Semicontinuous Banach spaces for Schrödinger's Eq. with Dirac-δ' potential
aa r X i v : . [ m a t h - ph ] D ec SEMICONTINUOUS BANACH SPACES FOR SCHRÖDINGER’S EQ.WITH DIRAC- δ POTENTIAL
B. BUTTON
Abstract.
Schródinger’s equation with distributional δ , or δ potentials has been wellstudied in the past. There are challenges in simultaneously addressing some of the inher-ent issues of the system: The functional operator cannot exist entirely within the standard L Hilbert spaces. On differentiable manifolds, the domain of the free kinetic energy op-erator is in the space of harmonic forms. Locally, by the Hodge decomposition theoremand the standard distributional calculus, the space of functionals of a δ or δ potentialmust be orthogonal to the free kinetic energy operator. Restricting to semicontinuoustopologies presents opportunities to address these, and other issues. We develop, in greatdetail, a formalism of Banach spaces with semicontinuous topologies, and their propertiesare extensively defined and studied. For C p R q functions, the spaces are indistinguishable.The semicontinuous analogs of the L P spaces, are nontrivial and result in a dense topologi-cally continuous embedding of the semicontinuous L p spaces into the semicontinuous C p R q spaces. Here, certain classes of distributions may be inverted in terms of their primitivefunctions. Also many operators are inherently self adjoint. We define equivalence rela-tions between the cohomology classes of distributions and derivatives of their associatedprimitives on local sections of R . Here Hamilton’s equations are canonical, and define aconnection on the fibers of the base space. Semicontinuity provides a resolution to theabove domain and interaction problems, and easily integrable Feynman functional. Wearrive at a compatible domain which is Krein ( H ) over disjoint components of R . Thesubspaces of H are isomorphic to the semicontinuous Hilbert spaces of the Hamiltonian. Introduction
The study of quantum mechanics necessitates the study of the self adjoint Hamiltonianoperator on some Hilbert space, H “ L p R n q for example. In one dimension, for some wavefunction ψ P H , the Hamiltonian (energy operator) acting on ψ is Schödinger’s differential Date : September 7, 2018. equation given by,(1) ´ i ddt ψ “ ˆ Hψ “ Eψ, where E is the eigenvalue of the energy operator, ψ “ ψ p x, t q . Since this is usually definedon L , the standard physics inner product notation for this coupled integral-differentialequation is the Dirac bra-ket, x ψ ˚ | ´ ddt | ψ y “ x ψ ˚ | ˆ H | ψ y . If the energy of the system isconstant in time, then ˆ H is given as(2) ˆ H “ ˆ P ` V p ˆ x q where the ˆ P is the Hermitian momentum operator is ˆ P “ ´ i ~ ? m BB x . Here m is the massof a point-like particle and V p ˆ x q is a time independent potential. We will denote the freeHamiltonian operator p V p ˆ x q “ q by ˆ H f .Over the past couple of decades, there has been a considerable amount of work done byboth the physics and mathematics community for cases where the potential is highly singular,and in particular point supported, such as the Dirac- δ or derivatives of the Dirac- δ . Thusone is left to make sense of a Hamiltonian operator of the sort ˆ H “ ´ ~ m B B x ` δ p x q (3)or, ˆ H “ ´ ~ m B B x ` δ p x q . (4)We will work explicitly on the extended real line R , we use the notation for partial deriva-tives BB x to denote the differential operator. This will facilitate later discussions when wediscuss (4) in terms of differential forms on closed manifolds. Furthermore, not all resultswill be here, generalize to higher dimensions. Though it should be readily apparent whichresults admit higher dimensional generalizations.There are several considerations which motivates our study of semicontinuous spaces.The collective set of motivations aim for both mathematical rigor (as much as possible)and relevance to applications in physics. As such, the work here attempts to bridge thedifference between mathematical theory and theoretical physics. The results which follow EMICONTINUOUS BANACH SPACES FOR SCHRÖDINGER’S EQ. WITH DIRAC- δ POTENTIAL 3 are not quite those of the usual L p p R q theory, but they are not so different as to be com-pletely unrecognizable from it. The formalism developed herein has some unexpected, butpleasant properties. The properties are themselves noteworthy in their own right, but alsohave potential for use in low dimensional condensed matter systems (in particular graphenesheets) and could possibly produce non-trivial results in AdS/CFT or string theory itself.We will make some general comments in the summary regarding applications to interactionsin perturbative string theory, as well as other avenues for future investigations. As such,in Section 3, we choose to apply the semicontinuous spaces to analyze the basic quantumsystem defined by Eq. (4) in terms of Feynman’s functional integral. We reserve applicationsto specific to other systems, i.e. string theory and holography for future work. It is withthese considerations in mind, as well as the notable differences of δ and δ potentials in 2and 3 dimensions [4] and specifically, the use of spherical/polar coordinates, that we do notto generalize beyond the discussion beyond one dimension. However, provided that our pos-tulates hold, there is no a priori reason we expect that higher dimensional generalizationswill necessarily fail. The issue (as is the case with all regularization methods), is whetheror not the regularized system is representative of the initial system.From a pure mathematics perspective, we address two particularly troublesome issuesregarding the system defined by Eq. (4), from which, we may define a method to enableone to more completely utilize the Feynman path integral in similar cases. We construct aformalism which is sufficient to accommodate for the problems that,(1) Although the space of test functions for δ is dense in the space of L p p R q functions,the space of test functions for δ (and derivatives of δ ) is not equivalent to the entiretyof any L p space, for any ď p ă 8 .(2) In general, the kinetic energy operator and the "potentials" of Eq. (3) or Eq. (4) donot act as maps from the same base space to the same target space.To elaborate on (1), methods of approximations or limiting sequences which approach δ or δ potentials have meaning in L except at the limit point itself, where the L functionsactually become true singular distributions. Closure in functional spaces (and thus selfadjoint-ness) is lost. Extensions to L or even Sobolev subspaces are not sufficient in suchcircumstances[33, Thm. 8.27]. B. BUTTON
Further elaborations on (2) have two complementary facets. As a distribution (i.e. at thelimit point of some approximation scheme for δ, δ ), the "potential" becomes a map fromthe space of Schwartz functions ( S ) to the real numbers, whereas the differential operatoris a linear transformation from one functional vector space to another. The complementaryissue arises if one attempts to place the singular Hamiltonians on a differentiable manifold.Here, one may view the systems above as linear functionals on the space of differentialforms. Local arguments for singular operators on differential forms are still subject to theHodge decomposition theorem. As a result, the second order differential operator and thesingular potential must belong to orthogonal spaces. The vector on which the kinetic energyoperator acts, is necessarily orthogonal to that of the potential, unless they are the same p -form degree. This prohibits the system from being a truly interacting system. We willdiscuss each of these in more detail shortly.Also, each component of our formalism is necessary in the sense that the formalism is selfconsistent, while simultaneously addressing both (1) and (2), as well as some other pointswhich we will encounter along the way. The space of singular distributions do not havethe same notion of "domain" as linear operators or linear transformations. In an abuse oflanguage, we will often reference the collective domain of the linear functional, (3) or (4).The main results of the paper span the sets of different tools used to address each ofthe above points. Topological measure spaces are constructed in such a way as to addressthe particular domain incompatibility between the distribution and the differential operatorcomponents of Eq. (4). We define spaces of semi-continuous function(al)s which cannotdistinguish between C p R q and L p p R q functions. Therefore we have no need to extend ourresults to subsets of L p spaces for self adjointness. A positive consequence of our constructionis that many operators are inherently self adjoint. The mapping to semicontinuous spacesproduces subspaces of semicontinuous L p functions which are orthogonal in regards to leftversus right semi-continuity. The defined semi-continuous measure spaces allows the freedomto define an equivalence class between δ and derivatives of functions which may be consideredas a primitive function of δ , such as the Heaviside function θ . This in turn affords one theability to define a mapping of δ , via equivalence class identifications, to the cohomologyclass of harmonic forms. Here, the system becomes a genuinely interacting system. Itis shown that the equivalence class mapping is locally canonical. Our example discussed EMICONTINUOUS BANACH SPACES FOR SCHRÖDINGER’S EQ. WITH DIRAC- δ POTENTIAL 5 in Section 3, the formalism is applied to Feynman’s path integral. The net result of theformalism, collectively broadens the applicability of the Feynman path integral to includeexponentiation of full Hamiltonian. The resulting function space is Krein. It is known thatKrein spaces have subspaces which are isomorphic to L p spaces of functions.1.1. Notation and Conventions.
Here we pause in order to state the conventions andnotation used throughout the paper. In later sections we will discuss the Hamiltonian asa functional on a differentiable (psuedo-Riemannian) manifold, where differential geometricstructures are relevant, and it will be necessary to make distinctions between various dif-ferential operators. Thus, given a differentiable function f then, BB x f : “ dx B f B x is a covector(1-form) in some cotangent space at the point x , whose component is B f B x . Exterior differen-tiation and codifferentiation will exclusively be denoted by d f and δ f , respectively. Then, B B x will be implicitly defined by the Laplace-Beltrami operator; ∆ : “ δd ` dδ . We willadopt the general notation of D when regarding differentiable set definitions or where wewish regard differentiation more colloquially, and will assume the applicable derivative tobe implicit. For functions (say f ) used in equations, we generally denote derivatives withrespect to their arguments as f p x q , and f for distributional/functional derivatives, andcommonly use either interchangeably when there is no danger of ambiguity.The space of Schwarz functions, the space of smooth functions such that f p x q and all itsderivatives go zero faster than x n for all n P Z , is denoted by S . The topological dual of S , the space of tempered distributions, is denoted by S . We shall also commonly, but notexclusively, denote distributions in a manner similar to: f δ x “ f p q , for some f P S , and theDirac- δ with point support at x .A Borel measurable space over some universal set p Σ, B q “ B p Σ q with total variationmeasure, µ (or sometimes ν ), defines a measure space p Σ, B , µ q “ p B p Σ q , µ q . We denotethe space of bounded linear functions over the previously given measure space by L p B , µ q .We almost exclusively have Σ “ R , and so in this case we will omit the universal setfrom the notation. We denote the left (respectively right) semicontinuous spaces of boundedlinear functions by with left-semicontinuous measure µ L (resp. right-semicontinuous measure µ R ) by L L p B , µ L q (resp. L R p B , µ R ). Left semicontinuity (resp. right semicontinuity) isdefined to be the continuous one-sided measure approaching from the left (resp. right).For example, if r a, b s P R , then the left-semicontinuous Borel measure will be given by B. BUTTON µ L : “ µ pp a, b qq Y µ t b u “ µ pp a, b sq , where the half-open interval notation is as expected.These are just Stieltjes measures over half-open Borel sets.The standard Lebesgue measure denoted as λ . Measures of functions under Lebesgueequivalence class identifications are generally understood with respect to the usual Lebesgue-Stieltjes measure. Left (resp. right) continuous Lebesgue (Lebesgue-Stieltjes (L-S)) mea-sures are denoted by λ L (resp. λ R ). Generally, whether we have measures µ L,R or λ L,R (where "
L, R " refers collectively to left or right semicontinuous measures), sets of measurezero under Lebesgue measure, can have non-zero measure under L-S measures. We will bemore precise about the semicontinuous measures and measure spaces in Section 2 below.1.2.
Further Comments on the Hamiltonian Functional.
