We discuss the Schrödinger equation with singular potentials. Our focus is non-relativistic Schrödinger operators H with scalar potentials V defined on Rd, hence covering such quantum systems as atoms, molecules, and subatomic particles whether free, bound, or localized. By a “singular potential” V, we refer to the case when the corresponding Schrödinger operators H, with their natural minimal domain in L2(Rd), are not essentially self-adjoint. Since V is assumed real valued, the corresponding Hermitian symmetric operator H commutes with the conjugation in L2(Rd), and so (by von Neumann’s theorem), H has deficiency indices (n, n). The case of singular potentials V refers to when n > 0. Hence, by von Neumann’s theory, we know the full variety of all the self-adjoint extensions. Since the Trotter formula is restricted to the case when n = 0, and here n > 0, two questions arise: (i) existence of the Trotter limit and (ii) the nature of this limit. We answer (i) affirmatively. Our answer to (ii) is that when n > 0, the Trotter limit is a strongly continuous contraction semigroup; so it is not time-reversible.
In this paper we shall introduce a generalized equi-continuity of a family of semigroups and prove a new type of Trotter-Kato Theorem, applicable to the weak convergence of semigroups.In [13], we prove the existence of non-unitary solutions to formally self-adjoint Schr\"odinger equations.In that paper, we need the Trotter-Kato Theorem for the weak convergence.However, various versions of the Trotter-Kato Theorem in locally convex spaces already published are not applicable to the weak convergence as far as the authors knows.Therefore we shall give a generalized form of the Trotter-Kato Theorem in Yosida [18].Next we shall define a kind of operator-valued integration and define the Feynman path integrals of Riemann integral type.It seems that it is one of the best possible conditions of the existence of the path integrals of Riemann integral type for Schr\"odinger equation with singular potentials.Our class of potentials is wide
enough;the real measurable potential
U should be continuous except a closed set of measure zero.それどころか，内積を取る相手が完全系を構成するかどうかについても，あまり厳密に考えて いません (実際に，そうした論文を読んだことがあります). 証明が厳密でなくても，物理的な 直観に照らして正当 (と感じられる) ならば充分であり，そのうち数学の得意な人がちゃんとし た証明をしてくれると期待している訳です．J \S 2. Trotter-Kato Theorem for Weak Covergence We shall introduce a generalized equicontinuity of a family of semigroups and prove a new type of Trotter-Kato Theorem, applicable to the weak convergence of semigroups.We begin by introducing some terminology and notation and present those aspects of the basic theory which are required in subsequent subsections.\S 2.1.Filter Definition 2.1.Given a set
E , a partial ordering
⊂ can be defined on the powerset
P(E) by subset inclusion.Define a filter
F on
E as a subset of
P(E) with the following properties:Definition 2.4.We say that filter base
B converges to
x , denoted by
Barrowx , if for every neighborhood
U of
x , there is a
B∈B such that
B⊂U .In this case,
x is called a limit of
B and
B is called a convergent filter base.Lemma 2.5.For every neighbourhood base
U(x) of
x , it follows that
U(x)arrowx .Lemma 2.6.
X is a Hausdorff space ifand only ifevery filter base on
X has at most one limit.For details concerning the filter, we refer to Bourbaki [1].\S 2.2.Locally Convex Topologies Definition2.7.A linear topological space
X over the complex number field
C is called a locally convex linear topological space, or, in short, a locally convex space, if and only if its open sets
∋O contains a convex, balanced and absorbing open set.Let
M⊂X .Then
Exact solutions describing a fall of a particle to the centre of a non-regularized singular potential in classical and quantum cases are obtained and compared. We inspect the quantum problem with the help of the conventional Schrödinger’s equation. During the fall, the wave function spatial localization area contracts into a single zero-dimensional point. For the fall-admitting potentials, the Hamiltonian is non-Hermitian. Because of that, the wave function norm occurs time-dependent. It demands an extension to this case of the continuity equation and rules for mean value calculations. Surprisingly, the quantum and classical solutions exhibit striking similarities. In particular, both are self-similar as the particle energy equals zero. The characteristic spatial scales of the quantum and classical self-similar solutions obey the same temporal dependence. We present arguments indicating that these self-similar solutions are attractors to a broader class of solutions, describing the fall at finite energy of the particle.
