We study the scattering in a quantum star graph with a Fülöp-Tsutsui coupling in its vertex and with external potentials on the edges. We find certain special couplings, for which the probability of the transmission between two given lines of the graph is strongly influenced by the potential put on another line. On the basis of this phenomenon we design a tunable quantum band-pass spectra...
A set of gauge invariants are identified for the gauge theory of quantum anholonomies, which comprise both the Berry phase and an exotic anholonomy in eigenspaces. We examine these invariants for hierarchical families of quantum circuits whose qubit size can be arbitrarily large. It is also found that a hierarchical family of quantum circuits generally involves an NP-complete problem.
A quantum-like description of human decision process is developed, and a heuristic argument supporting the theory as sound phenomenology is given. It is shown to be capable of quantitatively explaining the conjunction fallacy in the same footing as the violation of sure-thing principle.
We study Hermitian unitary matrices with the following property: There exist and such that the entries of satisfy and for all , . We derive necessary conditions on the ratio and show that they are very restrictive except for the case when is eve...
We design two simple quantum devices applicable as an adjustable quantum spectral filter and as a flux controller. Their function is based upon the threshold resonance in a Fülöp-Tsutsui type star graph with an external potential added on one of the lines. Adjustment of the potential strength directly controls the quantum flow from the input to the output line. This is the first example to ...
Boundary conditions in quantum graph vertices are generally given in terms of a unitary matrix . Observing that if has at most two eigenvalues, then the scattering matrix of the vertex is a linear combination of the identity matrix and a fixed Hermitian unitary matrix, we construct vertex couplings with this property: For all momenta , the transmission probability from t...
An adiabatic cycle of parameters in a quantum system can yield the quantum anholonomy, nontrivial changes not just in phase of the states, but also in eigenvalues and eigenstates. Such exotic anholonomies imply that an adiabatic cycle transpose an eigenstate into another eigenstate, even under the absence of spectral degeneracy. We show that an arbitrary large quantum circuit can exhibit the ...
We demonstrate how the inverse scattering problem of a quantum star graph can be solved by means of diagonalization of Hermitian unitary matrix when the vertex coupling is of the scale invariant (or Fulop-Tsutsui) form. This enables the construction of quantum graphs with desired properties in a tailor-made fashion. In particular, the quantum vertices with equitransmitting scattering matrices a...
We discuss formulations of boundary conditions in a quantum graph vertex and demonstrate that the so-called -form can be further reduced up to a form more effective in certain applications: In particular, in identifying the number of independent parameters for given ranks of two connection matrices, or in calculating the scattering matrix when both matrices are singular. The new form of b...
The quantum decision theory is examined in its simplest form of two-condition two-choice setting. A set of inequalities to be satisfied by any quantum conditional probability describing the decision process is derived. Experimental data indicating the breakdown of classical explanations are critically examined with quantum theory using the full set of quantum phases.
The inverse scattering problem of a quantum star graph is shown to be solvable as a diagonalization problem of Hermitian unitary matrix when the connection condition is given by scale invariant Tsutsui-Fulop form. This enables the construction of quantum graphs with desired properties in tailor-made fashion. The quantum vertices with uniform and reflectionless scatterings are examined, and t...
This electronic lecture covers the theory of "quantum graph", which is one of the main subject under current research at the Mathematical Engineering Laboratory/ Theoretical Physics Group at Kochi University of Technology. The lecture is based on various review articles by the author which have appeared on Scientific journals in Japanese. The original audience of this web lecture was the new g...
This electronic lecture note on the cellular automaton models of traffic flow is an outgrowth of the materials pepared for the seminars and study meetings of the Mathematical Engineering Laboratory (aka, Theoretical Physics Group) at the Graduate School of Kochi University of Technology. It has been made accessible to KUT students who might be interested in joining us. Now it is open to gener...
This eNote is a translation of my lecture note prepared for a course in the first and second quaters of academic year 2003 at Graduate School, Kochi University of Technology. This is made open in the hope to attract general public to the emerging field of mathematical theory of evolution, that is the evolutionary game theory. (* Translation is done in great hurry, so grammatical and spelling eo...
