In the light of the generalized Sturm-Liouville theorem, the Levinson theorem for the Dirac equation in two dimensions is established as a relation between the total number $n_{j}$ of the bound states and the sum of the phase shifts $η_{j}(\pm M)$ of the scattering states with the angular momentum $j$: $$η_{j}(M)+η_{j}(-M)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$$ $$~~~=\left\{\begin{array}{ll} (n_{j}+1)π&{\rm when~a~half~bound~state~occurs~at}~E=M ~~{\rm and}~~ j=3/2~{\rm or}~-1/2\\ (n_{j}+1)π&{\rm when~a~half~bound~state~occurs~at}~E=-M~~{\rm and}~~ j=1/2~{\rm or}~-3/2\\ n_{j}π~&{\rm the~rest~cases} . \end{array} \right. $$
\noindent The critical case, where the Dirac equation has a finite zero-momentum solution, is analyzed in detail. A zero-momentum solution is called a half bound state if its wave function is finite but does not decay fast enough at infinity to be square integrable.
The problem of localized states in 1D systems with the relativistic spectrum, namely, graphene stripes and carbon nanotubes, has been analytically studied. The bound state as a superposition of two chiral states is completely described by their relative phase which is the foundation of the variable phase method (VPM) developed herein. Basing on our VPM, we formulate and prove the relativistic Levinson theorem. The problem of bound state can be reduced to the analysis of closed trajectories of some vector field. Remarkably, the Levinson theorem appears as the Poincare indices theorem for these …
A two-dimensional analogue of Levinson's theorem for nonrelativistic quantum mechanics is established, which relates the phase shift at threshold(zero momentum) for the $m$th partial wave to the total number of bound states with angular momentum $m\hbar(m=0,1,2,...)$ in an attractive central field.
We studied two possible approaches to one-dimensional Levinson's theorem for Schödinger equation. The first one, we restrict the 3-dimensional theorem. The other one, the theorem proposed by Dong, Ma and Klauss \cite{Dong}. We find failures in each approach for a one-dimensional reflectionless potential. In order to see this, we explicitly evaluate the phase shift using Schrödinger equation using solution procedure proposed by Jaffe \cite{Jaffe}.
The Levinson theorem for two-dimensional scattering is generalized for potentials with inverse square singularities. By this theorem, the number of bound states in a given m-th partial wave is related to the phase shift and the singularity strength of the potential. For the m-wave phase shift the asymptotic behaviour is calculated for short wavelengths.
We apply the recently generalized Levinson theorem for potentials with inverse square singularities [Sheka et al, Phys.Rev.A, v.68, 012707 (2003)] to Aharonov-Bohm systems in two-dimensions. By this theorem, the number of bound states in a given m-th partial wave is related to the phase shift and the magnetic flux. The results are applied to 2D soliton-magnon scattering.
The Levinson theorem for nonrelativistic quantum mechanics in two spatial dimensions is generalized to Dirac particles moving in a central field. The theorem relates the total number of bound states with angular momentum $j$ ($j=\pm 1/2, \pm 3/2, ... $), $n_j$, to the phase shifts $η_j(\pm E_k)$ of scattering states at zero momentum as follows: $η_j(μ)+η_j(-μ)= n_jπ$.
In the light of the Sturm-Liouville theorem, the Levinson theorem for the Schrödinger equation with both local and non-local cylindrically symmetric potentials is studied. It is proved that the two-dimensional Levinson theorem holds for the case with both local and non-local cylindrically symmetric cutoff potentials, which is not necessarily separable. In addition, the problems related to the positive-energy bound states and the physically redundant state are also discussed in this paper.
The scattering of Dirac particles by symmetric potentials in one dimension is studied. A Levinson theorem is established. By this theorem, the number of bound states with even (odd) parity, $n_+$ ($n_-$), is related to the phase shifts $η_+(\pm E_k)$ [$η_-(\pm E_k)$] of scattering states with the same parity at zero momentum as follows: $$η_\pm(μ)+η_\pm(-μ)\pm{π\over 2}[\sin^2η_\pm(μ) -\sin^2η_\pm(-μ)]=n_\pmπ.$$ The theorem is verified by several simple examples.
