Stefan Giller - Academia.edu (original) (raw)

Papers by Stefan Giller

A method of fundamental solutions has been used to show its effectiveness in solving some well kn... more A method of fundamental solutions has been used to show its effectiveness in solving some well known problems of 1D quantum mechanics (barrier penetrations, over-barrier reflections, resonance states), i.e. those in which we look for exponentially small contributions to semiclassical expansions for considered quantities. Its usefulness for adiabatic transitions in two energy level systems is also mentioned.

Semiclassical wave functions in billiards based on the Maslov-Fedoriuk approach are constructed. ... more Semiclassical wave functions in billiards based on the Maslov-Fedoriuk approach are constructed. They are defined on classical constructions called skeletons which are the billiards generalization of Arnold's tori. Skeletons in the rational polygon billiards considered in the phase space can be closed with a definite genus or can be open being a cylinder-like or Möbius-like bands. The skeleton formulation is applied to calculate semiclassical wave functions and the corresponding energy spectra for the integrable and pseudointegrable billiards as well as in the limiting forms in some cases of chaotic ones. The superscars of Bogomolny and Schmit are shown to be simply singular semiclassical solutions of the eigenvalue problem in the billiards well built on the singular skeletons in the billiards with flat boundaries in both the integrable and the pseudointegrable billiards as well as in the chaotic cases of such billiards.

The rational billiards (RB) are classically pseudointegrable (P.J. Richens, M.V. Berry, Physica D... more The rational billiards (RB) are classically pseudointegrable (P.J. Richens, M.V. Berry, Physica D2, 495 (1981), Stefan Giller, arXiv: 1912.04155 [quant-ph]), i.e. their trajectories in the phase space lie on multi-tori of a genus g defined by 2g independent periods. Each such a multi-torus can be unfolded into elementary polygon pattern (EPP)-a smallest system of mirror images of RB obtained by their consecutive reflections by their sides and containing all different images of RB. A rational billiards Riemann surface (RBRS) corresponding to each RB is then an infinite mosaic made by a periodic distribution of EPP. Periods of RBRS are directly related to periodic orbits of RB. It is shown that any stationary solutions (SS) to the Schrödinger equation (SE) in RB can be extended on the whole RBRS. The extended stationary wave functions (ESS) are then periodic on RBRS with its periods. Conversely, for each system of boundary conditions (i.e. the Dirichlet or the the Neumann ones or their mixture) consistent with EPP one can find so called stationary pre-solutions (SPS) of the Schrödinger equation defined on RBRS and respecting its periodic structure together with their energy spectra. Using SPS one can easily construct SS of RB for most boundary conditions on it by a trivial algebra over SPS. It proves therefore that the energy spectra defined by the boundary conditions for SS corresponding to each RB are totally determined by 2g independent periods of RBRS being homogeneous functions of these periods. RBRS can be considered as a classical construction but in fact it can be done exclusively due to the rationality of the polygon billiards considered. Therefore the approach developed in the present paper can be seen as a new way in obtaining SS to SE in RB. On the other hand our results can be considered also as a generalization on the pseudointegrable systems of the well known semiclassical result corresponding to the integrable rational billiards being however general and exact. SPS can be constructed explicitly for a class of RB which EPP can be decomposed into a set of periodic orbit channel (POC) parallel to each other (POCDRB). For such a class of RB in which POCs find their natural place in the quantization of RB the respective RBRS can be built as a standard multi-sheeted Riemann surface (finitely sheeted in the case of doubly rational billiards (DRPB) and infinitely sheeted in other ones) with a periodic structure. For POCDRB a discussion of the existence of the superscar states (SSS)

The methods of the high energy semiclassical quantization in the rational polygon billiards used ... more The methods of the high energy semiclassical quantization in the rational polygon billiards used in our earlier papers are generalized to an arbitrary rational multi-connected polygon billiards i.e. to the billiards which is a rational polygon with other rational polygons inside them "rotated" with respect to the "mother" ones by rational angles. The respective procedure is described fully and its most important aspects are discussed. This generalization allows us to apply the method to arbitrary billiards with curved boundaries and with multi-connected areas where the respective semiclassical quantization is determined by the shortest periodic orbits of the billiards. As an example of the latter case the Sinai-like billiards is considered which is the right angle triangle with one of its acute angles equal to π/6 and with the circular hole in it.