If Eq. (3) is to act on a wavefunction ψ , as in Eq. (1), the resulting expression is ill-defined as a differential equation. Inparticular BB x is a differential operator, which is a linear transformation D : L Ñ L , where L denotes an arbitrary differentiable manifold, vector or linear functional space with domain D L . In particular, the kinetic energy operator is the mapping D “ p D ˝ D q : L Ñ L Ñ L .By implicitly assuming D „ ∆ , then D will map p -forms in L Ñ p -forms in L such that,for local neighborhoods U, V of some base spaces
M, N respectively, we require L p M q| U – L p N q| V . However δ x only has rigorous meaning either as a tempered distribution ( δ P S )or as a measure ( µ δ P B p Σ q ), with Σ some measurable σ -algebra manifold. But whatsort of object (or linear space) is Eq. (4) acting on? A rigid interpretation of Eq. (4),implies that BB x δ p x q “ dx B δ p x qB x , which makes the "potential" a 1-form. But then we couldhave no scalar wave function solutions which are mapped to a common space (althoughdirect product/sum spaces can be constructed). By the Hodge decomposition theorem, thesolution space of harmonic -forms is orthogonal to the solution space of the codifferentialon 1-forms. In this case, the interaction between the free kinetic energy operator and thepotential become independent, and thus completely decouple by orthogonality.This is particularly troublesome with respect to Feynman path integrals, where one wouldlike to have solutions to the functional integral of the form(5) ż D q i D p i ψ ˚ p q, p, t q exp i ş dt H p q,q i ,p i ; t q ψ p q i , p i ; t i q “ δ p q ´ q i q δ p t ´ t i q , EMICONTINUOUS BANACH SPACES FOR SCHRÖDINGER’S EQ. WITH DIRAC- δ POTENTIAL 7 where H p q, q i , p i ; t q is the Hamiltonian density functional, and ψ p q q are required to satisfy(to at least first order in t ) ψ ˚ p q, t q ˆ H | ψ p q i ; t i qy “ δ p q ´ q i q δ p t ´ t i q . Classically, this isinterpreted as x ψ ˚ | ˆ H | ψ y “ .Assuming that ψ is a 0-form, by the Hodge decomposition theorem, then ψ is a harmonic -form for the kinetic energy operator. The addition of any potential term ˆ V p q q is necessarilyeither cohomologous with ψ , or must lie in a functional space orthogonal to ψ . The latterrequires the wave function to be of the form Ψ “ ψ ‘ φ , with ψ a 0-form and ψ a 1-form.Let F , F be the space of 0 and 1 forms on T ˚ p L q . Then Ψ P F ‘ F , and F K F . Theproblem now becomes that the wave function Ψ solves the equation x Ψ ˚ | ˆ H | Ψ y “ λ , with λ ą (here λ is an eigenvalue). It cannot solve the equation that we intended it to solve( i.e. λ “ ), and any potential function ˆ V p q q must therefor be a vector in a space which isorthogonal to the free operator.Moreover, Hodge’s orthogonality condition implies that ˆ H cannot even act on a two non-cohomologous functions originating from the same function space. Indeed, a wave functionsolution Ψ , must be comprised of the direct sum of two wave function in orthogonal spaces p Ψ “ ψ | L ‘ φ | L q ! Clearly, the rigid interpretation of the "potential" as a 1-form is not theintent behind Eq. (3). We will return to this point again in Section 3.Specifying ˆ V “ δ as either a distribution or a measure, determines whether ψ P S ` R ˘ or ψ P B p Σ q . Clearly S ` R ˘ X B p Σ q ‰ H . In the former, the space of distributions isthe space of continuous linear functionals T : S p R q Ñ C (or R ). As a measure in thelatter case, µ δ : B p Σ q Ñ C (or R ). However in quantum mechanics, we typically regard ψ P H Ă L p ` R ˘ . Since we impose a self-duality condition; ψ ˚ “ ψ . Then the Hölderinequality formally restricts H P L ` R ˘ . However D δ x Ć L p ` R ˘ for any ď p ď 8 ,so D δ x Ć H ! Within this context, the question of the domain for Eq. (3) ( D ˆ H ) cannotbe meaningfully addressed. The standard approach of extension parameters is of littlehelp. Limiting sequences approaching the δ (or δ ) distribution can be constructed with L functions on compact subsets of R . However the limit point of the sequence inevitably hasa domain which cannot be extended be L p R q . Thus no self adjoint extension which is aresult of limits of sequences exists, as they are not closed in L .In the above paragraphs, we outlined a number of inconsistencies with regard to Eqs. (3)and (4). Neither one is a differential equation. Even worse, D δ x R H Ă L p R q and δ x is B. BUTTON not even a measure. On a differentiable manifold, BB x δ p x q defines 1-form, which implies thata free scalar solution and the interaction solution space are decoupled, and introduces anundefined inner product for any non-zero constants ψ . A similar point is raised in [31, Secs.3 and 4]. We assume that Eq. (3) originates from the variation of some linear functional(Lagrangian or Hamiltonian) density, H :(6) δδψ ˚ ż x P R H p ψ ˚ , ψ q dx “ ż x P R ψ ˚ ´ ˆ Hψ ¯ dx “ x ψ ˚ , ˆ Hψ y ´ E x ψ ˚ , ψ y “ , Obviously we require that ψ be self-dual in Eq. (6). We also require that ˆ H , at a min-imum be essentially self adjoint, with bounded operator norm: (cid:13)(cid:13)(cid:13) x ψ ˚ , ˆ Hψ y (cid:13)(cid:13)(cid:13) ă 8 , with ψ P D δ x : x ψ ˚ , ˆ Hψ y “ x ψ ˚ , ˆ Hψ y ˚ . Quantum mechanics demands that Eq. (6) admit a def-inition on some dense subset of L ` R ˘ (or a suitably equivalent notion thereof), otherwisecolloquially speaking, we break quantum mechanics. The above minimal requirements aremet by assuming ˆ H to be of Schatten class. The semicontinuous topological vector spaceson R with Riemann(Lebesgue)-Stieltjes measure ( C L,R ), which are defined in Section 2, arelocally compact Hausdorff spaces in D ˆ H . That ˆ H is compact, trivially follows by the Alexan-droff compactification for any wave function ψ p x q P C L,R p R q such that ˆ H : C L,R p R q Ñ R .We argue that the domain incompatibility, is inherently topological in nature. The δ x functional requires continuity at the point of support for any function on which it acts. Thisimmediately places the a solution in some subspace of C ` R ˘ for D δ x . Thus any solutionshould be in some subset of C ` R ˘ such that it admits a self adjoint extension to L ` R ˘ .Singular differential systems have been well studied in terms of nonholonomic geometricmechanics. The works of Faddeev and
Vershik (in particular [15, 42]) allow us to work withdistributions on (co)tangent spaces, which exist as geometrical objects in their own right ( i.e. vectors as jets or germs of fields, and 1-forms as modules over jets or germs). The inherentdifferential structure of the (co)tangent spaces will be particularly useful in later sections,where Eq.(4) will be defined on a configuration space endowed with a symplectomorphismstructure. Below we construct a general formalism which aims to bridge this gap betweenthe space of test functions for δ and the free kinetic energy operators more satisfactorily.There are many approaches available in order to tame singular Hamiltonian (and La-grangian) systems. There is also an overwhelming number of papers which apply functionalcalculus methods to point-supported interactions of Schrödinger operators. A small subset EMICONTINUOUS BANACH SPACES FOR SCHRÖDINGER’S EQ. WITH DIRAC- δ POTENTIAL 9 of these cover Green’s Function methods, deficiency indices, and extensions to L or Sobolevspaces (dense subspaces of L ) [30, 31, 2, 3, 28, 18, 13, 24, 17]. The seminal works by Al-beverio et al , spanning over three decades, is summarized in [4, and refs therein], and worthparticular mention due to the multitude of systems analyzed using the method of self ad-joint extensions of symmetric operators. In particular, the study of propagators of quantummechanical Hamiltonians with regular, and singular potentials in many spatial dimensions.Approximation methods determine estimates for boundedness and well-posedness in finitedifference Schrödinger equation (as well as the NLS, and semi-relativistic variants). See[4, 5, 37, 12, 25, 16], and references therein for instance.If one looks to non-perturbative methods for handling point supported interactions suchas Eq. 4, nonlinear distributional solutions are invariably the only other tool at one’s dis-posal. On the other hand, there has been a tremendous amount of work done in the fieldsof nonlinear functional analysis, with special attention to point interactions. Colombeaualgebras are a considerably intricate and abstract formalism which have had some success inrecent years with the construction of generalized functional algebras. The difficulty inherentin Colombeau algebras is matched only by their potential for use in large classes of functionspaces. For works on the general theory of Colombeau algebras see [10, 29, 19]. An inter-esting exposition on a modern generalization which simplifies some of the formalism, anddiscusses the current challenges of Colombeau algebras is [27]. Recently [21] has appeared,which outlines a generalization for algebras of operators and distributions.Here we will not need to employ such generalized formalisms, though there is certainlysome overlap with the afore mentioned in all cases. The approach here is a constructionfrom first principals, in terms of functional methods on topological vector spaces with par-ticular measure properties, differentiable manifolds[15, 42, 37], and the spaces of integrabledistributions [38, 40, 39].It is worth making a particular mention of the works of Johnson and Lapidus [20, and refstherein], which became known to the author only after the completion of the initial draft ofthis work. The work here contains some parallels of Johnson and Lapidus [20] in terms ofthe usage of the L-S measures in Feynman’s path integral. However, in this work, we buildBorel measurable and Banach spaces on the foundation of half-open Borel generating setson R – S . In this setting, L-S measures are in some sense, very natural measures for such semicontinuous Banach spaces. Specifically, L-S measures have been precisely chosen tocoincide with the generating sets of the underlying topological vector space (TVS). This hascertain benefits in the analysis below, and which are not generally possible in the standardHilbert (or Banach) spaces. For instance, we have a topology compatible with notions ofmaking identifications of certain (even singular) distributions with the derivatives of theirprimitive distributions, in particular see [38]. The ability to make these identifications, offersan intriguing option which may potentially (if generalizable in a meaningful way) expandthe tools available to define and evaluate Feynman integrals including those with singularmeasure potentials. Another benefit of our construction is that many operators are naturallyself adjoint on the semicontinuous manifold spaces defined below.1.3. Gauge integrals, primitive functions, and Krein spaces.
We would like to havea mapping I such that if ˆ H P L ˚ where I : L ˚ Ñ L Ñ L , where ˚ denotes the dual spaceof continuous linear functionals. In this case then it is at least, in principle, possible to havesome D D “ D δ . It is well known that the generalizations of the Riemann and Lebesgueintegrals are the class of gauge integrals (and in particular the Henstock-Kurzweil (HK) andStieltjes classes,[6, 1, 9, 8, 7]), which are known to integrate functions which are derivativesof unique primitive functions.Of particular relevance to our discussion here are the regulated classes of gauge integralswith Lebesgue-Stieltjes measure [38, 40, 39], as well as Krein function spaces[22, 23, 36,14]. The gauge integrals define a gauge function within subintervals I Ă R , and a taggedpartition of I . The uniqueness of primitive functions obtained from integrable functions anddistributions affords one the luxury of straight forward identifications of domains of certainclasses of functionals (particularly the space of Schwarz functions over some topologicalmetric space, T p X q ) where the inversion of distributional derivatives is possible. These arethe classes of integrable distributions.A function γ which maps some interval r a, b s to R is called a gauge on r a, b s if γ p x q ą for all x P r a, b s . A tagged partition is a finite set of pairs of closed intervals and tag points in R , P “ tpr x n ´ , x n s , ˜ x n qu Nn “ for some n P N , with ˜ x n P r x n ´ , x n s for each ď n ď N and ´8 “ x ă x ă x . . . ă x N “ 8 . The finite pair set consisting of a tagged partition andtag points is denoted by p P , t ˜ x u Nn “ q . For a particular γ p x q , a tagged partition p P , t ˜ x u Nn “ q is said to be γ - fine if every subinterval r x n ´ , x n s satisfies x n ´ x n ´ ă γ p x q . Therefore a EMICONTINUOUS BANACH SPACES FOR SCHRÖDINGER’S EQ. WITH DIRAC- δ POTENTIAL 11 gauge γ on R together with a tagged partition, maps to open intervals in R and for each x in some subinterval r a, b s Ă R , then γ p x q is an open interval containing x .In particular we will generally consider normed linear topological metric spaces generatedby the collection of all half-open Borel sets over the extended real line R “ r´8 , , which isalso one possible way to define a compactification of R . Krein spaces are useful due to theirstructure, since they can include regular subspaces which are isomorphic to Hilbert spacesof quantum mechanics. Krein spaces will evolve naturally out of the formalism developedhere.The organization of the paper is the following. In Section 2, we introduce and definethe spaces of semicontinuous functions. We begin with a discussion of some idiosyncrasiesregarding certain definitions of specific tempered distributions. The Heaviside distributionand general classes of step functions are discussed in detail. These discussions serve as themotivations which follow in the latter sections of Section 2, where we define the measurespaces of half-open Borel topologies, the spaces of continuous measurable (and thereforebounded) functions C p R q and the isomorphic left/right semicontinuous topological spaces C L,R p R q defined over the half-open Borel set topologies. We then discuss the L p analogs tothe semicontinuous spaces C L,R , which are defined for non-atomic semicontinuous L p func-tions, excluding sets (and collections of sets) of Lebesgue measure zero, such as fat Cantorsets or Cantor-Lebesgue measure. With a refined notion of the standard L p equivalence classidentifications, we construct the semicontinuous quotient spaces L pL,R which are continuouswith respect to the half-open Borel measure space topologies for ď p ď 8 . Under theseequivalence class identifications, we achieve a continuous and dense partial embedding ofnon-atomic L pL,R functions into the spaces of C L,R . This considerably enlarges the classes offunctions in C L,R which are topologically continuous with respect to the base Borel topolo-gies. Theorems are proved regarding the topologically continuous dual spaces of all of thedefined function spaces. The topological and norm closure of the dual spaces of semicontin-uous functions over half-open Borel topologies with the k ¨ k sup is given by the function spacesof semicontinuous bounded variation BV L,R with k ¨ k BV L,R which are finitely additive overall collections of compact subsets of R . The benefit of this construction is that with the half-open L-Sj measures, there is a one-to-one mapping of L L,R functions to the semicontinuousspaces of bounded variations, which can be generalized to the Riemann-Stieltjes measures for gauge integrals. The Radon-Nikodym theorem provides a description of the semicontin-uous spaces of absolutely continuous p AC L,R q functions in terms of the second fundamentaltheorem of calculus, which is reflexive and includes primitives for semicontinuous integrabledistributions. The spaces of semicontinuous integrable distributions are the finitely additivemeasures of BV L,R with the corresponding k ¨ k BV L,R . We note that by continuity in thesespaces, the k ¨ k L,R ;sup (equivalent to the k ¨ k L L,R ) bounds all k ¨ k L pL,R , for all p . In this mannerwe have containment of the norms k ¨ k L pL,R Ď k ¨ k L,R ;sup “ k ¨ k L,R ;sup Y k ¨ k BV L,R , with equalityin the case of k ¨ k L L,R .In Section 3 we will discuss the Hamiltonian in terms of differential geometric structuresand utilize the semicontinuous spaces of functions to analyze the Hamiltonian functionalequation Eq. (4). Placing the Hamiltonian on a differential manifold will yield a geometricapproach, which will give further support to the functional approach that we are proposing.In terms of the semi-continuous topological spaces, we will find the corresponding Hilbertspace H for the Dirac- δ system, which is separable, and admits a semicontinuous orthog-onal decomposition such that H “ H L ‘ H R . Therefore H is measure valued projectivespace. The Hilbert space is defined as Sobolev spaces of semicontinuous functions such thatthe Hamiltonian is bounded in the operator norm. Essentially D p ˆ H is the semicontinuousfunctions with ˆ H Schatten class. These are simply ´ L p R q X C L,R ;0 p R q ¯ Ă C L,R p R q , wherethe bar denotes the set closure of C L,R p R q .In Section 4 we summarize the structure and properties of the indefinite Krien spaces de-noted by H , which contains subspaces that are isomorphic to the Hilbert space H . It is shownthat the Hilbert space and its associated negative norm antispace correspond to the sign ofthe coupling term to the δ potential. The Hilbert space topology is the strong topology of H , and therefore H inherits the orthogonal decomposition from the Hilbert space/antispacestates in addition to the orthogonal decomposition in terms of semicontinuous functionsfrom H . We close the paper with a short summary and concluding remarks on future worksin progress. EMICONTINUOUS BANACH SPACES FOR SCHRÖDINGER’S EQ. WITH DIRAC- δ POTENTIAL 13 Spaces of Semi-Continuous Functions
Many points discussed in the previous section will become relevant if we consider theRiemann(Lebesgue)-Stieltjes integral of semi-continuous functions/distributions with mea-sures defined by the Borel sets of half open intervals over R . We want our space to bereflexive so that the weak ˚ equals the norm topology. Our strategy will be to use the halfopen topologies on R , along with set inclusion/exclusion definitions in order to define a mea-sure. Let ` R , B , µ ˘ denote our measure space over the extended real line. We also define anormed linear space, L ` R , k ¨ k ˘ : “ L ` R , D µ p f q | f or some f, k f k P B ˘ . Thus we will havea Banach space B over R . Then Riemann-Stieltjes integration will be semicontinuous withrespect to the norm inherited through the weak ˚ topology on L .In particular we will have a Banach space with isometric isomorphic dual over the spaceof C p R q Ĺ L p -space. It is well known that L p is the completion of C p R q with respect tothe L p -norm. We will show that with restriction to the solution space of Eq. (4) that wemay extend our space to a subspace of L p p R , d q for p “ . This extension will be necessaryin order that the domain of kinetic energy operator and the space of test functions for thepotential agree. It is also well known the domain of the free Hamiltonian operator D H f , isessentially self adjoint on L p R q . Let k be Fourier conjugate variable to x , then the uniqueself adjoint extension is the subspace of L p R q functions with Fourier transforms quadraticin k . Therefore we need to show that D H Ď D H f and that D H is self adjoint on its domain.Our approach will rely heavily on arguments of continuity, both algebraic and topological.We first start with the class of step functions χ over finite intervals, which are dense in L p for all p . This is a natural bridge between L p and C p R q , as the set C p R q is dense C p R q , and L p p R , } ¨ } p q “ C p C p R , } ¨ } sup qq , the norm closure of C p R , } ¨ } sup q . Thus any f P C p R q canbe approximated as some sequence of step functions, t χ n u P L p p R q , and therefore Lebesgue(or gauge Lebesgue-Stieltjes, HK-Stieltjes) integration in L p is sequentially equivalent toRiemann-Stieltjes integration in C p R q .2.1. Spaces of semi-continuous functions.