Levinson's theorem for the Schrödinger equation with a cylindrically symmetric potential in two dimensions is re-established by the Sturm-Liouville theorem. The critical case, where the Schrödinger equation has a finite zero-energy solution, is analyzed in detail. It is shown that, in comparison with Levinson's theorem in non-critical case, the half bound state for $P$ wave, in which the wave function for the zero-energy solution does not decay fast enough at infinity to be square integrable, will cause the phase shift of $P$ wave at zero energy to increase an additional $π$.
We deduce Levinson\'{}s theorem in non-relativistic quantum mechanics in one dimension as a sum rule for the spectral density constructed from asymptotic data. We assume a self-adjoint hamiltonian which guarantees completeness; the potential needs not to be isotropic and a zero-energy resonance is automatically taken into account. Peculiarities of this one-dimension case are explained because of the ``critical'' character of the free case $u(x) = 0$, in the sense that any atractive potential forms at least a bound state. We believe this method is more general and direct than the usual one in w…
The two-dimensional Levinson theorem for the Klein-Gordon equation with a cylindrically symmetric potential $V(r)$ is established. It is shown that $N_{m}π=π(n_{m}^{+}-n_{m}^{-})= [δ_{m}(M)+β_{1}]-[δ_{m}(-M)+β_{2}]$, where $N_{m}$ denotes the difference between the number of bound states of the particle $n_{m}^{+}$ and the ones of antiparticle $n_{m}^{-}$ with a fixed angular momentum $m$, and the $δ_{m}$ is named phase shifts. The constants $β_{1}$ and $β_{2}$ are introduced to symbol the critical cases where the half bound states occur at $E=\pm M$.
Levinson's theorem for the one-dimensional Schrödinger equation with a symmetric potential, which decays at infinity faster than $x^{-2}$, is established by the Sturm-Liouville theorem. The critical case, where the Schrödinger equation has a finite zero-energy solution, is also analyzed. It is demonstrated that the number of bound states with even (odd) parity $n_{+}$ ($n_{-}$) is related to the phase shift $η_{+}(0)[η_{-}(0)]$ of the scattering states with the same parity at zero momentum as $η_{+}(0)+π/2=n_{+}π, η_{-}(0)=n_{-}π$, for the non-critical case, $η_{+}(0)=n_{+}π, η_{-}(0)-π/2=n_{-…
We formulate scattering in one dimension due to the coupled Schrödinger equation in terms of the $S$ matrix, the unitarity of which leads to constraints on the scattering amplitudes. Levinson's theorem is seen to have the form $η(0) = π(n_b + 1/2 n - 1/2 N)$, where $η(0)$ is the phase of the $S$ matrix at zero energy, $n_b$ the number of bound states with nonzero binding energy, $n$ the number of half-bound states, and $N$ the number of coupled equations. In view of the effects due to the half-bound states, the threshold behaviour of the scattering amplitudes is investigated in general, and is…
We prove an analog of Levinson's theorem for scattering on a weighted (m+1)-vertex graph with a semi-infinite path attached to one of its vertices. In particular, we show that the number of bound states in such a scattering problem is equal to m minus half the winding number of the phase of the reflection coefficient (where each so-called half-bound state is counted as half a bound state).
We prove Levinson's theorem for scattering on an (m+n)-vertex graph with n semi-infinite paths each attached to a different vertex, generalizing a previous result for the case n=1. This theorem counts the number of bound states in terms of the winding of the determinant of the S-matrix. We also provide a proof that the bound states and incoming scattering states of the Hamiltonian together form a complete basis for the Hilbert space, generalizing another result for the case n=1.