In this paper a description of the small-ħ limit of loci of zeros of fundamental solutions for po... more In this paper a description of the small-ħ limit of loci of zeros of fundamental solutions for polynomial potentials is given. The considered cases of the potentials are bounded to the ones which provided us with simple turning points only. Among the latter potentials still several cases of Stokes graphs the potentials provide us with are distinguished, i.e. the general non-critical Stokes graphs, the general critical ones but with only single internal Stokes line and the Stokes graphs corresponding to arbitrary multiple-well real even degree polynomial potentials with internal Stokes lines distributed on the real axis only. All these cases are considered in their both versions of the quantized and not quantized ħ. In particular due to the fact that the small-ħ limit is semiclassical it is shown that loci of roots of fundamental solutions in the cases considered are collected along Stokes lines. There are infinitely many roots of fundamental solutions on such lines escaping to infin...

Two types of gauge transformations of noncommutative pure gauge theory are discussed. It is shown... more Two types of gauge transformations of noncommutative pure gauge theory are discussed. It is shown that Yang-Mills theory with the so called twisted gauge symmetry is consistent provided it also enjoys the standard noncommutative-gauge symmetry.

It is argued that the twisted gauge theory is consistent provided it exhibits also the standard n... more It is argued that the twisted gauge theory is consistent provided it exhibits also the standard noncommutative gauge symmetry. PACS number(s):

arXiv: Mathematical Physics, 2018

It is argued that the high energy semiclassical wave functions (SWF) in an arbitrary billiards ca... more It is argued that the high energy semiclassical wave functions (SWF) in an arbitrary billiards can be built by approximating the billiards by a respective polygon one. The latter billiards is determined by a finite number of periodic orbits of the original one limited by their lengths beginning with the shortest ones and which are common for both the billiards. The phenomenon of scars and superscars (Heller, E.J., {\it Phys. Rev. Lett.} {\bf 53},(1984) 1515) are then naturally incorporated into such a construction being a limit of periodic orbit channels (POCs) considered by Bogomolny and Schmit ({\it Phys. Rev. Lett.} {\bf 92} (2004) 244102). The Bunimovich stadium billiards is considered as an example of such an approach.

We consider the deformed Poincare group describing the space-time symmetry of noncommutative fiel... more We consider the deformed Poincare group describing the space-time symmetry of noncommutative field theory. It is shown that the deformed symmetry is equivalent to explicit symmetry breaking. PACS number(s): 11.10-z, 11.30Cp

Journal of Mathematical Physics, 2018

The superscars phenomena (Heller, E.J., Phys. Rev. Lett. 53, (1984) 1515) in the rational polygon... more The superscars phenomena (Heller, E.J., Phys. Rev. Lett. 53, (1984) 1515) in the rational polygon billiards (RPB) are analysed using the high energy semiclassical wave functions (SWF) built on classical trajectories forming skeletons. Considering examples of the pseudointegrable billiards such as the Bogomolny-Schmit triangle, the parallelogram and the L-shape billiards as well as the integrable rectangular one the constructed SWFs allow us to verify the idea of Bogomolny and Schmit (Phys. Rev. Lett. 92 (2004) 244102) of SWFs (superscars) propagating along periodic orbit channels (POC) and vanishing outside of them. It is shown that the superscars effects in RPB appear as natural properties of SWFs built on the periodic skeletons. The latter skeletons are commonly present in RPB and are always composed of POCs. The SWFs built on the periodic skeletons satisfy all the basic principles of the quantum mechanics contrary to the superscar states of Bogomolny and Schmit which break them. Therefore the superscars effects need not to invoke the idea of the superscar states of Bogomolny and Schmit at least in the cases considered in our paper.