Consider the semi-continuous step functionsdefined from subsets of the collection of all Borel generating sets of half open intervals in R .The generalization to R n is straight forward. For example, the Heaviside distribution (Fig.1) ´ ´ ´ xH R p x q (a) Right continuous Heavi-side ´ ´ ´ xH L p x q (b) Left continuous Heaviside
Figure 1.
One-Sided Continuous Heaviside Functionscan be uniquely defined as a semi-continuous function both from the left p H L “ H p , q andfrom the right p H R “ H r , q . However uniqueness is lost with the Lebesgue measure.A natural question one may ask is, why are half-open topologies necessarily helpful?One benefit is that in a sense, uniqueness is gained. Here, there is only one regulated,semi-continuous Heaviside function on each measurable space ` R , B p¨ , ¨s ˘ , and ` R , B r¨ , ¨q ˘ .With respect to the corresponding defining topologies, H L and H R are unique topologicallycontinuous L functions.The above sense of uniqueness, along with the function spaces to be defined in the fol-lowing sections, will admit maps from S to spaces of regular distributions for distributionsthat have primitive functions, such as δ . Such mappings, when they exist, permit well de-fined functional notions of integration by parts and exponentiation. This would be of usefor physicists who work with Feynman’s functional integral, which motivates the particularchoice of application discussed in Section 3. In essence, the functional tools at ones disposal,is enlarged with respect to such classes of distributions. Furthermore, the S topology uti-lized herein, is naturally compatible with world sheet topologies of closed (super)strings andwe posit the potential existence for non-trivial applications there as well.In order to better motivate our later definitions and conventions, we first discuss a trivialexample which still highlights particular nuances in semicontinuous spaces. Consider thesemi-continuous Heaviside distributions in Fig.1. Note that H L and H R are regulated inthe sense of [38]. When paired with some φ P S , these meet the standard definition of adistribution on S such that T H L,R : “ ş R H L,R p x q φ p x q dx , or rather T H L,R : S Ñ R ` . Theydefine a semi-positive definite mapping from the space of Schwartz functions to the field of EMICONTINUOUS BANACH SPACES FOR SCHRÖDINGER’S EQ. WITH DIRAC- δ POTENTIAL 15 scalars R ` . Now consider the signum (sgn) distribution. One typically encounters variousdefinitions in textbooks. For example(7) sgn p x q : “ $’&’% , x ą ´ , x ă or the semicontinuous variants. Another way to define the sgn distribution is to use the distri-butional identity defined by a linear combination of the completely discontinuous Heavisidedistribution : H p x q “ if x ą and H p x q “ if x ă , sgn p x q “ H p x q ´ H p´ x q . (8)Indeed, take φ p x q P S , which necessarily implies φ p x q P S . With Lebesgue measure and theusual topology on R , then(9) x sgn , φ y “ ż R sgn p x q φ p x q dx “ ż φ p x q dx ` ż ´8 p´ q φ p x q dx “ x H p x q ´ H p´ x q , φ p x qy“ ´ φ p q The equivalence definitions of sgn in Eqs.(7) and (8) is a distributional identity only. Howevertwo sequences, t g n u , t f m u say, converge to the same distribution p T g Ø T f q if and only ifthey converge pointwise in the dual topology. In the case of the semi-continuous Heavisidedistributions H L or H R (Fig.1) in Eq.(8) with a Stieltjes measure on the half-open measuretopology, the above distributional identity in Eq.(9) does not hold.This can be seen in the following (see also [38, 40]). Take φ p x q P S . From point reflection p x Ñ ´ x q and the definition of H L p x q in Fig.1 we find(10) H L p´ x q “ $’&’% , x ă , x ě Though this distributional identity does not hold pointwise with respect to arbitrary measure.
Then for sgn p x q as in Eq.(8), but using H L p x q and a left semicontinuous L-S measure instead(here λ L p dx q „ dx ), we have the distributional result(11) x sgn p x q , φ p x qy “ ż R sgn p x q φ p x q dx “ ż ´8 p H L p x q ´ H L p´ x qq φ p x q dx ` lim ǫ Ñ ` ż ǫ p H L p x q ´ H L p´ x qq φ p x q dx “ ż ´8 ´ H L p´ x q φ p x q dx ` lim ǫ Ñ ` ż ǫ H L p x q φ p x q dx “ p ¨ ´ φ p q ` H L p´8q φ p´8qq ` φ p8q ´ lim ǫ Ñ ` φ p ǫ q“ lim ǫ Ñ ` ż ǫ H L p x q φ p x q dx “ x H L p x q , φ p x qy In this case we have the distributional identity of sgn p x q “ H L p x q . Analogous calculationsyield the same result using H R p x q or H L p x q in Eq. (8) and the corresponding right or leftsemi-continuous half-open topologies . An analogous calculation to (11) using the standardLebesgue measure instead of the L-S measure, one obtains zero! The topology is clearlyimportant regarding distributional identities, and sets of measure zero are now relevant.We may view the discrepancies between the last example and the distributional "iden-tity" Eq. (8) resulting from the lack of reflection symmetry with the distributions H L and H R . Under reflections x Ñ ´ x , H L p x q and H R p x q do not maintain the direction of semi-continuity. Moreover, if we were to employ the distributional derivative after the first line inEq.(11), we would incorrectly conclude that x sgn p x q , φ p x qy “ ´ x δ p x q , φ p x qy , in agreementwith the expected distributional derivative of sgn. However, doing so after the last line inEq.(11), we arrive at x sgn p x q , φ p x qy “ ´ x δ p x q , φ p x qy .2.1.1. A glance at De Rham cohomology via homotopies.
In the space of distributions (or
De Rham cohomology), the two distributions are equivalent since they differ by a constant,however a homotopy analysis shows that they produce distinct Euler characters. This givesus a hint regarding the nature of the discontinuity. H L p x q and H R p x q are each discontinuous The half-open topologies here are measure norm topologies in terms of Lebesgue-Stieltjes measures. We say "topology" here because the L-S measure is chosen to be continuous with respect to the definingtopology. This does not occur in definitions where there is a discontinuity from both the left and the right directions,as in Eq.(7). This is a point which we will return to later.
EMICONTINUOUS BANACH SPACES FOR SCHRÖDINGER’S EQ. WITH DIRAC- δ POTENTIAL 17 at one point. However, using the standard Lebesgue measure, both fail to produce a non-zerodistributional identity.With Ex.(11), one may define a contractable homotopy map, such that it may be arepresentative of the 0- th De Rham class H p R q , of the piecewise semicontinuous 0-form H L p x q : dimH p R q “ dim p R q “ , where b is the Betti number. Define the homotopyparameter t , such that it is locally equivalent to the directional (left/right) limit of thesingular point in H L p x q and δ p x q . It follows that the Euler character (with δ p x q dx , a 1-from)is easily seen to be χ p R q “ p´ q p q ` p´ q p q “ “ χ p S q . In this case we have that thehomotopies sgn p x q and H L p x q are cohomologous, which implies that d p H L p x q ´ H L p´ x qq “ δ p x q dx is a C p R , R ` q projective diffeomorphism on R , in the sense that the support of d p H L p x q ´ H L p´ x qq P p ` , and not p´8 , .We repeat the analogous calculation with sgn p x q (Eq. (7) on R ) with the usual topologyand Lebesgue measure. The Euler character is χ p R q “ p´ q p q ` p´ q p q “ . sgn x maps R into two disconnected components p R ´ , R ` q . Here, d sgn p x q “ δ p x q dx , which isno longer a projection, as was the case above. As a homotopy, d sgn p x q must either mapto inequivalent De Rham groups depending on which measure topology one implements,or must violate the equivalence between the homotopy and De Rham groups. No matterthe case, the distinct Euler characters show that there is an inherent topological differencebetween sgn (as Eq. (7)) and linear combinations of H L (as in Ex. 11), and analogously for sgn defined from H R . R is homeomorphic to S . It is well known that for S , the two cohomology groupsof compact support are equivalent, H c p S q “ H c p S q “ R . Moreover, generally for somecompact manifold M , H pc p M q “ H p p M q . In terms of the above homotopies, a discontinuityat the origin of R is equivalent to a cut at a some point on S . For semicontinuous topologies,a discontinuity depends on the direction (orientation) in which the limit is taken, or rather,on the left/right half-open interval topology. The semicontinuous homotopy projects ontothe continuous path connected component of R .We see that we may identify the above topological distinction in the half-open topologieswith a violation of reflection symmetry, if we choose the convention to define the sgn dis-tribution(s) such that the reflection symmetry is maintained. We therefore define the left semi-continuous sgn function as(12) sgn L p x q : “ H L p x q ´ H R p´ x q , and similarly, the right semi-continuous sgn function as(13) sgn R p x q : “ H R p x q ´ H L p´ x q . Example µ L p dx q „ dx ) , sgn L yields the expected distributional equivalence,(14) x sgn L p x q , φ p x qy p´8 , “ ż ´8 sgn L p x q φ p x q dx ` lim ǫ Ñ ` ż ǫ sgn L p x q φ p x q dx “ ż ´8 p H L p x q ´ H R p´ x qq φ p x q dx ` lim ǫ Ñ ` ż ǫ p H L p x q ´ H R p´ x qq φ p x q dx “ p ¨ φ p q ´ ¨ φ p´8qq ´ p φ p q ´ φ p´8qq ` ˆ φ p8q ´ lim ǫ Ñ ` φ p ǫ q ˙ ´ ˆ ¨ φ p8q ´ ¨ lim ǫ Ñ ` φ p ǫ q ˙ “ ´ φ p q ´ φ p ` q“ ´ φ p q . Analogous calculations show, x sgn R p x q , φ p x qy p´8 , “ (15) x sgn L p x q , φ p x qy r´8 , “ (16) x sgn R p x q , φ p x qy r´8 , “ ´ φ p q . (17)The measure and measure space topologies in Eqs.(14)-(17) were chosen to coincide withthe semi-continuity of H L and H R , which has obvious generalizations to the entire class ofstep functions.Another homotopy analysis of the last example, shows that d sgn p x q is still a projectivediffeomorphism such that its support is in either p´8 , or r´8 , , depending on theparticular chosen left/right half-open topologies. Furthermore, for each non-zero result in We could have instead used the left semicontinuous L-S measure: λ L p dx q „ dx . We will establish theseequivalence classes below. EMICONTINUOUS BANACH SPACES FOR SCHRÖDINGER’S EQ. WITH DIRAC- δ POTENTIAL 19
Ex. 2.1.1, the Euler character is χ p R q “ , as we would like. Hence we confirm thatreflections of semicontinuous homotopic maps is a homeomorphism invariant of the Eulercharacter.Ex. 2.1.1 implies that weak equivalences with respect to classes of step functions maybe engineered such that reflection symmetry is maintained or broken, depending on one’sparticular preference. In what follows we will study topological spaces which preserve thereflection symmetry of the classes of semicontinuous step functions. In this sense, the classessemicontinuous of step functions are the simple function representatives of the maximallysymmetric classes of functions of these spaces.2.1.2. Spaces of semicontinuous functions in C ` R ˘ . With the above in mind, we make thefollowing definitions.
Definition 2.1.
Let B be the collection of all generating Borel sets on R , such that B is a σ -algebra with the usual topology. We denote by B L , all countable disjoint unions ofleft continuous half-open Borel sets p¨ , ¨s , taken as the generating sets for B L . Similarly wedenote by B R , the half-open right continuous generating Borel sets r¨ , ¨q . Clearly B L , B R Ď B are also σ -algebras on R . We define the σ -finite measure space X “ ` R , B , λ ˘ , with λ , thestandard Lebesgue measure on R . We also separately define the measure spaces for the leftand right half-open Borel sets as X µ L “ ` R , B L , µ L ˘ and X µ R “ ` R , B R , µ R ˘ respectively,where µ L “ µ p¨ , ¨s and µ R “ µ r¨ , ¨q are the appropriate ( sup , x¨ , ¨y ) norms. Note:
Since we are on R – S , half-open intervals are indeed open sets [34, Ch. 6.3a]. Thus we have no problems taking the left (resp. right) half-open intervals as generatingsets. It is also important to point out here that X µ L ,µ R are not considered a priori to be bitopological spaces. Though, it is surely possible to define such set structures. X µ L ,µ R isnotational convenience to collectively denote the distinct measure spaces X µ L and X µ R . Definition 2.2.
Let λ be the standard Lebesgue measure on R generated by any Borel set B . We define the measure inclusions for B L and B R to be that µ L ĺ λ , and µ R ĺ λ , suchthat µ L , µ R will be the finer or equivalent continuous topologies with respect to the Lebesguemeasure λ . For Lebesgue measure zero sets (LMZ sets), we regard all topologies and measurespaces X, X µ L , X µ R as equivalent. Obviously all topologies generated on R , the usual topologies ( i.e. half-open topologiesor other), are homeomorphic and generate paracompact subsets with respect to λ . Usinggenerating Borel sets, we have that for any finite half-open Y µ L ,µ R Ă X µ L ,µ R , then Y µ L ,µ R will also contain open, closed, and half-open subsets which are finer to Y µ L ,µ R . Similarly,for some finite Y Ă X , Y will contain finer closed, open, and half-open subsets. Thusfor any continuous mapping f , between the topological metric spaces X, X µ L ,µ R with therespective norm topologies, we assume that generally there is always a coarser or finergauge refinement such that f ´ exists and is also continuous. We assume the relativetopologies to be the weakest relative topologies such that f will be bijectively continuous. Orrather, for any Lebesgue measure λ , we may define µ L , µ R such that either may be extendedby an appropriate set of measure zero (with respect to the half-open topologies) giving µ L , µ R “ λ . The notable characteristic of this construction is that sets which contain finitejump discontinuities with respect to λ are defined such that a finite discontinuity becomesleft/right semicontinuous. In this respect, we regard µ L , µ R as continuous Lebesgue-Stieltjesmeasures on subsets of X , and λ as a continuous Lebesgue measure on subsets of X µ L ,µ R .Thus for LMZ sets, µ L p a q “ µ R p a q “ λ p a q , for some a P X , such that the LMZ set t a u isclosed in all spaces, X, X µ L ,µ R .It is well known that measure theory of functions and topology can be intimately linked[35].The advantage of the coinciding half-open Borel topologies and measure topologies, is thatwe have the ability to use the discrete reflection (or rather parity) symmetry to describe a(partial) continuous symmetry on the subspaces X µ L ,µ R separately. Lemma 2.3.
