Symmetry preserving self-adjoint extensions of Schrödinger operators with singular potentials

D. M. Gitman; A. G. Smirnov; I. V. Tyutin; B. L. Voronov

Home/Archive/arXiv

arXiv · 1012.2588

Symmetry preserving self-adjoint extensions of Schrödinger operators with singular potentials

D. M. Gitman·A. G. Smirnov·I. V. Tyutin·B. L. Voronov

View original source ↗Read source document ↗Export metadata ↓

Abstract

We develop a general technique for finding self-adjoint extensions of a symmetric operator that respect a given set of its symmetries. Problems of this type naturally arise when considering two- and three-dimensional Schrödinger operators with singular potentials. The approach is based on constructing a unitary transformation diagonalizing the symmetries and reducing the initial operator to the direct integral of a suitable family of partial operators. We prove that symmetry preserving self-adjoint extensions of the initial operator are in a one-to-one correspondence with measurable families of self-adjoint extensions of partial operators obtained by reduction. The general construction is applied to the three-dimensional Aharonov-Bohm Hamiltonian describing the electron in the magnetic field of an infinitely thin solenoid.

Explore related subjects

math-ph math.MP

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

D. M. Gitman, A. G. Smirnov, I. V. Tyutin, B. L. Voronov. 2010-12-12. Symmetry preserving self-adjoint extensions of Schrödinger operators with singular potentials. https://arxiv.org/abs/1012.2588

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Holography for bulk-boundary local topological order

In our previous article [arXiv:2307.12552], we introduced local topological order (LTO) axioms for quantum spin systems which allowed us to define a physical boundary (associated to a cut of the lattice) manifested by a net of boundary algebras in one dimension lower. This gives a formal setting for topological holography, where the braided tensor category of DHR bimodules of the physical boundary algebra captures the bulk topological order. In this article, we extend the LTO axioms to quantum spin systems equipped with a topological boundary (domain wall with the trivial phase), again producing a physical boundary algebra for the bulk-boundary system, whose category of (topological) boundary DHR bimodules recovers the topological boundary order. We perform this analysis in explicit detail for Levin-Wen and Walker-Wang bulk-boundary systems. Along the way, we introduce a 2D braided categorical net of algebras built from a unitary braided fusion category (UBFC). Such nets arise as boundary algebras of Walker-Wang models. We consider the canonical state on this braided categorical net corresponding to the standard topological boundary for the Walker-Wang model. Interestingly, in this state, the cone von Neumann algebras are type I with finite dimensional centers, in contrast with the type II and III cone von Neumann algebras from the Levin-Wen models studied in [arXiv:2307.12552]. The superselection sectors recover the underlying unitary category of our UBFC, and it was recently proven in [arXiv:2609.20725] that the superselection category also captures the fusion and braiding.

math-ph↗

arXiv2026

Semi-local observables, edge modes and quantum reference frames in quantum electromagnetism: an algebraic approach

Boundaries and corners of spacetime play a vital role in understanding physical concepts including entanglement entropy, the infrared problem in QFT and quantum gravity. Standard local quantum field theory struggles to accommodate such boundary-sensitive observables. In this paper we develop an algebraic framework for semi-local quantum electromagnetism on finite Cauchy lenses: a class of compact spacetimes with boundaries and corner. At the classical level, we establish a decomposition of the reduced covariant phase space into bulk closed-loop and surface sectors and demonstrate how the covariant phase space approach relates to the Peierls bracket construction commonly used in perturbative algebraic quantum field theory. Upon quantisation, we obtain a Weyl $C^{*}$-algebra of semi-local observables transforming non-trivially under large gauge transformations (those with non-trivial boundary contribution). To recover gauge invariance, we invoke the notion of quantum reference frames (QRFs) and construct a relativisation map, where we treat auxiliary surface degrees of freedom as QRFs for the large gauge transformations. The relativisation map is constructed directly on the level of $C^{*}$-algebras, making our construction state-independent. The QRF viewpoint on semi-local observables provides new tools for understanding gauge theories on manifolds with boundary, including the problem of gluing theories on Cauchy lenses with common boundaries.

math-ph↗

arXiv2026

A no-go theorem for irreversibility in arbitrary realizations of the collapse dynamics

We study finite dimensional quantum systems with arbitrary collapse events, establishing a structural no-go for operational irreversibility along arbitrary realizations of the collapse dynamics. More precisely, we prove that, for every choice of a physically admissible trajectory (i.e., collapse outcomes having nonzero Born weight) assigned to each state, there exists a nonempty topologically closed subset of the projective state space within which any two states can be connected with arbitrarily fine Fubini-Study precision and arbitrarily small integrated energetic cost. This shows that the preservation of information along observed realizations of outcomes guarantees islands of quasi-reversibility, while genuine irreversibility requires additional ingredients such as non-compactness or information erasure.

math-ph↗