Acta Physica Polonica B, 2015

A consistent scheme of semiclassical quantization in polygon billiards by wave function formalism... more A consistent scheme of semiclassical quantization in polygon billiards by wave function formalism is presented. It is argued that it is in the spirit of the semiclassical wave function formalism to make necessary rationalization of respective quantities accompanied the procedure of the semiclassical quantization in polygon billiards. Unfolding rational polygon billiards (RPB) into corresponding Riemann surfaces (RS) periodic structures of the latter are demonstrated with 2g independent periods on the respective multitori with g as their genuses. However it is the two dimensional real space of the real linear combinations of these periods which is just used for quantizing RPB's. Next a class of doubly rational polygon billiards (DRPB) is considered for which these real linear relations are rational and their semiclassical quantization by wave function formalism is presented. It is then shown that semiclassical quantization of both the classical momenta and the energy spectra are determined completely by periodic structure of the corresponding RS's. Each RS can be then reduced to elementary polygon patterns (EPP) as its basic periodic elements building it. Each such EPP can be glued to a torus of genus g. Semiclassical wave functions (SWF) are then constructed on EPP's. The SWF's for DRPB's appear to be exact and have forms of coherent sums of plane waves. They satisfy on the billiards boundaries well defined conditions-the Dirichlet, the Neumannn or the mixed ones. Not every mixing of such conditions is allowed however and the respective limitations can ignore some semiclassical states in the presented formalism. A respective incompleteness of SWF's provided by the method used in the paper is discussed. Families of DRPB's can form dens subsets of angle similar rational polygon billiards allowing for approximate semiclassical quantization of the latter. Next general rational polygons are quantized by approximating them by doubly rational ones. A natural extension of the formalism to irrational polygons is described shortly as well. When the semiclassical approximations constructed in the paper appear really as only approximations the latter are controlled by a general criteria of the eigenvalue theory. Finally a relation between the superscar solutions and SWF's constructed in the paper is also discussed.

In this paper a description of the small-h limit of loci of zeros of fundamental solutions for po... more In this paper a description of the small-h limit of loci of zeros of fundamental solutions for polynomial potentials is given. The considered cases of the potentials are bounded to the ones which provided us with simple turning points only. Among the latter potentials still several cases of Stokes graphs the potentials provide us with are distinguished, i.e. the general non-critical Stokes graphs, the general critical ones but with only single internal Stokes line and the Stokes graphs corresponding to arbitrary multiple-well real even degree polynomial potentials with internal Stokes lines distributed on the real axis only. All these cases are considered in their both versions of the quantized and not quantized h. In particular due to the fact that the small-h limit is semiclassical it is shown that loci of roots of fundamental solutions in the cases considered are collected along Stokes lines. There are infinitely many roots of fundamental solutions on such lines escaping to infinity and a finite number of them on internal Stokes lines.

Fourdimensional bicovariant differential *-calculus on quantum E(2) group is constructed. The rel... more Fourdimensional bicovariant differential *-calculus on quantum E(2) group is constructed. The relevant Lie algebra is obtained and covariant differential calculus on quantum plane is found.

The structure of Berrry's phase for time-reversal invariant systems is reviewed. The method o... more The structure of Berrry's phase for time-reversal invariant systems is reviewed. The method of constructing general «spin» Hamiltonians with quaternionic Berry's a holonomy is presented

The infinitesimal action of κ-Poincaré group on κ-Minkowski space is computed both for generators... more The infinitesimal action of κ-Poincaré group on κ-Minkowski space is computed both for generators of κ-Poincaré algebra and those of Woronowicz generalized Lie algebra. The notion of invariant operators is introduced and generalized Klein-Gordon equation is written out.

The infinitesimal action of κ-Poincaré group on κ-Minkowski space is computed both for generators... more The infinitesimal action of κ-Poincaré group on κ-Minkowski space is computed both for generators of κ-Poincaré algebra and those of Woronowicz generalized Lie algebra. The notion of invariant operators is introduced and generalized Klein-Gordon equation is written out.