The topological metric spaces X µ L and X µ R are isometric homeomorphismsunder the continuous identity mapping e , such that e : X – X L and e : X – X R . Moreover,the topological metric subspaces X L – X R are isometrically homeomorphic under e .Proof. : Trivial. e is a bicontinuous mapping of open (closed) subsets of the topological space p R , B q onto the spaces p R , B L q and p R , B R q . e is a distance preserving map in the respectivenorm topologies, and therefore X – X µ L and X – X µ R are isometric homeomorphisms.Transitively, X µ L – X µ R . (cid:3) Remark 1.
Clearly X is a homogeneous space under the identity map. We could have chosennot to separately define the spaces X µ L , X µ R , then trivially shown that they are isometrically EMICONTINUOUS BANACH SPACES FOR SCHRÖDINGER’S EQ. WITH DIRAC- δ POTENTIAL 21 homeomorphic. However it in what follows it is better to have these spaces separately defined.Indeed as we have seen in Ex.2.1.1 above, the class of step functions do not possess equalmeasures between X µ L and X µ R . At this point we have equivalence between the topological metric spaces under the identitymap in the respective norm topologies. However in light of Ex.2.1.1 equivalence betweenlinear function(al) spaces is clearly not possible for all function spaces. In particular forthe class of semicontinuous step functions, reflection symmetry (for the semicontinuoussgn functions , and thus linear combinations of H L and H R ) and semi-continuity are onlyisometrically preserved in the spaces of corresponding semicontinuous norm topology. Lemma 2.4.
Let C ` R ˘ denote the class of bounded, linear, continuous functions over R . For any function f defined on subsets C ` R ˘ , the normed linear vector spaces C µ L ” C ` R , µ L ˘ , C µ R ” C ` R , µ R ˘ , and C λ ” C ` R , λ ˘ are isometrically isomorphic. Moreover,they are Banach spaces.Proof. : Let f be any measurable, bounded, and continuous Borel function with domain D f such that D f Ď C p R q Ñ R . f is continuous if and only if lim x Ñ a ´ f p x q “ lim x Ñ a ` f p x q “ f p a q for every a P D f . The measures µ L , µ R , λ are mappings from homeomorphic topologicalspaces X µ L , X µ R , X to R respectively. Since the measures µ L , µ R ĺ λ almost everywhere,they are continuous. Thus for any continuous function f : C ` R , µ ˘ Ñ R , with µ either µ L , µ R , λ , and we have µ L p f q Ñ λ p f q in the strong norm topology, and similarly for µ R .Thus by continuity µ L p f q “ µ R p f q “ λ p f q , for all f with D f P C ` R ˘ . Since f is alinear, norm preserving, continuous mapping such that f : X µ L , X µ R , X Ñ R , f must alsobe injective, which implies that it has a kernel with ker p f q “ t u . Therefore, for each x P Rng p f q , there exists a g P C p R q where g “ f ´ p x q . Thus f is a bijective invertiblemap over X µ L , X µ R , and X . Therefore C µ L – C λ , and similarly for C µ R . By transitivity, C µ L – C µ R . C ` R ˘ inherits the norm from the metric on C λ , which is a norm-completelinear metric space. Therefore C λ is a Banach space of bounded continuous functions over R . It follows that C µ L , C µ R are also Banach spaces since each is isometrically isomorphic to C λ . (cid:3) as defined from Eqs.(14) and (13) Proof. :(Alt): Let be B p C λ , C µ L q denote the space of bounded, linear, continuous mappingsfrom C λ Ñ C µ L . Take f to be a bounded, linear, continuous function in C ` R ˘ . Bydefinition f is Lebesgue integrable, and hence is measurable in X with lim x Ñ a ` f p x q “ lim x Ñ a ´ f p x q “ f p a q . The identity map e , acts as an invertible, isometric, bijective lineartransformation. Take e ˝ f such that e ˝ f : C λ Ñ C µ L over the field of scalars R , implies f is bounded, linear and continuous in C µ L . Therefore C λ – Cµ L . This holds analogously for C µ R . By transitivity, C µ L – C µ R . C ` R ˘ inherits the norm from the metric on C λ , whichis a norm-complete linear metric space. Thus C λ is a Banach space of bounded continuousfunctions over R . It follows that C µ L , C µ R are also Banach spaces since each is isometricallyisomorphic to C λ . (cid:3) Corollary 2.5.
Since C λ , C µ L , and C µ R are isometrically isomorphic Banach spaces, wehave trivially have that B p C λ , C µ L ,µ R q and B p C µ L ,µ R , C λ q are also Banach spaces, where B p¨ , ¨q denotes the space of bounded continuous linear mappings. Remark 2.
The Banach spaces C µ L ,µ R are just two copies of the same Banach space C λ .This construction is by choice and rather inert for the class of continuous functions on R .We trivially have f P C λ if and only if f P C µ L X C µ R , by definition. However, this will notbe true for more general function(al) spaces. If we consider the measure spaces X µ L ,µ R and the linear function spaces C µ L ,µ R , wemay view them as the topological quotient spaces X { X µ L ,µ R and linear function spaces C λ { C µ L ,µ R . Definition 2.6.
We define the equivalence class structures X L “ X { X µ L and X R “ X { X µ R as the equivalence classes of left and right semi-continuous measure spaces over the measurespace X “ ` R , B , λ ˘ . Similarly,
Definition 2.7.
We define the Banach space equivalence class structures C L “ C λ { C µ L and C R “ C λ { C µ R as the equivalence classes of continuous functions which have equivalent left(resp. right) measures for all f P C λ “ C p X, λ q ðñ f P C L X C R , and µ L p f q “ µ R p f q “ λ p f q . EMICONTINUOUS BANACH SPACES FOR SCHRÖDINGER’S EQ. WITH DIRAC- δ POTENTIAL 23
Theorem 2.8.
The space of bounded linear continuous mappings B p C λ , C L,R q , and B p C L,R , C λ q are isometrically isomorphic Banach spaces with the uniform norm topologies. Moreover,they are locally convex and separable.Proof. : The normed linear metric space C λ is homogeneous. Any linear bounded continuousmapping f : C λ Ñ C L,R can be regarded as a continuous isometric mapping C λ Ñ Y Ď C λ .Since any homogeneous space is homeomorphic to itself, the mapping f preserves the normtopologies. Since f is a norm preserving continuous linear map, it is invertible and f ´ iscontinuous on Y . Therefore f is bijective. It follows that B p C λ , C L,R q , and B p C L,R , C λ q are isometrically isomorphic. C λ is complete linear metric space with the uniform norm andtherefore is a Banach space. Since C λ – C L,R , and C L,R are also Banach spaces, whichimply B p C λ , C L,R q and B p C L,R , C λ q are Banach spaces. The properties of convexity andseparability can be shown in the standard way, and are omitted in the proof. (cid:3) L p function spaces. We now move on to the spaces of Lebesgue integrable functions, p L p q , and their duals, the continuous linear functionals acting on L p functions. This will beslightly more intricate than the spaces of bounded linear continuous functions. The enlargedclass of absolutely integrable functions includes discontinuous functions. The Lebesgueand Lebesgue-Stieltjes measures which are built from collections of disjoint intervals of R has the effect of changing the measure of discontinuous functions equal almost everywherefor equivalence classes up to LMZ subsets. This matter complicates the construction ofequivalence class structures on the space of absolutely integrable functions, and leaves intactthe subspaces of continuous functions.Let us begin with the usual equivalence class identifications. For two functions f, g in thespace of absolutely integrable functions, we make the identification of equivalences classesof f, g such that r f s „ r g s if f “ g almost everywhere. Let L p p X q denote the quotient spaceof equivalence classes of the p -th power (with ď p ď 8 ) of absolutely integrable functionsover the measure space X , with Lebesgue measure λ .Here the function spaces L p p X, λ q and L p p X L,R q are not isometrically isomorphic. Forexample, take two elements from the class of step functions over the intervals r , s and p , s .The measure of H L in the latter interval is µ L p H L q p , s “ H L p q ´ H L p ` q “ ´ “ , whereas in the former interval λ p H L q r , s “ H L p q ´ H L p q “ ´ “ , and therefore µ L p H L q ‰ λ p H L q . Definition 2.9.
In the function space L ploc p X q with p ď p ď 8q , we define the subspaceof discontinuous functions by L pd p X q ” L ploc p X q z C λ p X q , where C λ p X q denotes the p -normcompletion of the subspace C λ p X q over all collections of intervals in L ploc p X q . Note: L pd p X q contains functions defined by collections of measure zero sets, however they are clearly notdense in X . Theorem 2.10.
Any non-LMZ function f P L pd p X q over some bounded interval I Ă R suchthat f is not left or right semicontinuous on I is completely discontinuous . Furthermore f has measure zero on the subspaces of semicontinuous L ploc p X q -functions in the p -normtopology, but not necessarily on L ploc (X) itself.Proof. : Let L pµ L ,µ R ,loc p X L,R q denote the subspaces of left, respectively right, semicontinu-ous L ploc functions over all intervals I Ď R . We may then define the quotient space equiva-lence classes L ploc p X q{ L pµ L ,µ R ,loc p X L,R q ” L pL,R,loc . By def. 2.9, L pd p X q “ L ploc p X qz C λ p X q – L ploc p X qz ` C L X C R ˘ – ` L ploc p X qz C L ˘ Y ` L ploc z C R ˘ , by Lemma 2.4. For any bounded interval I Ă R , and any semicontinuous function g on I , g has the domain D g “ L pL,R,loc Y C L,R Ă L ploc . The equivalence classes defined above are just the L ploc -norm completion of C L,R , whichare bounded by the uniform p sup q norms in C L,R and equivalent to the L -norm in L ploc .Therefore D g “ cl ` C L,R ˘ “ L pL,R,loc , where cl p¨q denotes the closure of the subspaces oflocally semicontinuous functions over all collections of subintervals I i Ă R . Thus, semi-continuity of the linear quotient subspaces L pL,R,loc implies that L pd p X q X L pL,R,loc “ H .Therefore f has measure µ p f q “ µ pHq “ . (cid:3) For the moment we merely state the following corollary.
Corollary 2.11.
Let f, I be as in Theorem 2.10. If there exists an I Ă I Ă R (alsofor I Ą I ) which allows f to be continuous (or semicontinuous ) over I , then there is arefinement of the measure gauge ˜ µ “ µ I such ˜ µ p f q ‰ . If the refinement is accomplishedby adding (or removing) a single point to I (i.e. a set of measure zero), we say that " f is µ L - extendable (analogously µ R - extendable )" (and respectively " µ L p µ R q - restrictable ").Proof. : See Theorem 2.15 below. (cid:3) EMICONTINUOUS BANACH SPACES FOR SCHRÖDINGER’S EQ. WITH DIRAC- δ POTENTIAL 25
Remark 3.
Our convention will be to generally apply the term extendable for both of theextension and restriction cases when there is no chance of confusion from the immediatecontext. If a given set is extendable, we take it as implicit, that this also implies that the setis restrictable, unless otherwise stated.
Corollary 2.12. µ L,R -extendable implies regularity.Proof. : For any locally measurable function over X L,R the addition or removal of a set ofmeasure zero at an endpoint will extend any half-open measure µ L,R uniquely such that itis a regular Borel measure measure. Then choose a measure gauge refinement such that ˜ µ L,R “ µ ˚ or ˜ µ L,R “ µ ˚ and ˜ µ L,R ĺ λ . The uniqueness lies in the designation of thestrongest measure topology for which continuity between X L,R and X holds. (cid:3) Proposition 2.13. : Properties of L pµ L p X L q and L pµ R p X R q For ď p ď 8 , the linear measure subspaces L pµ L p X L q and L pµ R p X R q have the followingproperties.(1) L pµ L p X L q and L pµ R p X R q are isomorphic but not isometric.(2) Each are separable seminormed Banach subspaces of L p p X q .(3) L pµ L p X L q and L pµ R p X R q are complete linear semi-normed sub-manifolds of L p p X q ,with L pµ L p X L q K L pµ R p X R q , in the p -norm topology.Proof. :(1) There exists some linear transformation f : L pµ L p X L q Ñ L pµ R p X R q . Since f is alinear mapping over the same field of scalars in one dimension, they are isomorphic.To see that they are not isometric, see Ex. 2.1.1.(2) L pµ L p X L q and L pµ R p X R q are formed by countable disjoint collections of Borel sets,any of which can be taken as a base for R . Hence they are separable. For anyfunction f over either space, the identity map e will be an isometric bijection of f into a subspace of L p p X q . Since L p p X q is a Banach space, so are L pµ L p X L q and L pµ R p X R q . However, L pµ L p X L q and L pµ R p X R q have only a semi-norm in the p -normtopology, as they do not have quotient space identifications to make them a truenormed linear subspace of L p p X q .(3) Completeness is shown below in Theorem 2.15. From 2, they are each semi-normedBanach spaces, and therefore linear. Orthogonality is as follows. There exists linear transformations f : L pµ L p X L q Ñ L pµ R p X R q and g : L pµ R p X R q Ñ L pµ L p X L q over a baseinterval I b Ă R which includes a discontinuity, such that f, g are injective, µ L,I b p f q ‰ µ R,I b p g q ‰ , and f “ g almost everywhere on I b . Since we have f “ g almosteverywhere, this implies that there exists an equivalence class over the measurespace X : r f s „ r g s such that λ I b pr f sq “ λ I b pr g sq on L p p X q . By Theorem 2.10, itfollows that f is completely discontinuous in L p p X q z L pµ L p X L q – L pµ R p X R qY L pd p X q ,and therefore µ R,I b p f q “ . This holds analogously for g , with µ L p g q “ . Since L pµ L p X L q and L pµ R p X R q are Banach subspaces with the norm topology, they aresemi-normed spaces. To see that they are sub-manifolds, take the class of stepfunctions which is dense in L p p X q . For any χ I “r a,b q , χ I “r c,d q P χ R over R , with a, d ă b, c P R , we have χ I ` χ I “ χ I Y I P χ R . For any scalar a P R , then itfollows that aχ I P χ R . This holds for χ L as well. (cid:3) For any two measurable functions f, g in L pµ L p X L q or L pµ R p X R q , we cannot form theequivalence class identifications r f s „ r g s with f “ g almost everywhere. We have seen thatit is possible to have f “ g a.e., but µ p f q ‰ µ p g q or even µ p f q ‰ but µ p g q “ . We mustbe more discriminatory in defining the quotient space equivalence class identifications. Definition 2.14. : L pL p X L q and L pR p X R q quotient spacesWe define the quotient space spaces of L pL ” L p p X q { L pµ L p X L q and L pR ” L p p X q { L pµ R p X R q by identifying any two functions, say f, g P L pL where f “ g almost everywhere and if either f or g is µ L -extendable such that µ L p f q “ µ L p g q P L pL . We denote the left semicontinuousequivalence identifications by r f s „ L r g s . We analogously define the right semicontinuousequivalence classes and denote the right semicontinuous equivalence identifications by r f s „ R r g s . Theorem 2.15.
