Izumi Tsutsui

Connection conditions and the spectral family under singular potentials

To describe a quantum system whose potential is divergent at one point, one must provide proper connection conditions for the wavefunctions at the singularity. Generalizing the scheme used for point interactions in one dimension, we present a set of connection conditions which are well defined even if the wavefunctions and/or their derivatives are divergent at the singularity. Our generalized scheme covers the entire U(2) family of quantizations (self-adjoint Hamiltonians) admitted for the singular system. We use this scheme to examine the spectra of the Coulomb potential V(x) = −e2/|x| and the harmonic oscillator with square inverse potential V(x) = (mω2/2)x2 + g/x2, and thereby provide a general perspective for these models which have previously been treated with restrictive connection conditions resulting in conflicting spectra. We further show that, for any parity invariant singular potential V(−x) = V(x), the spectrum is determined solely by the eigenvalues of the characteristic matrix U U(2).

Möbius structure of the spectral space of Schrödinger operators with point interaction

The Schrodinger operator with point interaction in one dimension has a U(2) family of self-adjoint extensions. We study the spectrum of the operator and show that (i) the spectrum is uniquely determined by the eigenvalues of the matrix U∈U(2) that characterizes the extension, and that (ii) the space of distinct spectra is given by the orbifold T2/Z2 which is a Mobius strip with boundary. We employ a parametrization of U(2) that admits a direct physical interpretation and furnishes a coherent framework to realize the spectral duality and anholonomy recently found. This allows us to find that (iii) physically distinct point interactions form a three-parameter quotient space of the U(2) family.

Spectral Properties on a Circle with a Singularity

We investigate the spectral and symmetry properties of a quantum particle moving on a circle with a pointlike singularity (or point interaction). We find that, within the U (2) family of the quantum mechanically allowed distinct singularities, a U (1) equivalence (of duality-type) exists, and accordingly the space of distinct spectra is U (1) ×[ SU (2) / U (1)], topologically a filled torus. We explore the relationship of special subfamilies of the U (2) family to corresponding symmetries, and identify the singularities that admit an N = 2 supersymmetry. Subfamilies that are distinguished in the spectral properties or the WKB exactness are also pointed out. The spectral and symmetry properties are also studied in the context of the circle with two singularities, which provides a useful scheme to discuss the symmetry properties on a general basis.

Duality symmetry and plane waves in non-commutative electrodynamics

Abstract We generalise the electric–magnetic duality in standard Maxwell theory to its non-commutative version. Both spacespace and spacetime non-commutativity are necessary. The duality symmetry is then extended to a general class of non-commutative gauge theories that goes beyond non-commutative electrodynamics. As an application of this symmetry, plane wave solutions are analysed. Dispersion relations following from these solutions show that, in the presence of spacetime non-commutativity, non-commutative electrodynamics admits two waves with distinct polarisations propagating at different velocities in the same direction.

Supersymmetric quantum mechanics with a point singularity

We study the possibility of supersymmetry (SUSY) in quantum mechanics in one dimension under the presence of a point singularity. The system considered is the free particle on a line or on the interval [−l,l] where the point singularity lies at x=0. In one dimension, the singularity is known to admit a U(2) family of different connection conditions which include as a special case the familiar one that arises under the Dirac delta δ(x)-potential. Similarly, each of the walls at x=±l admits a U(1) family of boundary conditions including the Dirichlet and the Neumann boundary conditions. Under these general connection/boundary conditions, the system is shown to possess an N=1 or N=2 SUSY for various choices of the singularity and the walls, and the SUSY is found to be ‘good’ or ‘broken’ depending on the choices made. We use the supercharge which allows for a constant shift in the energy, and argue that if the system is supersymmetric then the supercharge is self-adjoint on states that respect the connection/boundary conditions specified by the singularity.

Classical aspects of quantum walls in one dimension

We investigate the system of a particle moving on a half line x{>=}0 under the general walls at x=0 that are permitted quantum mechanically. These quantum walls, characterized by a parameter L, are shown to be realized as a limit of regularized potentials. We then study the classical aspects of the quantum walls by seeking a classical counterpart that admits the same time delay in scattering with the quantum wall, and also by examining the WKB exactness of the transition kernel based on the regularized potentials. It is shown that no classical counterpart exists for walls with L<0, and that the WKB exactness can hold only for L=0 and L={infinity}.

Supersymmetric quantum mechanics under point singularities

We provide a systematic study on the possibility of supersymmetry (SUSY) for one-dimensional quantum mechanical systems consisting of a pair of lines or intervals [−l, l] each having a point singularity. We consider the most general singularities and walls (boundaries) at x = ±l admitted quantum mechanically, using a U(2) family of parameters to specify one singularity and similarly a U(1) family of parameters to specify one wall. With these parameter freedoms, we find that for a certain subfamily the line systems acquire an N = 1 SUSY which can be enhanced to N = 4 if the parameters are further tuned, and that these SUSY are generically broken except for a special case. The interval systems, on the other hand, can accommodate N = 2 or N = 4 SUSY, broken or unbroken, and exhibit a rich variety of (degenerate) spectra. Our SUSY systems include the familiar SUSY systems with the Dirac δ(x)-potential, and hence are extensions of the known SUSY quantum mechanics to those with general point singularities and walls. The self-adjointness of the supercharge in relation to the self-adjointness of the Hamiltonian is also discussed.

Quantum Tunneling and Caustics under Inverse Square Potential

Abstract We examine the quantization of a harmonic oscillator with inverse square potential V(x)=(mω2/2) x2+g/x2 on the line −∞ 0, say, to the other x

Exchange Symmetry and Multipartite Entanglement

Entanglement of multipartite systems is studied based on exchange symmetry under the permutation group

Quantum mechanically induced Hopf term in the O(3) nonlinear sigma model

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