A method of fundamental solutions has been used to show its effectiveness in solving some well kn... more A method of fundamental solutions has been used to show its effectiveness in solving some well known problems of 1D quantum mechanics (barrier penetrations, over-barrier reflections, resonance states), i.e. those in which we look for exponentially small contributions to semiclassical expansions for considered quantities. Its usefulness for adiabatic transitions in two energy level systems is also mentioned.

Semiclassical wave functions in billiards based on the Maslov-Fedoriuk approach are constructed. ... more Semiclassical wave functions in billiards based on the Maslov-Fedoriuk approach are constructed. They are defined on classical constructions called skeletons which are the billiards generalization of Arnold's tori. Skeletons in the rational polygon billiards considered in the phase space can be closed with a definite genus or can be open being a cylinder-like or Möbius-like bands. The skeleton formulation is applied to calculate semiclassical wave functions and the corresponding energy spectra for the integrable and pseudointegrable billiards as well as in the limiting forms in some cases of chaotic ones. The superscars of Bogomolny and Schmit are shown to be simply singular semiclassical solutions of the eigenvalue problem in the billiards well built on the singular skeletons in the billiards with flat boundaries in both the integrable and the pseudointegrable billiards as well as in the chaotic cases of such billiards.

The rational billiards (RB) are classically pseudointegrable (P.J. Richens, M.V. Berry, Physica D... more The rational billiards (RB) are classically pseudointegrable (P.J. Richens, M.V. Berry, Physica D2, 495 (1981), Stefan Giller, arXiv: 1912.04155 [quant-ph]), i.e. their trajectories in the phase space lie on multi-tori of a genus g defined by 2g independent periods. Each such a multi-torus can be unfolded into elementary polygon pattern (EPP)-a smallest system of mirror images of RB obtained by their consecutive reflections by their sides and containing all different images of RB. A rational billiards Riemann surface (RBRS) corresponding to each RB is then an infinite mosaic made by a periodic distribution of EPP. Periods of RBRS are directly related to periodic orbits of RB. It is shown that any stationary solutions (SS) to the Schrödinger equation (SE) in RB can be extended on the whole RBRS. The extended stationary wave functions (ESS) are then periodic on RBRS with its periods. Conversely, for each system of boundary conditions (i.e. the Dirichlet or the the Neumann ones or their mixture) consistent with EPP one can find so called stationary pre-solutions (SPS) of the Schrödinger equation defined on RBRS and respecting its periodic structure together with their energy spectra. Using SPS one can easily construct SS of RB for most boundary conditions on it by a trivial algebra over SPS. It proves therefore that the energy spectra defined by the boundary conditions for SS corresponding to each RB are totally determined by 2g independent periods of RBRS being homogeneous functions of these periods. RBRS can be considered as a classical construction but in fact it can be done exclusively due to the rationality of the polygon billiards considered. Therefore the approach developed in the present paper can be seen as a new way in obtaining SS to SE in RB. On the other hand our results can be considered also as a generalization on the pseudointegrable systems of the well known semiclassical result corresponding to the integrable rational billiards being however general and exact. SPS can be constructed explicitly for a class of RB which EPP can be decomposed into a set of periodic orbit channel (POC) parallel to each other (POCDRB). For such a class of RB in which POCs find their natural place in the quantization of RB the respective RBRS can be built as a standard multi-sheeted Riemann surface (finitely sheeted in the case of doubly rational billiards (DRPB) and infinitely sheeted in other ones) with a periodic structure. For POCDRB a discussion of the existence of the superscar states (SSS)

The methods of the high energy semiclassical quantization in the rational polygon billiards used ... more The methods of the high energy semiclassical quantization in the rational polygon billiards used in our earlier papers are generalized to an arbitrary rational multi-connected polygon billiards i.e. to the billiards which is a rational polygon with other rational polygons inside them "rotated" with respect to the "mother" ones by rational angles. The respective procedure is described fully and its most important aspects are discussed. This generalization allows us to apply the method to arbitrary billiards with curved boundaries and with multi-connected areas where the respective semiclassical quantization is determined by the shortest periodic orbits of the billiards. As an example of the latter case the Sinai-like billiards is considered which is the right angle triangle with one of its acute angles equal to π/6 and with the circular hole in it.