Each semicontinuous measure space is a complete dense subspace of semi-continuous functions of L ploc p X q , which is also semicontinuous with respect to the uni-form norm topologies on the measure spaces X L,R . The p -norm is the completion of thesemicontinuous subspaces with respect to the uniform metric norms on the measure spaces X L,R . We denote the semicontinuous subspaces of L ploc p X q by L pL,loc and L pR,loc respectively. L pL,loc and L pR,loc are quotient space equivalence classes which extend uniquely over all of EMICONTINUOUS BANACH SPACES FOR SCHRÖDINGER’S EQ. WITH DIRAC- δ POTENTIAL 27 L p p X q , such that L pL,loc ” L ploc p X q{ L pµ L ,loc p X L q and L pR,loc ” L ploc p X q{ L pµ R ,loc p X R q , where L pµ L ,µ R ,loc p X L,R q are the µ L , µ R -measure subspaces of L ploc p X q . Furthermore, under these ex-tensions, all L p -functions are either piecewise semicontinuous or completely (left and right)continuous.Proof. :2.15 The density is straightforward. Take the class of step functions χ I , over someinterval I Ď R . We tacitly assume the interval I to be congruent with the measure spacetopology over which χ I is associated within the immediate adjacent text. It is well knownthat χ I is dense in L p and there for dense in all L p subsets. Take t χ I u P L ploc p X q for I “ Y I i Ď R I i , where the unions are taken over all disjoint collections of Borel generating setsin X . Thus for any f P L ploc p X I q , with I a bounded subset, there is a Cauchy sequence ofstep functions over I such that t χ I n u Ñ f a.e. as n Ñ 8 . Take the gauge for the Lebesguemeasure to be the strongest topology for which given any ǫ ą , there is a finite sub-coverof I “ Y m I i Ď R I i , congruent with the metric topology on X , such that for any x P X and n ą m , then | χ I n p x q ´ f p x q| ď ǫ as n Ñ 8 , and the sub-cover I contains the limit pointof t χ I n u “ ˝ f p x q . For any p -th power, the sequence t χ I n u Ñ f pointwise monotonically foreach sub-interval. By linearity, the sequence is continuous (and also sequentially compact).Continuity (of a sequentially compact set) and the sub-additivity of the measure, implyuniform convergence and therefore the series of the subsequences Σ n i | χ n i | p is term-wise p -summable. It follows that t χ I n u Ñ f uniformly in the p -norm topology, and thereforeit is complete in L p p X q . Similar to definitions 2.6 and 2.7, the spaces of L pµ L ,µ R ,loc p X L,R q mappings are absolutely integrable over the measure spaces X { X µ L ,µ R , with µ L , µ R ĺ λ forall choice of gauges such that the mapping f : L ploc p X q Ñ L pL,R,loc p X L,R q also is topologicallycontinuous. Therefore the continuity for any pre-image f ´ : L pµ L ,µ R ,loc p X L,R q Ñ L ploc p X q is continuous in the stronger Lebesgue-Stieltjes measure topologies, µ L , µ R , as well as λ .By Theorem 2.10, the semi-continuity of the linear mappings L pµ L ,µ R ,loc p X L,R q imply that L pd p X q X L pµ L ,µ R ,loc p X L,R q “ H however , they may share intervals with common interiors.Let f P L pd p X q be µ L,R -extendable. We denote the subset of all µ L,R -extendable functionsas L pd,ext . For any Cauchy t χ I u and therefore also any f P L pd,ext , f can be uniquely extendedby of a set of measure zero such that L pd,ext “ L pL,R,loc . Thus, any extension by a collection ofsets of measure zero, will be a unique extension from L pd to the left or right semicontinuousfunction spaces, such that for any completely discontinuous function f over the interval p a, b q , then f p a,b q L “ f p a,b qYr b s P L ploc p X L q and similarly, f p a,b q R “ f p a,b qYr a s P L ploc p X R q .Since it is generally true that L p p Y q Ą L ploc p Y q for some arbitrary measure space Y , thenthe quotient space of semicontinuous functions L pL,loc can be extended to all of L p p X q up toan equivalence relation almost everywhere. The process can be repeated for any piecewisedefined function on L p p X q defined over all collections of bounded subintervals of R . Inthis way, all subintervals I i Ă R can be reduced to arbitrarily small but countable lengths.Therefore the subspace L pd p X q may be reduced to functions taking values from collections ofnowhere dense sets. Since we regard µ L “ µ R “ λ on sets of measure zero, every L p -functionmay be regarded as either semicontinuous or completely (left and right) continuous almosteverywhere. (cid:3) Proposition 2.16. : Properties of L pL,R For ď p ď 8 , the quotient subspaces L pL p X L q and L pR p X R q have the following propertiesin L p p X q .(1) L pL p X L q and L pR p X R q are isomorphic, but not isometric.(2) Each are separable normed Banach sub-spaces of L p p X q .(3) L pL p X L q and L pR p X R q are complete sub-manifolds of L p p X q with L pL p X L q K L pR p X R q .Proof. : Analogous to Proposition 2.13. After making the identifications of equivalenceclasses defined by the quotient spaces in Def. 2.14, we have bona fide norms on L pL p X L q and L pR p X R q . (cid:3) At this point we have constructed a nice formalism of sub-manifolds within the classical L p p X q space of functions. It is interesting to point out that the Banach spaces L pL,R includethe Banach spaces of C L,R p X L,R q . This can be seen by viewing the spaces L pL,R p X L,R q asjust the p -norm completions of C L,R p X L,R q in L p p X q . However, we seem to have muchmore. The µ L,R -extendable functions are also mappings into subspaces C L,R p X L,R q as well.Thus we have a partial embedding from subspaces of L p to subspaces of C λ p X q . Theorem 2.17. : C L,R p X L,R q Ð â L pL,R p X L,R q Let f be a non-atomic, completely discontinuous, and piecewise defined function on a sub-space of either L pL p X L q or L pR p X R q . If f is µ L,R -extendable, then the left (right) extendedmappings f L , f R are partial embeddings of equivalence classes from L pd,ext p X L q into sub-spaces C L p X L q (or collections of left semicontinuous intervals of C L p X L q ) and L pd,ext p X R q EMICONTINUOUS BANACH SPACES FOR SCHRÖDINGER’S EQ. WITH DIRAC- δ POTENTIAL 29 into subspaces of C R p X R q (or collections of right semicontinuous intervals of C R p X R q ) withthe relative uniform } ¨ } sup topology on C L , C R respectively. Moreover, C ` R ˘ Ĺ C L,R p X L,R q in general.Proof. : (Left case)Let f be non-atomic, completely discontinuous, and piecewise in L pd,ext . Then f is µ L -extendable such that f Ñ f L P L pL p X L q , for example, a collection of left semicontinuousstep functions. f L need only be non-zero on some collection of left semicontinuous intervals I L “ Y i p¨ , ¨s i Ă R , provided that on each subinterval p¨ , ¨s i where f ‰ , f is continuousand, f “ on the remaining left semicontinuous intervals. Since functions which admitleft semicontinuity are regarded as continuous in C L p X L q , restricting to the L L p X L q normis equivalent to the uniform } ¨ } sup norm on C L p X L q . In the Hölder extremal case, left(semi)continuity then implies that } f } ď } f } is L -norm bounded wherever f ‰ . Hence f L is Lebesgue-Stieltjes integrable, which implies it is Riemann-Stieltjes integrable, andnorm bounded by the relative } ¨ } sup in C L p X L q . Therefore we have a complete normedspace. The right semicontinuous case is analogous. By Theorem 2.15, up to functionsdefined on sets of measure zero, we have a topologically continuous partial embedding offunctions of L p p X L q ã Ñ C L,R p X L,R q . In particular for p ă 8 , this shows that the classesof semicontinuous step functions χ L,R , which are dense in L p p X q , are similarly dense in C L,R p X L,R q , though χ L,R R C ` R ˘ in general. This implies that C ` R ˘ Ĺ C L,R p X L,R q ingeneral. (cid:3) We mention that as a result of the imposed topological and measure space congruence,Theorem 2.17 implies that the embedding holds both algebraically and topologically. Thiswill have interesting implications regarding the dual spaces of L pL,R . For example, it is wellknown that the dual of L is not generally L , but does contain L as a subspace[35, 41].The dual of L may be characterized such that L – L Y BV , where BV is the spaceof functions of bounded variation with } ¨ } BV -norm. BV includes the subset of finitelyadditive signed and countably additive measures. Let B p Σ q be the set of bounded Σ -measurable functions, and µ , a measure on the space B p Σ, µ q . The measurable space, B p Σ q ˚ – BV p Σ q is the continuous dual of B p Σ q . It follows that B p Σ, µ q ˚ – BV p Σ, µ q holds with respect to the } ¨ } sup -norm, if and only if B p Σ, µ q is continuous in the sup -norm topology. With the essential sup -norm L p µ q – B p Σ q { N µ , where N µ is the closedsubspace of all bounded null measurable functions[41]. It follows that L – N K µ , theorthogonal complement of N µ , which is the space of all finitely additive measures on Σ thatare absolutely continuous in measure ( µ -a.c.). If the measure space B p Σ, µ q is σ -finite (andtherefore separable), then we can identify L p µ q ˚ – L p µ q . Taking the dual once more, wehave that L p µ q Ă L p µ q ˚˚ – L p µ q ˚ by the Radon-Nikodym theorem. Given Theorem 2.17and the class of step functions: χ P BV , this suggests that there is also some non-trivialembedding of BV p X L,R q ã Ñ C L,R p X L,R q ˚ , provided any finitely additive measure µ P BV is µ L,R -extendable.2.2.
Linear transformations on the semicontinuous Banach spaces.
In 2.1.3 we es-tablished that L pL,R p X L,R q are sub-manifolds of L p p X q , however we would also like to definethe transformations acting on them. These are Banach spaces (subspaces and subalgebras)of continuous linear transformations and operators, which we denote as B and B L,R . Definition 2.18. : Banach spaces of linear transformationsThe space of linear transformations from L pL,R p X L,R q Ñ L pL,R ´ X L,R ¯ is also a Banachspace with the p -norm topology. We denote the Banach space of left (resp. right) lineartransformations by B L ` L pL p X L q , L pL p X L q ˘ and B R ` L pR p X R q , L pR p X R q ˘ . Note:
Reflectionsymmetry (parity) is not preserved, and is the result of an odd number of parity violat-ing transformations (i.e. transformations composed of an odd number of reflections). Wedistinctly denote parity violating linear transformations by B L Ñ R ` L pL p X L q , L pR p X R q ˘ , or B R Ñ L ` L pR p X R q , L pL p X L q ˘ , respectively. Remark 4.
