically inadmissible macroscopic A5-symmetry of the domain**D**. The process of this A5-symmetrization of**D**is governed by the (not avoided) Z4-symmetrical inhomogeneous Klein-Gordon-like Equation (35). The solutions of this equation are in such an uncontrollable manner perturbed (due to inadmissible destructive composition of the LET and the tunnelling process) that the macroscopic objects (like a pierced semi-product in our case) are structurally disintegrated and damaged in a way, indicating the Z4 action on them. This action is (technologically) incorrectly permitted by a removing of the Mannesmann effect and its evolution via a (premature) mechanical removing of the central part of the metal semi-product (continuously casted billet). We have demonstrated it by a representative Figure 1 for the steel 42CrMo4.- (d)
**The goal of the processes**ad (a) and ad (b) is an avoidance of ad (c) connected with the target y_{1}. In the both admissible situations of controllability, a specific type of configuration of the groups is considered.—It is represented by the partial coupling b between the finitely generated abelian group ($\mathit{G},+$) and the cyclic group Z4 (without any isomorphism between them, as it would be the case at their complete mutual connection) with respect to a complete separation of Z4 and the alternating unsolvable simple group A5. This separation is existentially conditioned and preserved at the same time by such a tunnelling of the domain**D**that prevents any emergence or origin of its symmetry via the central emergence of the cavity origin, typical for it. In the Mannesmann process we call it the Mannesmann effect, when its evolution is mediated via the piercing plug tunnelling the corresponding barrier in a possible (controllable) perturbation in its shape-configuration.- (e)
**The control parameter k cannot be either implemented, or considered**. This case concerns the FEM of numerical simulation of the modelled process. The complete lack of the control parameter is the reason, why they are not suitable for the modelling of the process identified within the system (1). As we have concretely shown in the Figure 2, the law of the volume preservation is hardly violated during an attempt to simulate the piercing process, without any meaningful dependence on the used piercing plug shapes. This “simulation” therefore can reach only up to the piercing plug functionality. Some advantage follows from it, namely when such FEM-model is strictly examined by the mathematical model, especially as far as the mesh density choice and a range of removing of the Mannesmann effect are concerned, it can eventually roughly indicate a suitability of the used piercing plug from its resistance to move within the hole-point of view, how we can also observe in the Figure 2.

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## Conflicts of Interest

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**Figure 1.**Four ring-like internal surface-integrity violations with respect to Z4 in uncontrolled. Instance at $k=0.$

**Figure 2.**Failings of the law of the volume constancy at a FEM-simulation of Mannesmann process with a mechanically removed centre of billet and corresponding typical effective strain maps.

j | k_{j} | l | p_{j} | Serial Number S_{j} of P_{j} | $\mathbf{Order}\text{}\mathbf{of}\text{}\mathbf{the}\text{}\mathbf{Group}\text{}{\mathit{C}}_{{\mathit{p}}_{\mathit{j}}}^{{\mathit{l}}_{\mathit{j}}}$ | Number of Electrons = 2(2l_{j} + 1) | Characteristic (Periodic) Occurrence in Nature (Atoms) |
---|---|---|---|---|---|---|---|

1 | 91/6535 | 1 | 17 | 7 | 17^{1} | 6 | C |

2 | 77/63356 | 2 | 31 | 11 | 31^{2} | 10 | Ne |

3 | 9/17924 | 3 | 47 | 15 | 47^{3} | 14 | Si |

4 | 32/26229 | 4 | 67 | 19 | 67^{4} | 18 | Ar |

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**MDPI and ACS Style**

Perna, T.
On One Controllability of the Schrödinger Equation as Coupled with the Atomic-Level Mannesmann Effect. *Symmetry* **2020**, *12*, 1301.
https://doi.org/10.3390/sym12081301

**AMA Style**

Perna T.
On One Controllability of the Schrödinger Equation as Coupled with the Atomic-Level Mannesmann Effect. *Symmetry*. 2020; 12(8):1301.
https://doi.org/10.3390/sym12081301

**Chicago/Turabian Style**

Perna, Tomáš.
2020. "On One Controllability of the Schrödinger Equation as Coupled with the Atomic-Level Mannesmann Effect" *Symmetry* 12, no. 8: 1301.
https://doi.org/10.3390/sym12081301