A recently proposed reference potential approach to the inverse Schrödinger problem is further developed. As previously, theoretical developments are demonstrated on example of diatomic xenon molecule in its ground electronic state. An exactly solvable reference potential for this quantum system is used, which enables to solve the related energy eigenvalue problem exactly. Moreover, the full energy dependence of the phase shift can also be calculated analytically, and as a particular result, full agreement with Levinson theorem has been achieved and explicitly demonstrated. In principle, this …
The variable-phase approach is applied to scattering and bound states in an attractive Coulomb potential, statically screened by a two-dimensional (2D) electron gas. A 2D formulation of Levinson's theorem is used for bound-state counting and a hitherto undiscovered, simple relationship between the screening length and the number of bound states is found. As the screening length is increased, sets of bound states with differing quantum numbers appear degenerately.
We consider the Dirac equation in one space dimension in the presence of a symmetric potential well. We connect the scattering phase shifts at E=+m and E=-m to the number of states that have left the positive energy continuum or joined the negative energy continuum respectively as the potential is turned on from zero.
We replay to the critique by Sudarshan and Shaji of our argument of impossibility to obtain a non-relativistic proof of the spin-statistics theorem in the Galilean frame.
Here we explore the possibility to obtain a non-relativistic proof of the spin-statistics theorem. First, we examine the structure of axioms and theorems involved in a relativistic Schwinger-like proof of the spin-statistics relation. Second, starting from this structure we identify the relativistic assumptions. Last, we reformulate these assumptions in order to obtain a Galilean proof. We conclude that the spin-statistics theorem cannot be deduced into the framework of Galilean quantum field theories for three space dimensions because two of the assumptions needed to prove the theorem are inc…
Ehrenfest theorem is proven in relativistic quantum theory of charged particles, moving under the influence of an external electromagnetic field. In order to extend the classic Ehrenfest result to the relativistic domain we bypassed the problems with the relativistic position operator by deriving directly Newton's second law. Our approach is characterized by its universality. The detailed form of the wave equation is not needed. All that is required is the existence of the conserved electric four-current built from the particle wave function. The derivation is based on the conservation laws fo…
In Bell inequality tests, the evolution of the wavefunction is not covariant, i.e. not invariant under velocity boost that change the time ordering of events, but the laws that govern the probability distribution of possible results are. In this note I investigate what this could mean and whether there could be some covariant "real quantum stuff". This clarifies the implication of the Free Will Theorem and of relativistic spontaneous localization models based on the flash ontology (rGRWf). Some implications for the concept of time(s) are spelled out.
A realistic physical axiomatic approach of the relativistic quantum field theory is presented. Following the action principle of Schwinger, a covariant and general formulation is obtained. The correspondence principle is not invoked and the commutation relations are not postulated but deduced. The most important theorems such as spin-statistics, and CPT are proved. The theory is constructed form the notion of basic field and system of basic fields. In comparison with others formulations, in our realistic approach fields are regarded as real things with symmetry properties. Finally, the general…
The Schrodinger equation for stationary states in a central potential is studied in an arbitrary number of spatial dimensions, say q. After transformation into an equivalent equation, where the coefficient of the first derivative vanishes, it is shown that in such equation the coefficient of the second inverse power of r is an even function of a parameter, say lambda, depending on a linear combination of q and of the angular momentum quantum number, say l. Thus, the case of complex values of lambda, which is useful in scattering theory, involves, in general, both a complex value of the paramet…
We review the argument that latent image formation is a measurement in which the state vector collapses, requiring an enhanced noise parameter in objective reduction models. Tentative observation of a residual noise at this level, plus several experimental bounds, imply that the noise must be colored (i.e., non-white), and hence frame dependent and non-relativistic. Thus a relativistic objective reduction model, even if achievable in principle, would be incompatible with experiment, the best one can do is the non-relativistic CSL model. This negative conclusion has a positive aspect, in that t…