In general the space of linear transformations B p X, Y q which maps the arbitrarymeasure space X Ñ Y is a Banach space if and only if Y is a Banach space, where givensome f P B , f p Y q inherits the relative topology from Y [35] . Since L pL,R p X L,R q are Banachspaces, we take this as a definition for B , and B L,R . Definition 2.19. : Reflection (parity) MapLet f L , f R be semicontinuous over some interval I Ă R , with f L P B L p I p¨ , ¨s qq and f R P B R p I r¨ , ¨q q , and ˆ Π be the parity operator define on elements x P X, X
L,R such that ˆ Π : x ÞÑ EMICONTINUOUS BANACH SPACES FOR SCHRÖDINGER’S EQ. WITH DIRAC- δ POTENTIAL 31 ´ x . Therefore ˆ Π : X L,R Ñ X R,L , where
L, R designates that ˆ Π : X L Ñ X R and R, L des-ignates ˆ Π : X R Ñ X L . For functions f P B , B L,R , ˆ Π acts through composition p ˆ Πf qp x q ”p f ˝ ˆ Π qp x q “ f p´ x q . If f L , f R are not invariant under reflections, then ˆ Πf L defines a map-ping B L Ñ R ` L pL p I p¨ , ¨s q , L pR p I r¨ , ¨q q ˘ and ˆ Πf R defines a mapping B R Ñ L ` L pR p I r¨ , ¨q q , L pL p I p¨ , ¨s q ˘ . Corollary 2.20. : Measures of ˆ Π Let f L , f R be as in def. 2.19, in particular not invariant with respect to ˆ Π . It follows that µ L p πf L q p I p¨ , ¨s q “ µ R p πf R q p I r¨ , ¨q q “ .Proof. : A consequence of ˆ Πf L P L pd p X, µ L q and ˆ Πf R P L pd p X, µ R q , when f L , f R are notinvariant under reflections of the domain coordinate. (cid:3) We see that ˆ Π can act as a linear operator on functions of B L,R , provided it acts onfunctions which are invariant under reflections. Otherwise, ˆ Π acts as a linear transformation,mapping subsets of B L,R to subsets B R,L .We also have that B L and B R are left and right semicontinuous Banach subalgebras.Let f, g be families of left and right semicontinuous functions and a P R . Then f ` f : B L ˆ B L Ñ B L and af : B L ˆ R Ñ B L are well defined and closed under addition andscalar multiplication. The same holds for the family of right semicontinuous functions, g .Moreover, if we take f , f P L pL p X L q , it follows that may define pointwise multiplicationover their common domains ( D f ¨ f “ D f X D f ‰ tHu ), as f and f are continuous in L pL p X L q and therefore, f ` f : L pL p X L q ˆ L pL p X L q Ñ L pL p X L q . We may take this onestep further to define f ` g : L pL p X L q ˆ L pR p X R q Ñ L pL p X L q ‘ L pR p X R q , as well as f g : L pL p X L q ˆ L pR p X R q Ñ L p p X q . This is really nothing new, as the sum and products of twoabsolutely integrable functions is also integrable[35, 41]. However subspaces of L pL,R p X L,R q having embeddings in C ` R ˘ can have non-trivial implications for the continuous dual spaceof linear functionals, to which we now turn.2.3. The duals of L p p X q and L pL,R p X L,R q . We now wish to consider the dual space ofour Banach spaces. These spaces are the spaces of continuous linear functionals on L p p X q and the semicontinuous submanifolds L pL,R p X L,R q . Let p, q P N : 1 ă p, q ă 8 . Fromthe Hölder inequality, it is well known that L p ` R ˘ ˚ – L q ` R ˘ where p, q are conjugatepairs such that p ` q “ . Since our universal space is L p X q where X “ ` R , B , λ ˘ ,the well known duals apply for ď p ď 8 remain true. Therefore in addition to dual spaces stated for ă p ă 8 , and since the Lebesgue measure λ is σ -finite, we know that L p X q ˚ – L p X q . It follows that L p X q ˚˚ – L p X q ˚ Ą L p X q Y BV p X q , where wehave identified BV as the set which contains all finitely additive signed Borel measures andall countable finitely additive signed Borel measures which are absolutely continuous ( µ -a.c.) as a subspace[41]. See the discussion after the proof of Theorem 2.17. Our measurespace is generated by the collection of all Lebesgue measures over R , which is equivalentto the measure space generated by all collections of all Borel measures. Therefore we canmake the identification that B p B , λ q “ L p X q ˚ , where B p B , λ q is the Banach space ofall Lebesgue measurable functions , but with the total variation metric norm | ν | p Y q ” sup t ř ni “ | ν p Y i q| ˇˇ Y i P B disjoint , Y “ Y ni “ Y i u . Note that the definition of the | ν | p Y q doesnot require σ -additivity, so ν is finite if | ν | p Y q ă 8 . ν is therefore a map ν : B Ñ C p R q ,and is referred to as a (complex) content . It follows that | ν | is a positive content [41]. Wedo not need to say anything further regarding L p p X q ˚ for ď p ď 8 , and now turn ourattention towards finding the duals for L pL,R p X L,R q .2.3.1. L pL,R p X L,R q ˚ , with ď p ă 8 . Given that we are working with L p submanifoldsdefined on semicontinuous metric-norm quotient spaces of L p ` R ˘ , these dual spaces requireslightly more care in their definitions. First we make the following definition. Definition 2.21. : Continuous linear functionals on L p p X q , p ă 8 For the Banach space L p p X q ” L p p X, λ q with the σ -finite Lebesgue measure λ , p ă 8 , and q , the dual conjugate to p such that p ` q “ , then we define the space of continuous linearfunctionals L q p X q to be the collection of all maps g P L q p X q ÞÑ l g P L p p X q ˚ given by (18) l g p f q “ ż X gf dµ p g q We state the well known theorem for the Banach spaces L p p X q : Theorem 2.22. : L q p X q propertiesLet L p p X q , l g , p, q be as in def. 2.21. For ď p ă 8 , then the collection of all mappingsdefined by Eq. (18) is an isometric isomorphism, and thus L p p X q ˚ – L q p X q , and reflexivefor ă p ă 8 . If p “ 8 , then the mapping l g is isometric, but not isomorphic. Here we note that B p B , λ q “ B p B , µ q , is the Borel µ -norm completion of the measure space. But } ¨ } L “} ¨ } sup , which is the metric norm for µ . EMICONTINUOUS BANACH SPACES FOR SCHRÖDINGER’S EQ. WITH DIRAC- δ POTENTIAL 33
Proof. : See [41, theorem 11.1 and corollary 11.2]. (cid:3)
Remark 5.
The isometric isomorphism for p “ 8 is discussed in 2.3 With def. 2.21 and Theorem 2.22, we may now define the dual of L pL,R p X L,R q for ď p ă8 . Theorem 2.23. : The dual space of L pL,R p X L,R q , with ď p ă 8 Let p, q be the Hölder conjugate pairs, with ď p ă 8 . The continuous isometric isomorphicdual to the semicontinuous Banach spaces L pL,R p X L,R q is given L qL,R p X L,R q , for p “ 8 , theyare only isometric. Again, for ă p ă 8 , the spaces L pL,R p X L,R q are reflexive.Proof. :( ` L pL p X L q ˘ ˚ ) We recall that the Banach spaces are equivalence classes define bythe topologies of quotient space L pL p X L q ” L p p X { X L , µ L q „ L p p X, λ q { L p p X L , µ L q wherethe metric norm is inherited from the relative topology of X L ” ` R , B , µ L ˘ and µ L ĺ λ . L pR p X R q was analogously defined. They are also closed subspaces (submanifolds) of L p p X q , see discussion following Corollary 2.20. Since for an arbitrary Banach space Y with closed subspace M Y , the dual of the quotient space p Y { M Y q is p Y { M Y q ˚ – t l P Y ˚ ˇˇ M Y Ď ker p l qu , the analogous result must hold here. Thus we have that L pL p X L q ˚ –t l P L p p X q ˚ ˇˇ L p p X L , µ L q Ď ker p l qu . But p L p p X q { L p p X L , µ L qq ˚ – ` L pL p X L q ˘ K , which isthe annihilator set of L pL p X L q . Therefore ` L pL p X L q ˘ K “ t l P X ˚ ˇˇ l p x q “ @ x P L pL p X L qu .But this is just the set of functionals l , which takes g P L q p X q ÞÑ l g P L p p X q ˚ , for whichany f P L pL p X L q is null for the functional l g p f q “ ş X gf d p¨q “ . Since L pL p X L q K L pR p X R q by Propositions 2.13, 2.16, this implies that the annihilator set is just the union of the setof mappings with measure µ R and the zero functional r s . Thus by the Hölder inequality, ` L pL p X L q ˘ ˚ – L qL p X L q , with p ` q “ . By Theorem 2.22 L pL p X L q – L qL p X L q : they areisometrically isomorphic, and for ă p ă 8 , they are reflexive. For p “ 8 , they areisometric. The proof of ` L pR p X R q ˘ ˚ follows analogously. (cid:3) The dual space of L L,R p X L,R q . Now we discuss the dual space of L L,R p X L,R q . Inshort, this will be similar to what is to be expected from L p X q ˚ , modulo minor modi-fications regarding the quotient space constructions of L L,R p X L,R q . To reassure ourselvesthat all the relevant details are taken into consideration, we will proceed constructively. It is rather harmless to assume that L L,R p X L,R q ˚ Ă L p X q ˚ – BV p X q , consistent withstandard results from analysis. This leads us to the following. Theorem 2.24. : Let µ L,R left and right semicontinuous Borel measures, and f L , f R beleft and right continuous functions respectively in BV ` R ˘ . There is a one-to-one correspon-dence between functions f L P BV ` R ˘ , and f R P BV ` R ˘ which are left, respectively rightcontinuous, and normalized by f L p q “ , f R p q “ , and complex Borel measures µ L andrespectively µ R on R such that f L is the left continuous distribution function of µ L definedby (19) f L p x q ´ to ´ ÐÝÝÝÑ µ L p x q ” $’’’’’&’’’’’% ´ µ L pp x, sq , x ă , , x “ ,µ L pp , x sq , x ą , and similarly, f R is the right continuous distribution function of µ R defined by (20) f R p x q ´ to ´ ÐÝÝÝÑ µ R p x q ” $’’’’’&’’’’’% ´ µ R pr x, qq , x ă , , x “ ,µ R pr , x qq , x ą , It follows that the distribution functions of the total variations of µ L , µ R are respectivelydefined by (21) | µ L | p a q “ lim x Ñ a ´ V p ,x s p f L q “ V p ,a s p f L q , and (22) | µ R | p a q “ lim x Ñ ` V r x,a q p f R q “ V r ,a q p f R q . Proof. : µ R , pÑq Each right continuous complex measure df R can be identified with a function f R P BV . As-sume f R is normalized. Then by construction f R is equal to the right continuous distributionfunction. pÐq Let dµ R be a complex measure with distribution function µ R . For each a ă b P R , which EMICONTINUOUS BANACH SPACES FOR SCHRÖDINGER’S EQ. WITH DIRAC- δ POTENTIAL 35 has the interval partition P “ t a “ x , . . . , x n “ b u . It follows that the total variation, V r a,b q p µ R q “ sup P V p P, µ R q “ sup P ř ni | µ R pr x i ´ , x i qq| ď | µ R | pr a, b qq , which is of boundedvariation. This can also be extended to all Borel sets. First consider a measure µ p x q withtotal variation V r ,x q p µ R q . Now µ is inner regular with respect to µ R , and thus valid for allopen subsets of a compact interval I Ă R . Extend this to all Borel sets by outer regularity.It then follows that µ “ | µ R | , which implies that | µ R | p x q “ V r ,x s p f q . The case for leftcontinuous Borel measures follows analogously. (cid:3) So we have for any Borel measure, a unique left, and a unique right continuous functionin BV . As before, we form quotient spaces for the left and right continuous measure spaces X L,R , such that functions that are equal almost everywhere in measure, with respect to theleft and right Borel measures are identified.
Definition 2.25.
For the measure spaces X L,R , we denote the left and right semicontinuoussets of BV over X L,R respectively by defining the quotient spaces BV L ” BV L p X L q „ B p X, λ q { BV p X L , µ L q and BV R ” BV R p X R q „ BV p X, λ q { BV p X R , µ R q . These quotientspaces identify functions which are almost everywhere equivalent and continuous with respectto µ L and µ R respectively, such that for f L P BV L and f R P BV R , µ L p f L q , µ R p f R q ‰ and µ L p f R q “ µ R p f L q “ . Let I “ r a, b s Ă R be a bounded interval. It is a well known result from analysis thatthe set p BV r I s , } f } BV q is a Banach space, with norm defined } f } BV ” | f p a q| ` V I p f q . Nowthat we have BV L,R defined, we may see that they are also Banach spaces bounded aboveby the } f } BV -norm. We will return to this momentarily. For now let us exploit the freedomgranted us by continuity of our Banach spaces.We recall that any Borel measure µ is absolutely continuous ( µ -a.c.) with respect toLebesgue measure λ , if and only if its distribution function is locally absolutely continuous ( i.e. absolutely continuous on every compact sub-interval). The consequence of this is andthe Radon-Nikodym derivative, is that µ is differentiable a.e., such that(23) µ p x q “ µ p q ` ż x µ p y q dy, µ integrable, and ş R | µ p y q| dy “ | µ | p R q . However this is just the fundamental theorem ofcalculus, which provides an alternative definition µ -a.c. functions. Since we have a one-to-one correspondence between half-open Borel measures and semicontinuous functions of BV ,we may then characterize the half-open Borel measures in terms of some unique primitivefunction associated with the integral of Eq. (23). Theorem 2.26. : On the quotient spaces of BV L,R p X L,R q , any semicontinuous function isabsolutely continuous with respect to Lebesgue measure.Proof. : Recall that µ L,R ĺ λ by construction. Then each Borel measure µ L,R is uniquelyassociated with some primitive left p f L q or right p f R q continuous function. (cid:3) Theorem 2.26, and the preceding discussion gives us everything that we need to completethe discussion for continuous dual of L L,R p X L,R q . Theorem 2.27. : L L,R p X L,R q ˚ L L,R p X L,R q ˚ – BV L,R p X L,R , } ¨ } BV q , where } ¨ } BV is the norm completion of BV L,R p X L,R q .Moreover, the bi-dual of L L,R p X L,R q is precisely the set L L,R,loc p X L,R q ã Ñ C L,R,c p X L,R q ,where the embedding is continuous and dense.Proof. : Here we implicitly assume that we are on the measure spaces X L or X R , and omittheir explicit mention in the Banach spaces. We start with ´ L L,R ¯ ˚ – L L,R . Therefore wehave the inclusions(24) L L,R Ă ` L L,R ˘ ˚ – L L,R
Taking the dual, we have from Theorem 2.26 that ´ L L,R ¯ ˚ – BV L,R . The dual of this gives(25) BV L,R
Ă p BV L,R q ˚ – ` L L,R ˘ ˚˚ Ă ` L L,R ˘ ˚ – BV L,R . EMICONTINUOUS BANACH SPACES FOR SCHRÖDINGER’S EQ. WITH DIRAC- δ POTENTIAL 37
Therefore BV L,R is self-dual. Eq. (24) also implies(26) L L,R Ă ` L L,R ˘ ˚˚ Ă ` L L,R ˘ ˚ – L L,R ùñ L L,R Ă ` L L,R ˘ ˚˚ Ă ` L L,R ˘ ˚ – BV L,R ùñ L L,R Ă ` L L,R ˘ ˚˚˚ – ` L L,R ˘ ˚˚ Ă p BV L,R q ˚ – BV L,R ùñ L L,R
Ă p BV L,R q ˚ Ă ` L L,R ˘ ˚˚ – p BV L,R q ˚ – BV L,R ùñ L L,R Ă ` L L,R ˘ ˚˚ – BV L,R – p BV L,R q ˚ The last line above allows us to identify the locally Lebesgue measurable functions with the } ¨ } sup -norm as a subset of the dual to BV L,R functions, which are the continuous functionsover all compact intervals of R , denoted as C L,R,c p X L,R q . We can see this by noting that forall inclusions above, each set inclusion is dense with respect to the corresponding superset.Next, the L L,R,loc functions can be µ L,R -extended by Theorem 2.17. Since L L,R,loc has } ¨ } sup -norm, and(27) } ¨ } p ď } ¨ } sup ď } ¨ } BV “ } ¨ } ` } V L,R } sup , where } V L,R } sup denotes the supremum of the left/right variation. Hence for ď p ď 8 , wehave } ¨ } L pL,R,loc is bounded by } ¨ } BV . It follows that the BV -norm is the norm completionfor ´ L pL,R,loc ¯ ˚ and therefore, for all L pL,R . Thus L pL,R,loc ã Ñ C L,R,c continuously. (cid:3)
Remark 6.
After the initial posting of this work to the arXiv, the author was made aware ofthe work of Johnson and Lapidus, by a form student of M. Lapidus. Particularly, the normabove is very similar in form to the mixed-norm defined in [20, Ch. 15.2] . However thereare some differences, which differ mainly in their respective origins based on how the linearfunction spaces are fundamentally structured. We will not discuss these details further here. The Dirac- δ System
We will now utilize the formalism developed in 2 to analyze the quantum system describedby Eq. (4), which we reproduce below. The system under investigation here is given bythe quantum mechanical Hamiltonian in 1-dimension described by Schrödinger’s equation.In what follows, we will only discuss the so called "interaction Hamiltonian" , where thepotential is assumed to contribute to the functional equation: ˆ V p x q ‰ . In order to make the equation well defined, we consider as a linear functional given by thefirst integral equation derived from Eq. (6). If the system is Hamiltonian (at least locally),there exists a vector flow whose first integral is the solution to Hamilton’s equations. Thus,integration is implicit in the construction of the functional equation.(28) ˆ H “ m ˆ P ` V p ˆ x q“ ´ ~ m B B x ` α B δ p x qB x . ˆ P is the one dimensional momentum operator, i ~ BB x . α is a coupling constant with unspec-ified sign ( | α | ą ) and units of length ˆ energy .3.1. Differential Geometry and the Hamiltonian Operator.