In this paper a description of the small-ħ limit of loci of zeros of fundamental solutions for po... more In this paper a description of the small-ħ limit of loci of zeros of fundamental solutions for polynomial potentials is given. The considered cases of the potentials are bounded to the ones which provided us with simple turning points only. Among the latter potentials still several cases of Stokes graphs the potentials provide us with are distinguished, i.e. the general non-critical Stokes graphs, the general critical ones but with only single internal Stokes line and the Stokes graphs corresponding to arbitrary multiple-well real even degree polynomial potentials with internal Stokes lines distributed on the real axis only. All these cases are considered in their both versions of the quantized and not quantized ħ. In particular due to the fact that the small-ħ limit is semiclassical it is shown that loci of roots of fundamental solutions in the cases considered are collected along Stokes lines. There are infinitely many roots of fundamental solutions on such lines escaping to infin...

Two types of gauge transformations of noncommutative pure gauge theory are discussed. It is shown... more Two types of gauge transformations of noncommutative pure gauge theory are discussed. It is shown that Yang-Mills theory with the so called twisted gauge symmetry is consistent provided it also enjoys the standard noncommutative-gauge symmetry.

It is argued that the twisted gauge theory is consistent provided it exhibits also the standard n... more It is argued that the twisted gauge theory is consistent provided it exhibits also the standard noncommutative gauge symmetry. PACS number(s):

arXiv: Mathematical Physics, 2018

It is argued that the high energy semiclassical wave functions (SWF) in an arbitrary billiards ca... more It is argued that the high energy semiclassical wave functions (SWF) in an arbitrary billiards can be built by approximating the billiards by a respective polygon one. The latter billiards is determined by a finite number of periodic orbits of the original one limited by their lengths beginning with the shortest ones and which are common for both the billiards. The phenomenon of scars and superscars (Heller, E.J., {\it Phys. Rev. Lett.} {\bf 53},(1984) 1515) are then naturally incorporated into such a construction being a limit of periodic orbit channels (POCs) considered by Bogomolny and Schmit ({\it Phys. Rev. Lett.} {\bf 92} (2004) 244102). The Bunimovich stadium billiards is considered as an example of such an approach.

We consider the deformed Poincare group describing the space-time symmetry of noncommutative fiel... more We consider the deformed Poincare group describing the space-time symmetry of noncommutative field theory. It is shown that the deformed symmetry is equivalent to explicit symmetry breaking. PACS number(s): 11.10-z, 11.30Cp

Journal of Mathematical Physics, 2018

The superscars phenomena (Heller, E.J., Phys. Rev. Lett. 53, (1984) 1515) in the rational polygon... more The superscars phenomena (Heller, E.J., Phys. Rev. Lett. 53, (1984) 1515) in the rational polygon billiards (RPB) are analysed using the high energy semiclassical wave functions (SWF) built on classical trajectories forming skeletons. Considering examples of the pseudointegrable billiards such as the Bogomolny-Schmit triangle, the parallelogram and the L-shape billiards as well as the integrable rectangular one the constructed SWFs allow us to verify the idea of Bogomolny and Schmit (Phys. Rev. Lett. 92 (2004) 244102) of SWFs (superscars) propagating along periodic orbit channels (POC) and vanishing outside of them. It is shown that the superscars effects in RPB appear as natural properties of SWFs built on the periodic skeletons. The latter skeletons are commonly present in RPB and are always composed of POCs. The SWFs built on the periodic skeletons satisfy all the basic principles of the quantum mechanics contrary to the superscar states of Bogomolny and Schmit which break them. Therefore the superscars effects need not to invoke the idea of the superscar states of Bogomolny and Schmit at least in the cases considered in our paper.