Schrodinger operators with very singular potentials fail to produce reasonable spectra, because the usual boundary conditions are insufficient to make them self-adjoint. If one regularizes a singular potential at a length , the bound state energies diverge as → 0. A meaningful spectrum can be restored by picking a self-adjoint extension of the operator, by renormalizing the theory, or by allowing nonunitary time evolution. We show that renormalizations of the 1/r potential fall into two classes: those that are asymptotically equivalent to self-adjoint extensions, and those that reduce to treatments of delta functions. We also apply the apparatus of self-adjoint extensions and renormalization to clarify aspects of anomalous symmetry breaking, supersymmetric quantum mechanics, and one-dimensional quantum mechanics. Lines composed upon reading a thesis abstract by S. Gopalakrishnan Think a moment, dear reader, of that long-legged fly, The sprung meniscus underfoot, And what asymptotes invisible to well-trained eyes Still break the surface at its root. How a singular potential picks tight-bounded energy Apart till it no longer recognizes Its own mirrored face, its extensions, or symmetry; Unless one cautiously renormalizes. You have tried in these pensive pages to show the stuff Such theory’s made of, its quiddity When logic is applied to the diamond rough, And quantum notes ring purely. May Apollo’s clarity pierce through your reasoning And self-adjointed truth and fact Elevate your thought to that sense so pleasing When truth recovers nature’s pact.
Efficiently simulating many-body quantum systems with Coulomb interactions is a fundamental question in quantum physics, quantum chemistry, and quantum computing, yet it presents unique challenges: the Hamiltonian is an unbounded operator; its Hilbert space dimension grows exponentially with particle number; and the Coulomb potential is singular, long-ranged, and non-smooth, violating the regularity assumptions of many prior state-of-the-art many-body simulation analyses. In this work, we establish rigorous error bounds for Trotter formulas applied to these systems. Our first main result shows that for general initial conditions in the domain of the Hamiltonian, second-order Trotter achieves a
1/4 convergence rate with explicit polynomial dependence of the error prefactor on the particle number. The polynomial dependence on system size suggests that the algorithm remains quantumly efficient, even without introducing any regularization of the Coulomb singularity. Such a worst-case rate has been observed in prior work for the hydrogen ground state, demonstrating its physical relevance. We further establish one-step local error lower bounds of order
5/4 for both formulas, showing that the local exponent is sharp. Our second main result identifies a set of physically meaningful conditions on the initial state under which the convergence rates improve, with the full first-order and second-order rates recovered at sufficiently high angular momentum. Our theoretical findings are consistent with prior numerical observations. Our third main result establishes improved convergence rates for many-body fermionic Coulomb systems with initial data of different Sobolev regularity. In particular, even without imposing any additional regularity beyond the natural domain of the Hamiltonian, both the first- and second-order Trotter formulas converge with rate
1/2, rather than the general
1/4 rate.
Quantum tunneling of a nonrelativistic particle through a singular potential barrier V is studied on the line. The Hamiltonian is a self-adjoint extension of the operator H1=−d2/dx2+V(x). If H1 is essentially self-adjoint on its natural domain, the tunneling is forbidden for a class of potentials that includes all semiclassically impenetrable barriers. If H1 is not essentially self-adjoint, the Friedrichs extension of H1 yields no tunneling for another class of potentials which again includes the semiclassically impenetrable ones. In general, the occurrence of tunneling is not excluded and depends on the self-adjoint extension we choose as the Hamiltonian of our problem. As an example, we evaluate the transmission coefficient for all self-adjoint extensions of the operator H1 referring to V(x)=gx−2 with 0<g< 3/4 .
We present a mathematically rigorous quantum-mechanical treatment of a one-dimensional non-relativistic motion of a particle in the potential field , For g 2 >0 and g 1 <0, the potential is known as the Kratzer potential V K ( x ) and is usually used to describe molecular energy and structure, interactions between different molecules and interactions between non-bonded atoms. We construct all self-adjoint Schrödinger operators with the potential V ( x ) and represent rigorous solutions of the corresponding spectral problems. Solving the first part of the problem, we use a method of specifying self-adjoint extensions by (asymptotic) self-adjoint boundary conditions. Solving spectral problems, we follow Krein's method of guiding functionals. This work is a continuation of our previous works devoted to the Coulomb, Calogero and Aharonov–Bohm potentials.