For the moment we recallsome generalities of Hamiltonian systems on differentiable manifolds in order to redefineEq. (28) in terms of operators acting on them. Let M be the compactified real line in n -dimensions. The Hamiltonian functional ˆ H defines a map ˆ H : T M Ñ M , and thus ˆ H P T ˚ M – T M . We identify the generalized coordinates p q , . . . , q n q on M as the configurationspace of the manifold, such that for x P M , then x “ x p q i q . Then ˆ H p M q is a linear functionalon the tangent bundle T M of M. Let X p M q denote the space of a vector fields in T M , and F p p M q denote the space of p -forms on M .The solutions to Eq. (28) are defined on the space of p -forms F p M q “ À np “ F p p M q , interms of the scalar product. For two forms φ, ψ P F p p M q , the scalar product is(29) x ˆ Hψ, φ y “ ż M ˆ Hψ ^ ˚ φ “ Eψ r˚ φ s , where E is the eigenvalue of ˆ H , and ˚ the Hodge dual. In the case of vacuum to vacuumtransitions, then ψ “ φ , for a vacuum state ψ and x ψ, ψ y ě for all ψ . In terms of scalarsolutions, then ψ P F p M q .The dimensionless free kinetic energy operator ˆ H f , is equivalent to the Laplace-Beltrami(Laplacian) operator, which defines a map from F p p M q Ñ F p p M q , for ď p ď n . For a C p M q scalar function f P F p M q , d f p q q : T q i M Ñ T f p q i q M, ď i ď n EMICONTINUOUS BANACH SPACES FOR SCHRÖDINGER’S EQ. WITH DIRAC- δ POTENTIAL 39 such that at the point q i , d f p q i q P T ˚ q i M , the cotangent bundle, and T f p q i q M “ T q i p T M q is the tangent space of T M at q i . Any free vacuum to vacuum scalar wave function whichsatisfies Eq (28) with ˆ V p x q “ is restricted to the class of harmonic 0-forms, H . Thismust also be the case for ˆ V p x q ‰ , otherwise by the Hodge decomposition theorem, any ψ P F p p M q with p ą will necessarily be orthogonal to H , resulting in a decoupled ( i.e. non-interacting) solution set .In order for the Hamiltonian to admit an interactive scalar solutions and avoid the intro-duction of the 1-form "potential", dx B δ p x qB x , we take Eq. (28) to be defined as(30) ˆ H : “ ~ m ∆ ` α p˚ d δ p x qq , On R , ˚ d δ p x q “ ˚ ` BB x δ p x q ˘ “ B δ p x qB x is a 0-form rather than the component of a 1-form. Itfollows that ˆ H defines a map, ˆ H : F p Ñ F p , from which Eq. (28) follows directly. Therefore,we take Eq (28) to implicitly have the intent defined by Eq. (30).For the remainder of this subsection, it will be convenient to set all the constants above to1, and discuss Eq. (30) with respect to a general potential θ P F p M q rather than specificallyhaving ˆ V p x q “ ˚ B δ p x qB x . We will also restrict our discussion to M “ R , so P p M q “ M .Then we have ˆ H given by(31) ˆ H Ñ ∆ ` ˚p d θ q . In order to discuss possible harmonic solutions to Eq. (31), we first need to define somedifferential equivalence class relations. Equivalence class identifications may seem somewhatunnecessary. However, because the function equivalence classes established in Section 2.1.3exclude identifications on sets of LMZ, and such identifications must be established underalternative associations. This is important for limiting processes, such as derivatives, con-vergence of regularized sequences of nets to distributions, and sheafs. In these cases, thereis some form of ǫ neighborhood for which we wish to include the limit point, ǫ “ . Withoutsuch equivalence class identifications it is not necessarily true, that a sequence of approxi-mating functions which ordinarily converge to a distribution at a measure zero limit point,may be identified with a distribution. We exclude cases such as tensor products of R ˆ . . . ˆ R For example, take f p x q “ δ p x q and g ǫ p x q “ tan ´ p xǫ q , and ş dk e ikx ¨ e ǫ | k | “ ǫx ` ǫ .In this case the regulated Dirac- δ , given by δ ǫ p x q “ ǫx ` ǫ , may be integrated to produce g ǫ p x q . Since g ǫ p x q Ñ sgn p x q as ǫ Ñ , and thus δ ´ ǫ p x q “ g ǫ p x q . However, one would liketo have f ´ p x q “ δ ´ p x q “ sgn p x q at the limit point of ǫ “ , as in Ex. 2.1.1. Theequivalence class identifications permit such connections to be established at ǫ “ , withoutexplicitly mapping LMZ sets under the function(al) equivalence classes of Section 2.1.3.Suppose that in some open neighborhood x P U Ă M (with a given topological space M ),we have d f p x q “ g p x q P M which defines a class of differentiable equivalences : d f „ x g . Wecan repeat this process for any x P U Ă U . Furthermore, on the spaces of vector fields anddifferential forms, the equivalence relation „ x defines the stalk F x : “ d f x of the presheaf onopen neighborhoods of U . For M a differentiable manifold, this is the space of jets of order k , J kx p M, M q . This is particularly true for distributions on the tangent spaces (vector fields) ofopen subsets of U Ă T x M and codistributions on open subsets of V Ă T p T x M q “ T ˚ x M . If itso happens that f P H p R q , then g P H p R q . It follows that d ˚ g “ dδ f P H p R q whichis coexact and coclosed, and establishes another differential equivalence relation. Thus, wehave a second order equivalence relation for some h „ x δ g „ x δd f , on all open subsets of V . If a sheaf is established, then we have a form of uniqueness given by the equivalencerelation. We now give the formal definition. Definition 3.1. : Let α be a 0-form for which d α “ θ is exact. In the category of differentialforms, the equivalence class of differentiable maps f, g P U Ă M , for an open neighborhood U of a differentiable manifold M at the point x , is the set of germs, given by f x : “ r f s x „ x r g s .Denote the equivalence class r f s on the space of differentiable forms by f Ñ α and Ñ d ,then this equivalence class defines a germ with primitive θ : r d α s x „ x r θ s in the stalk F x of the presheaf F , @ U Ă U P M . Similarly for vector fields, X x denotes the stalk of theequivalence class of vector fields such that, for X, Y P X , then r X s x „ x r Y s for the presheaf X . If the equivalence relations hold at each x P U , @ U Ă U , then F is a sheaf. Then X ,p and F ,p denotes the space of p -dimensional vector fields and the dual space of p -dimensionalforms respectively. EMICONTINUOUS BANACH SPACES FOR SCHRÖDINGER’S EQ. WITH DIRAC- δ POTENTIAL 41
Recently a diffeomorphism invariant full sheaf property was established in [26, Def.17-Prop.19] utilizing Colombeau algebras. There, a similar set of identifications to those madein Def. 3.1 are established generally for regularizible generalized functions.With the previous definition, we now consider θ P F x in Eq. (31) and look for localharmonic forms defined r θ s x P f x Ă F p M q . Proposition 3.2. : Let ω P H p R q , the space of harmonic 0-forms on R , and boundedon R . The Hamiltonian (Eq. (31) ) admits non-trivial local solutions (on R q , if and onlyif the 0-form θ of the potential (given by ˆ V p x q “ ˚p d θ q is exact and closed with respect tosome exact 1-form r d α s x , such that ˚r d α s x “ r θ s x , and r θ s x is coexact and coclosed. Bynon-trivial, we mean that θ ‰ and θ P H .Proof. : pñq The forward direction is trivial. Assume that r θ s P H . Since ω is harmonic,then the scalar product x ˆ Hω, ω y becomes x˚p d θ q ω, ω y . If r θ s P H , then we necessarily have d r θ s “ δ r θ s “ . Therefore d r θ s is closed and coclosed. This implies that there exists some0-form α , such that d α P F and ˚ d α “ r θ s P F . So r θ s is exact and trivially coexact, sincefor all β P F , δβ “ . pðq Let r θ s „ x ˚r d α s , with d α an exact 1-form. Then the inner product becomes(32) xp˚ d r θ sq ω, ω y “ xp˚ d ˚ r d s α q ω, ω y“ xp δ r d α sq ω, ω y“ xp ∆ α q ω, ω y We may rewrite ( ∆ α q ω in the last line above as(33) p ∆ α q ω “ ∆ p αω q ´ α ∆ ω ´ d α δ ω ´ δ α d ω. The last two terms in the previous line vanish trivially by the fact that α, ω P F , so δ α “ and similarly for the term with δ ω . Further more, the terms are necessarily orthogonal to H by the Hodge decomposition theorem. Therefore, the last line above reduces to,(34) xp ∆ α q ω, ω y “ x ∆ p αω q , ω y ´ x α ∆ ω, ω y“ x αω, ∆ ω y ´ x α ∆ ω, ω y . The final line above follows because ∆ is self adjoint. Since ω is harmonic both terms aboveare in H , and therefore we have a solution in H for the left hand side. (cid:3) Remark 7.
Let us comment on the above proposition. First, the proposition does not implythat θ P H p M q , but rather that θ is equivalent to a derivative of an exact form α , which isharmonic. Thus the above proposition first maps to θ to its primitive function (its f x germequivalence class), which is harmonic. Second, we note that at no point do require anythingregarding the smoothness of α , only that it be differentiable. The proof does not depend uponbeing able to use integration by parts. Therefore this includes distributional derivatives forwhich an equivalence in f can be established. Although we did not directly invert the linearfunction θ , the semicontinuous Banach spaces here do allow for this option. However, wedo need to be in a linear space where this option exists in order to justify the equivalenceclasses above. Corollary 3.3. : Let ˆ H be given as in Eq. (31) , θ, α P F p R q such that r θ s x „ x ˚r d α s asin Prop. (3.1) , and either ω P H X C p R q or x αω, ω y ă 8 . Then ˆ H reduces to (35) ˆ H “ ∆ ´ α ∆ “ p ´ α q ∆ . Proof. : This is almost a trivial consequence of Prop. (3.1) and ω P H X C p R q . InEq. (34), x ∆ p αω q , ω y becomes a boundary term in the inner product by(36) x ∆ p αω q , ω y “ x ∆ p αω q , ω y ` x αω, ∆ ω y“ ∆ px αω, ω yq . Taking ∆ inside the integral produces the boundary term, which must vanish because ω vanishes. Alternatively, x¨ , ¨y P R , then ∆ x¨ , ¨y “ trivially. Therefore, we may drop thecondition that ω P C p R q provided that x αω, ω y ă 8 . The rest obviously follows. Thefactor of 1/2 in Eq. (35) arises from moving to the local generalized coordinates in the adjointmap of the canonical cotangent projection, i.e. Hamilton’s equations on T ˚ q,p p T ˚ p R qq . Wewill discuss Hamilton’s equations in the next section. (cid:3) Hamilton’s Equations on the Cotangent Bundle.
