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

Abstract.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 R 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. *

show abstract

“…which provides a physical length scale to the system [10]. The condition (2.16) can then be solved by…”

Section: Supersymmetry On a Line With Point Singularitymentioning

confidence: 99%

Supersymmetric quantum mechanics with a point singularity

Uchino

Tsutsui

2003

Nuclear Physics B

Self Cite

View full text Add to dashboard Cite

show abstract

“…On the theoretical side, they have been found to exhibit a number of intriguing features, many of which have been seen before only in connection with quantum field theories. Examples include renormalization [1,2,3,4,5], Landau poles [6], anomalous symmetry breaking [5], duality [7,8,9], supersymmetry [9] and spectral anholonomy [9,10,11]. On the experimental side, the rapid developments of nanotechnology forecast that nano-scale quantum devices can be designed and manufactured into desired specifications.…”

Section: Introductionmentioning

confidence: 99%

Classical aspects of quantum walls in one dimension

2002

Self Cite

View full text Add to dashboard Cite

Abstract.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 which 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 = ∞. *

show abstract

“…In particular, the levels of the mirror-S 3 singlets depend on neither of the parameters ξ and ζ and hence are independent on the choice of V in U = V σ 3 V −1 . In fact, these are a consequence of the spectral preserving SU(2) (or its subgroup U(1)) transformations [29] which are found in the family of inequivalent quantizations we are considering here.…”

Section: Spectral Preserving Su (2) and The Angular Spectrummentioning

confidence: 99%

“…It can be shown [29] that by choosing c i appropriately one finds σ such that σ U σ = V −1 UV for any V ∈ SU(2). In other words, the transformation generated by Q in (5.12) yields the change 14) without altering the spectrum.…”

Section: Spectral Preserving Su (2) and The Angular Spectrummentioning

confidence: 99%

Inequivalent quantizations of the N=3 Calogero model with scale and mirror-S3 symmetry

Yonezawa

Tsutsui

2006

Journal of Mathematical Physics

Self Cite

View full text Add to dashboard Cite

We study the inequivalent quantizations of the N = 3 Calogero model by separation of variables, in which the model decomposes into the angular and the radial parts. Our inequivalent quantizations respect the 'mirror-S 3 ' invariance (which realizes the symmetry under the cyclic permutations of the particles) and the scale invariance in the limit of vanishing harmonic potential. We find a two-parameter family of novel quantizations in the angular part and classify the eigenstates in terms of the irreducible representations of the S 3 group. The scale invariance restricts the quantization in the radial part uniquely, except for the eigenstates coupled to the lowest two angular levels for which two types of boundary conditions are allowed independently from all upper levels. It is also found that the eigenvalues corresponding to the singlet representations of the S 3 are universal (parameter-independent) in the family, whereas those corresponding to the doublets of the S 3 are dependent on one of the parameters. These properties are shown to be a consequence of the spectral preserving SU (2) (or its subrgoup U (1)) transformations allowed in the family of inequivalent quantizations.

show abstract

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

Cited by 45 publications

References 12 publications

Supersymmetric quantum mechanics with a point singularity

Supersymmetric quantum mechanics with a point singularity

Classical aspects of quantum walls in one dimension

Inequivalent quantizations of the N=3 Calogero model with scale and mirror-S3 symmetry

Contact Info

Product

Resources

About