18 research outputs found
Moebius Structure of the Spectral Space of Schroedinger Operators with Point Interaction
The Schroedinger 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 belonging to U(2) that characterizes the extension, and that (ii) the
space of distinct spectra is given by the orbifold T^2/Z_2 which is a Moebius
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.Comment: 16 pages, 2 figure
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 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 = infinity.Comment: TeX, 21 pages, 4 figures. v2: some parts of the text improved, new
and improved figure
Quantum contact interactions
The existence of several exotic phenomena, such as duality and spectral
anholonomy is pointed out in one-dimensional quantum wire with a single defect.
The topological structure in the spectral space which is behind these phenomena
is identified.Comment: A lecture presented at the 2nd Winter Institute on Foundations of
Quantum Theory and Quantum Optics (WINST02), Jan. 2-11, 2002, S.N.Bose
Institute, Calcutta, India: 8 pages latex with Indian Acad. Sci. style fil
Boundary effect of a partition in a quantum well
The paper wishes to demonstrate that, in quantum systems with boundaries,
different boundary conditions can lead to remarkably different physical
behaviour. Our seemingly innocent setting is a one dimensional potential well
that is divided into two halves by a thin separating wall. The two half wells
are populated by the same type and number of particles and are kept at the same
temperature. The only difference is in the boundary condition imposed at the
two sides of the separating wall, which is the Dirichlet condition from the
left and the Neumann condition from the right. The resulting different energy
spectra cause a difference in the quantum statistically emerging pressure on
the two sides. The net force acting on the separating wall proves to be nonzero
at any temperature and, after a weak decrease in the low temperature domain, to
increase and diverge with a square-root-of-temperature asymptotics for high
temperatures. These observations hold for both bosonic and fermionic type
particles, but with quantitative differences. We work out several analytic
approximations to explain these differences and the various aspects of the
found unexpectedly complex picture.Comment: LaTeX (with iopart.cls, iopart10.clo and iopart12.clo), 28 pages, 17
figure
Quantum Force Induced on a Partition Wall in a Harmonic Potential
Boundary effects in quantum mechanics are examined by considering a partition
wall inserted at the centre of a harmonic oscillator system. We put an equal
number of particles on both sides of the impenetrable wall keeping the system
under finite temperatures. When the wall admits distinct boundary conditions on
the two sides, then a net force is induced on the wall. We study the
temperature behaviour of the induced force both analytically and numerically
under the combination of the Dirichlet and the Neumann conditions, and
determine its scaling property for two statistical cases of the particles:
fermions and bosons. We find that the force has a nonvanishing limit at zero
temperature T = 0 and exhibits scalings characteristic to the statistics of the
particles. We also see that for higher temperatures the force decreases
according to 1/sqrt{T}, in sharp contrast to the case of the infinite potential
well where it diverges according to sqrt{T}. The results suggest that, if such
a nontrivial partition wall can be realized, it may be used as a probe to
examine the profile of the potentials and the statistics of the particles
involved.Comment: 22 pages, 7 figures, typos corrected, references adde
Inequivalent quantizations of the three-particle Calogero model constructed by separation of variables
We quantize the 1-dimensional 3-body problem with harmonic and inverse square
pair potential by separating the Schr\"odinger equation following the classic
work of Calogero, but allowing all possible self-adjoint boundary conditions
for the angular and radial Hamiltonians. The inverse square coupling constant
is taken to be with and then the angular
Hamiltonian is shown to admit a 2-parameter family of inequivalent
quantizations compatible with the dihedral symmetry of its potential term
. These are parametrized by a matrix
satisfying , and in all cases we describe the
qualitative features of the angular eigenvalues and classify the eigenstates
under the symmetry and its subgroup generated by the particle
exchanges. The angular eigenvalue enters the radial Hamiltonian
through the potential allowing a 1-parameter family of
self-adjoint boundary conditions at if . For
our analysis of the radial Schr\"odinger equation is consistent with previous
results on the possible energy spectra, while for it shows that
the energy is not bounded from below rejecting those 's admitting such
eigenvalues as physically impermissible. The permissible self-adjoint angular
Hamiltonians include, for example, the cases ,
which are explicitly solvable and are presented in detail. The choice reproduces Calogero's quantization, while for the choice the
system is smoothly connected to the harmonic oscillator in the limit .Comment: 45 pages, 6 figures, LaTeX, v2: a reference and a note added, v3:
merely a constant is correcte
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) x [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.Comment: TeX, 26 pages. v2: one reference added and two update
Two fermion relativistic bound states: hyperfine shifts
We discuss the hyperfine shifts of the Positronium levels in a relativistic
framework, starting from a two fermion wave equation where, in addition to the
Coulomb potential, the magnetic interaction between spins is described by a
Breit term. We write the system of four first order differential equations
describing this model. We discuss its mathematical features, mainly in relation
to possible singularities that may appear at finite values of the radial
coordinate. We solve the boundary value problems both in the singular and non
singular cases and we develop a perturbation scheme, well suited for numerical
computations, that allows to calculate the hyperfine shifts for any level,
according to well established physical arguments that the Breit term must be
treated at the first perturbative order. We discuss our results, comparing them
with the corresponding values obtained from semi-classical expansions.Comment: 16 page
An approximation to couplings on graphs
We discuss a general parametrization for vertices of quantum graphs and show,
in particular, how the and coupling at an edge vertex
can be approximated by means of couplings of the type provided
the latter are properly scaled.Comment: 10 pages, LaTeX, 1 figure added, to be published in J. of Phys.
Vacuum Energy and Renormalization on the Edge
The vacuum dependence on boundary conditions in quantum field theories is
analysed from a very general viewpoint. From this perspective the
renormalization prescriptions not only imply the renormalization of the
couplings of the theory in the bulk but also the appearance of a flow in the
space of boundary conditions. For regular boundaries this flow has a large
variety of fixed points and no cyclic orbit. The family of fixed points
includes Neumann and Dirichlet boundary conditions. In one-dimensional field
theories pseudoperiodic and quasiperiodic boundary conditions are also RG fixed
points. Under these conditions massless bosonic free field theories are
conformally invariant. Among all fixed points only Neumann boundary conditions
are infrared stable fixed points. All other conformal invariant boundary
conditions become unstable under some relevant perturbations. In finite volumes
we analyse the dependence of the vacuum energy along the trajectories of the
renormalization group flow providing an interesting framework for dark energy
evolution. On the contrary, the renormalization group flow on the boundary does
not affect the leading behaviour of the entanglement entropy of the vacuum in
one-dimensional conformally invariant bosonic theories.Comment: 10 pages, 1 eps figur