A crucial task for secure communication networks is to determine the minimum of physical requirements to certify a cryptographic protocol. A widely accepted candidate for certification is the principle of relativistic causality which is equivalent to the disallowance of causal loops. Contrary to expectations, we demonstrate how correlations allowed by relativistic causality could be exploited to break security for a broad class of multi-party protocols (all modern protocols belong to this class). As we show, deep roots of this dramatic lack of security lies in the fact that unlike in previous …
In a recent paper, Conway and Kochen proposed what is now known as the "Free Will theorem" which, among other things, should prove the impossibility of combining GRW models with special relativity, i.e., of formulating relativistically invariant models of spontaneous wavefunction collapse. Since their argument basically amounts to a non-locality proof for any theory aiming at reproducing quantum correlations, and since it was clear since very a long time that any relativistic collapse model must be non-local in some way, we discuss why the theorem of Conway and Kochen does not affect the progr…
We illustrate the existence of single-excitation bound states for propagating photons interacting with $N$ two-level atoms. These bound states can be calculated from an effective spin model, and their existence relies on dissipation in the system. The appearance of these bound states is in a one-to-one correspondence with zeros in the single-photon transmission and with divergent bunching in the second-order photon-photon correlation function. We also formulate a dissipative version of Levinson's theorem for this system by looking at the relation between the number of bound states and the wind…
We transform the two-dimensional Dirac-Weyl equation, which governs the charge carriers in graphene, into a non-linear first-order differential equation for scattering phase shift, using the so-called variable phase method. This allows us to utilize the Levinson Theorem to find zero-energy bound states created electrostatically in realistic structures. These confined states are formed at critical potential strengths, which leads to us posit the use of `optimal traps' to combat the chiral tunneling found in graphene, which could be explored experimentally with an artificial network of point cha…
The Brillouin theorem has been generalized for the extended non-relativistic electronic Hamiltonian (Hkin+ Hne+ aHee) in relation to coupling strength parameter (a), as well as for the configuration interactions (CI) formalism in this respect. For a computation support, we have made a particular modification of the SCF part in the Gaussian package: essentially a single line was changed in an SCF algorithm, wherein the operator rij-1 was overwritten as 1/rij to a/rij, and a was used as input. The case a=0 generates an orto-normalized set of Slater determinants which can be used as a basis set f…
Alice gives Bob an unknown localized physical state at some point P. At some point Q in the causal future of P, Alice will ask Bob for the state back. Bob knows this, but does not know at which point Q until the request is made. Bob can satisfy Alice's summons, with arbitrarily short delay, for a quantum state in Galilean space-time or a classical state in Minkowski space-time. However, given an unknown quantum state in Minkowski space-time, he cannot generally fulfil her summons. This {\it no-summoning theorem} is a fundamental feature of, and intrinsic to, relativistic quantum theory. It fol…
Comparison theorems are established for the Dirac and Klein--Gordon equations. We suppose that V^{(1)}(r) and V^{(2)}(r) are two real attractive central potentials in d dimensions that support discrete Dirac eigenvalues E^{(1)}_{k_dν} and E^{(2)}_{k_dν}. We prove that if V^{(1)}(r) \leq V^{(2)}(r), then each of the corresponding discrete eigenvalue pairs is ordered E^{(1)}_{k_dν} \leq E^{(2)}_{k_dν}. This result generalizes an earlier more restrictive theorem that required the wave functions to be node free. For the the Klein--Gordon equation, similar reasoning also leads to a comparison theor…
Many of the "counterintuitive" features of relativistic quantum field theory have their formal root in the Reeh-Schlieder theorem, which in particular entails that local operations applied to the vacuum state can produce any state of the entire field. It is of great interest, then, that I.E. Segal and, more recently, G. Fleming (in a paper entitled "Reeh-Schlieder Meets Newton-Wigner") have proposed an alternative "Newton-Wigner" localization scheme that avoids the Reeh-Schlieder theorem. In this paper, I reconstruct the Newton-Wigner localization scheme and clarify the limited extent to which…