Acta Physica Polonica B, 2015

A consistent scheme of semiclassical quantization in polygon billiards by wave function formalism... more A consistent scheme of semiclassical quantization in polygon billiards by wave function formalism is presented. It is argued that it is in the spirit of the semiclassical wave function formalism to make necessary rationalization of respective quantities accompanied the procedure of the semiclassical quantization in polygon billiards. Unfolding rational polygon billiards (RPB) into corresponding Riemann surfaces (RS) periodic structures of the latter are demonstrated with 2g independent periods on the respective multitori with g as their genuses. However it is the two dimensional real space of the real linear combinations of these periods which is just used for quantizing RPB's. Next a class of doubly rational polygon billiards (DRPB) is considered for which these real linear relations are rational and their semiclassical quantization by wave function formalism is presented. It is then shown that semiclassical quantization of both the classical momenta and the energy spectra are determined completely by periodic structure of the corresponding RS's. Each RS can be then reduced to elementary polygon patterns (EPP) as its basic periodic elements building it. Each such EPP can be glued to a torus of genus g. Semiclassical wave functions (SWF) are then constructed on EPP's. The SWF's for DRPB's appear to be exact and have forms of coherent sums of plane waves. They satisfy on the billiards boundaries well defined conditions-the Dirichlet, the Neumannn or the mixed ones. Not every mixing of such conditions is allowed however and the respective limitations can ignore some semiclassical states in the presented formalism. A respective incompleteness of SWF's provided by the method used in the paper is discussed. Families of DRPB's can form dens subsets of angle similar rational polygon billiards allowing for approximate semiclassical quantization of the latter. Next general rational polygons are quantized by approximating them by doubly rational ones. A natural extension of the formalism to irrational polygons is described shortly as well. When the semiclassical approximations constructed in the paper appear really as only approximations the latter are controlled by a general criteria of the eigenvalue theory. Finally a relation between the superscar solutions and SWF's constructed in the paper is also discussed.

In this paper a description of the small-h limit of loci of zeros of fundamental solutions for po... more In this paper a description of the small-h limit of loci of zeros of fundamental solutions for polynomial potentials is given. The considered cases of the potentials are bounded to the ones which provided us with simple turning points only. Among the latter potentials still several cases of Stokes graphs the potentials provide us with are distinguished, i.e. the general non-critical Stokes graphs, the general critical ones but with only single internal Stokes line and the Stokes graphs corresponding to arbitrary multiple-well real even degree polynomial potentials with internal Stokes lines distributed on the real axis only. All these cases are considered in their both versions of the quantized and not quantized h. In particular due to the fact that the small-h limit is semiclassical it is shown that loci of roots of fundamental solutions in the cases considered are collected along Stokes lines. There are infinitely many roots of fundamental solutions on such lines escaping to infinity and a finite number of them on internal Stokes lines.

Fourdimensional bicovariant differential *-calculus on quantum E(2) group is constructed. The rel... more Fourdimensional bicovariant differential *-calculus on quantum E(2) group is constructed. The relevant Lie algebra is obtained and covariant differential calculus on quantum plane is found.

The structure of Berrry's phase for time-reversal invariant systems is reviewed. The method o... more The structure of Berrry's phase for time-reversal invariant systems is reviewed. The method of constructing general «spin» Hamiltonians with quaternionic Berry's a holonomy is presented

The infinitesimal action of κ-Poincaré group on κ-Minkowski space is computed both for generators... more The infinitesimal action of κ-Poincaré group on κ-Minkowski space is computed both for generators of κ-Poincaré algebra and those of Woronowicz generalized Lie algebra. The notion of invariant operators is introduced and generalized Klein-Gordon equation is written out.

The infinitesimal action of κ-Poincaré group on κ-Minkowski space is computed both for generators... more The infinitesimal action of κ-Poincaré group on κ-Minkowski space is computed both for generators of κ-Poincaré algebra and those of Woronowicz generalized Lie algebra. The notion of invariant operators is introduced and generalized Klein-Gordon equation is written out.