Recall the phase space of theHamiltonian system P p M q is a n -dimensional symplectic manifold, such p q i , p i q , ď i ď n EMICONTINUOUS BANACH SPACES FOR SCHRÖDINGER’S EQ. WITH DIRAC- δ POTENTIAL 43 is the phase space coordinates are identified through the preimage of canonical projectionon the cotangent bundle, assuming that p dq i , dp i q is the local basis for T ˚ q i ,p i M . Thus, π ´ p T ˚ q i M q “ T ˚p q i ,p i q M such that for π p T ˚ p q i ,p i q M q “ p q i , q . Let β be a p -form on T ˚ q M ,then π ˚ β is the pull-back which defines the p -form on T ˚ q,p M . This is just the standardfibration of M over the cotangent bundle, which is naturally endowed with the fundamentalsymplectic 2-form structure Ω “ ř i dp i ^ dq i .The Hamiltonian ˆ H , is a 0-form on the cotangent bundle. Thus d ˆ H is identified as thePoincareé 1-form on the cotangent bundle. In the generalized local coordinates p q i , p i q , theHamiltonian in natural units given by Eq. (28) is(37) H p q,p q “ p ` V p q q . Let us define two Hamiltonians H , H as H “ p ` B θ p q qB q , (38)and, H “ p ´ α p q qq p , (39)where we identify r θ s q „ q ˚r d α s as in Section 3.1. This implies ˚ d θ Ñ B θ B q in Eq. (38).Eq (39) is obtained through the defined equivalence class and the results found in Corr. 3.3,then ˚r d α s Ñ ? αp in the local coordinates. Note that this last statement is the origin ofthe factor of 1/2 which appears in Corr. 3.3. Our goal here is to show that Eqs. (38) and(39) produce equivalent sets of Hamilton’s equations, which we will now show. Proposition 3.4. : The Hamiltonian given by (39) defines a symplectomorphism of Eq. (38) ,which is a first integral along the flow generated by the vector field X “ ´ Ω ´ d H , where Ω “ dp ^ dq is the symplectic 2-form on the phase space P p R q “ T ˚ q,p R .Proof. : We need to show that a map φ : P Ñ P : H Ñ H is canonical. A canonicaltransformation preserves the Poisson brackets, and therefore is a symplectomorphism on P .If d H is closed along the vector field X , then H is a first integral of X . We begin by finding the differentials associated to each Hamiltonian, given by d H “ B H B q dq ` B H B p dp , and the corresponding equations of motion. Thus d H “ B θ B q dq ` pdp, (40)which produces the equations of motion(41) dqdt “ B H B p , dpdt “ ´ B H B q “ p “ ´ B B q ¨ θ Similarly for H , we have d H “ ´ B α B q p dq ` p ´ α q pdp, (42)which yields the equations of motion(43) dqdt “ p ´ α q p, dpdt “ ´ B α B q p Then φ : P Ñ P : q “ q , p Ñ p “ p : BB q r θ p q qs “ ? p r θ p q qs Ñ α p q q p , is the correspond-ing map φ : H Ñ H .It is true locally that the difference between the Poincaré 1-forms of H and H is acanonical transformation if the corresponding difference is exact. Let ωdt be the differencebetween the Poincaré 1-forms obtained from H and H respectively. Thus we have thetotal time differential as,(44) ωdt “ p dqdt dt ´ p dq dt dt “ B H B p dqdt ´ B H B p dq dt “ p dt ´ p ´ α q p dt “ p dt ´ p dt “ . Therefore the transformation is canonical, and also closed. The above is equivalent to thePoisson brackets t H , H u P.B. “ . Since the Poisson brackets are equal to zero, this impliesthat H is a constant (i.e a first integral) along some locally Hamiltonian vector flow, X . EMICONTINUOUS BANACH SPACES FOR SCHRÖDINGER’S EQ. WITH DIRAC- δ POTENTIAL 45
We determine X from,(45) X “ ´ Ω ´ d H “ ´p ´ α q p BB q ` B α B q p BB p . It follows that(46) d i X Ω “ dd H “ d ˆ p ´ α q pdp ´ B α B q p dq ˙ “ ´ ˆ B α B q ˙ pdq ^ dp ´ ˆ B α B q ˙ pdp ^ dq “ ´ ˆ B α B q ˙ pdq ^ dp ` ˆ B α B q ˙ pdq ^ dp “ Therefore the 1-form d H is closed. This is equivalent to the Lie derivative L X d H “ , which implies energy conservation. Therefore H is a first integral along the locallyHamiltonian vector field X “ ´ Ω ´ d H . (cid:3) Let us discuss the results of the previous two sections in a bit more detail. Clearly theseresults only apply locally, or ultra-locally. An important implicit assumption is that theequivalence classes exist and admit identifications of r θ s q „ q ˚ d α , which acts as identifi-cation of a functional with the derivative of its primitive functional. This is precariousespecially with respect to singular distributions. By construction, the results have been de-rived from the pull-back of some mapping φ ˚ to the cotangent bundle, which we can alwaysmake well defined locally. The equivalence class simply defines an identification betweenprincipal fiber in tangent space with the canonical projection of its lift to the cohomologyclass representatives in the cotangent bundle, fibrated over each point in the base[15].Ideally, we would like to push-forward to the tangent bundle, or the base space by φ ˚ ˆ H “ ˆ H ˝ φ ´ . ˆ H is a 0-form by definition. Therefore we must have φ ´ exist. Themapping φ implicitly assumes that we have invertible transformations r θ s φ ÐÝÑ φ ´ ˚r d α s . Theimplicit assumption of the existence of the inverse restricts this mapping to (sub)spaces onwhich they are defined. However in the last few sections, we spoke generally of θ , wherethe potential was defined by ˆ V p q q “ ˚ d θ . Thus, if r θ s is a globally defined smooth function without singularities, then φ is a globally defined and invertible map. The map φ given byProp. 3.4 is a fiber homomorphism on the cotangent bundle, by φ ˚ : T ˚ q,p p L ˚ p R q , Ω p θ p q q ,p q q Ñ T ˚ p L ˚ p R qq , Ω pr θ p q qs ,p q | U q , for some U Ă T ˚ L p R q , the space of linear functionals. In partic-ular, φ establishes a covariant connection on the space of jets as in [42]. Moreover, the fiberhomomorphism maintains the unique point q P R identification in the base space. In thissense, φ is involutive.However, if ˆ V p q q “ δ p q q as in Eq. (4), then this is not so. We must restrict φ to spaceswhere we can establish the equivalence relation of the Dirac delta with the derivative of itsprimitive. We saw that this is possible locally and uniquely in Section 2. De Rham’s theoremapplies locally, and results in a Pfaffian solution on a foliated submanifold of the phase space.In particular, p -forms are the spaces of linear functionals, which form a module over thecotangent bundle. The only derivations that map r F p s Ñ r F p s on a finite dimensional, C k ` manifold is zero, for ď k ă 8 (Corr. 4.2.39,[43]). Therefore, the defined equivalence classis non-trivial only for C maps over manifolds, which is precisely the space of distributions.Therefore, the established equivalence could only make sense if it relates distributions. Itthen follows that we have φ : T ˚ q,p p L ˚ p C q , Ω p θ p q q ,p q q Ñ T ˚ q,p p L ˚ p C ,L,R q , Ω pr θ p q qs ,p q q .Finally we remark that as a consequence of preserving the Poisson brackets, the map φ given in Prop. 3.4 defines a "Lie algebra" homomorphism on the phase space p P p R q , Ω q ,such that for ω P H , then H p P q Ñ diff p P, Ω q ; ω Ñ X ω , d ω “ i X ω Ω , with kernel theconstant functions on each maximal connected component. This essentially makes a claimregarding an "algebra" over the space of functionals (distributions), which is generally diffi-cult to define consistently. At the moment, we do not speculate on the algebraic implicationsof the above. As the mapping ( φ ) could be seen as an attempt to define an indefinite integralfor distributions (though we regard the mapping as a nuanced, but distinct process), andleave those investigations for future work.3.3. The Hamiltonian functional equation.
Let the ket state be an unspecified wavefunction represented by | ψ y . We assume a priori , that it is defined over a compatible domain,which remains to be determined. The configuration space (position) x , is continuouslyparametrized by an independent time parameter t , ensuring that the energy is a constantof motion with respect to time ( dEdt “ ). Therefore, we implicitly define the wave functionin Dirac’s notation, as the position x at time t , such that | ψ t y „ | x t y . We begin with EMICONTINUOUS BANACH SPACES FOR SCHRÖDINGER’S EQ. WITH DIRAC- δ POTENTIAL 47 infinitesimal time shifts of the wave function in the Heisenberg picture. The state ψ at time t ` δt (an infinitesimal time shift) is obtained from the state ψ at time t by the perturbativeexpansion [32] | ψ t ` δt y « | ψ t y ` i ~ δt ˆ H | ψ t y ` O p δt q . (47)This implies that the transition amplitude is given by x ψ t ` δt | ψ t y “ x ψ t | ψ t y ´ i ~ x ψ t | ˆ H | ψ t y δt ` O p δt q , (48)where x ψ t | “ x ψ p x q t | .The configuration states of the system at a time t , must obey the relations ψ „ | ψ t y (49) ψ ˚ „ x ψ t | (50) ˆ x | ψ t y “ x | ψ t y (51) x ψ t | ψ t y “ δ p x ´ x q (52) ż | ψ t y x ψ t | “ . (53)Eqs. (49) and (50), are identifications of the particle-state correspondence, Eq. (52) definesthe orthonormal Fourier basis with the Dirac- δ normalization, and Eq. (53) is the complete-ness relation.In what follows, we will drop the explicit O p δt q term, and tacitly assume it remainspresent. In terms of a continuous linear functional, the bra-kets must contain informationabout the measure (space), and must belong to some linear vector space. Therefore Eq. (48)has the interpretation as a continuous linear functional with measure µ px ψ t ` δt | ψ t yq “ , ofthe form(54) µ ` x ψ t ` δt | ψ t y ˘ “ ż " ψ ψ ´ i ~ δt ψ ´ ˆ Hψ ¯* t dµ “ We now consider the linear functional equation (Eq. (54)) defined over the quotient mea-sure spaces X L,R . We will explicitly work with the left continuous quotient space X L and note that the results will analogously apply in X R . We may then consider the linear func-tional to be defined by(55) x ψ t ` δt | ψ t y “ ż " ψ ψ ´ i ~ δt ψ ´ ˆ Hψ ¯* t dµ L Inserting the Hamiltonian operator from (28) into the transition amplitude (48) andkeeping linear terms in δt yields x ψ t ` δt | ψ t y “ x ψ t | ψ t y ´ i ~ δt x ψ t | ´ ~ m d dx ` αδ p x q | ψ t y ` . . . (56)For the moment, we work with the second term on the R.H.S. of (56). We wish to have inthe space of test functions for this linear functional to cover all of R or rather all the measurespace X L . We know the measure of X L is continuous with respect to the Lebesgue measure, λ . From 2.1.1 we see that we if we assume φ P S , then we have the weak equivalences(57) x δ p x q , φ p x qy “ ´ x δ p x q , φ p x qy “ ´ x sgn p x q , φ p x qy “ x sgn p x q , φ p x qy “ ´ φ p q , up to negligible terms involving powers of x multiplying δ p x q . We furthermore make theassumptions that | ψ y is self-dual (i.e. x ψ | : “ | ψ y ). We use the above weak equivalence tomake the identification of δ „ x ˚r d sgn L s as in Section 3.2. It is interesting to note that if indefinite integrals of distributions were indeed defined, the same result could be obtainedusing integration by parts two times on second term on the R.H.S. of (56). Remark 8.
It is well known that L-S measures do allow us to write the integral of δ x similarto ş R f p x q δ t dx u “ ş x ´8 f p y q dH p y q , for f P S and H p y q , the Heaviside distribution. In factTalvila [38, 40] discusses Banach spaces of integrable distributions, where the above is defineduniquely. The issue which arises as that such spaces are not, in general, separable. However,this is not an issue for the present case. Throughout our derivation above and below, weassume that the spatial variable x really represents some interval: x P p¨ , ¨s or x P r¨ , ¨q of R , and therefore admits a countable basis for the Hilbert space. The Banach spaces ofintegrable distributions are separable under such circumstances. However, this is no longertrue once we take the "continuum limit". This is often the standard approach by physicists,but only after completing the operational calculi steps. Thus, in theory, we could employ The terms of the form xδ p x q are discarded. These terms are either zero, or are orthogonal to the harmonicsolution space H . EMICONTINUOUS BANACH SPACES FOR SCHRÖDINGER’S EQ. WITH DIRAC- δ POTENTIAL 49 such methods as integration by parts and maintain the structure of a "functional" Hilbertspace. It is likely that we would even have some notion of a non-commuting Banach algebrasimilar to the family of disentangling algebras t A t u t ą of [20, Ch. 18] . However, we willleave such discussions for future work. The | ψ y has an expansion in terms of some Schauder basis, such that | ψ y “ Σ n | ψ n y ă 8 .Therefore we can say that the unbounded differential operators and the Dirac- δ are weaklybounded for wave functions which belong to a compatible function space. The calculationis sketched as follows,(58) x ψ t | ˆ H | ψ t y “ „ ´ ~ m d dx ` αδ p x q x ψ t | ψ t y , We have left continuity and apply the results of Prop. 3.4 to obtain an operator similarto [11], which is just a linear transformation on the functional space ,(59) x ψ t | ˆ H | ψ t y “ ż dµ L p x q „ ´ ~ m ` α L p x q d dx x ψ t | ψ t y . In order to reduce the accumulation of constants in (59), we relabel the constant termswith the definition a ” ~ ? m , and write the functional equation in the less cluttered form x ψ t | ˆ H | ψ t y “ ż dµ L p x q „ ´ ´ a ´ α L p x q ¯ d dx x ψ t | ψ t y (60) “ ż dµ L p x q ” a ´ α L p x q ı ˆ i ddx ˙ x ψ t | ψ t y , (61)where we have adsorbed the minus sign by restoring the i in the differential operators tomake them Hermitian.We may now write the transition amplitude of (56) to st order in δt as x ψ t ` δt | ψ t y “ ż dµ L p x q „ ´ i ~ δt ˆ H p x q x ψ t | ψ t y ` O p δt q , (62)where ˆ H p x q “ ´ a ´ α L p x q ¯ ˆ i ddx ˙ . (63)Eq. (63) then represents the transformed connection on the fibers of the cotangent bundle. Itis worth noting that the Fourier transform of Eq. (62) above, (and more generally Eq. (35)) Theorem 2.24 ensures that this map is well defined, as it is locally compact and Hausdorff on p´8 , . is very similar, and seemingly analogous to the difference equation results of [37, Eq. 4.4](with understandably different boundary conditions). Another notable similarity of theabove result, is to the Hartee equation for infinitely many particles, where the L well-posedness of which was discussed in terms of Strichartz estimates by [25].We close this section by noting that we may find the Lagrangian density function fromEq. (63), which defines an isomorphism of the fibers from the cotangent bundle to thetangent bundle. We recall the normalization condition of Eq. (52) and interpret the p x ´ x q factor as a velocity by writing it as(64) x ´ x “ dxdt δt “ xδt to first order in δt . By including a factor of ~ , we may also interpret the differential operatoras the momentum operator ˆ P such that(65) ˆ P “ i ~ ddx , and obeys the eigenvalue equation(66) ˆ P | ψ y “ p | ψ y with eigenvalue p . Equations (62) and (63) are then a Legendre transformation on the linearfunctional equation. Together they yield, x ψ t ` δt | ψ t y “ ż dµ L p x q " i ~ ż t f t i δt L * ` O p δt q , (67)where the Lagrangian density is L “ xp ´ ˆ ´ ~ m ` α L p x q ˙ ¨ p . (68)Obviously, analogous results are obtained for the measure space X R .Eq. (68) may be exponentiated, and inserted into the free Feynman functional integral,(69) x ψ t ` δt | ψ t y “ ż D x D p exp i ~ ş δt L , EMICONTINUOUS BANACH SPACES FOR SCHRÖDINGER’S EQ. WITH DIRAC- δ POTENTIAL 51 where D x is the Feynman measure. The full path integral may be evaluated by first inte-grating over the momentum. Then analytically continue by t Ñ iτ , which compactifies τ on S , and produces the convergent Gaussian integral.3.4. The domain (test function space) of the functional equation.
From Eqs. (62)and (63), the topological measure space X L , from which we demanded topological continuityin our solution space. Since a discontinuous function cannot be in D δ , this leaves us out ofthe space of L p p R q functions. However as we have seen above, the semicontinuous quotientspaces allows the partial embedding of measure extendable L p functions into C L,R p R q . Thisis an artifact of the Lebesgue-Stieltjes measure when continuity is restricted to semicontinu-ity. There are a few more interesting properties of this which we will comment on shortly.However with respect to X L,R , the Hamiltonian operator in Eq. (63) is topologically con-tinuous. Thus in the case of Lebesgue-Stieltjes measures, functions which are topologicallycontinuous with respect to X L,R , may now also be in D δ . It follows that we may define aSobolev space which has some L p -space functions, but are continuous with respect to X L,R .With Lebesgue-Stieltjes measures µ L,R , D δ includes the space of C L,R functions func-tions. In terms of the Hamiltonian operator of Eq. (63), D ˆ H “ t f | f P C L,R p X L,R qu : “ C L,R p X L,R q , or the class of µ L,R -extendable L p functions which are also µ L,R -measurable(integrable) on all compact subsets of X L,R . The dual space of C L,R p X L,R q is the spaces ofleft/right continuous measures of bounded variation, which is the completion of C L,R withrespect to the } ¨ } BV -norm. We note that the Hamiltonian operator (Eq. (63)) with the } ¨ } op -norm is not only bounded (weakly) by the BV L,R -norm completion of L L,R p X L,R q Ą L pL,R p X L,R q for ď p ď 8 , but in fact they are equivalent. Let φ P C L,R p X L,R q such that } φ } L L,R “ and } V p φ q} L L,R “ | a | } φ } L L,R , for some a P R and | a | ă 8 . Then(70) } ˆ Hφ } L L,R ď ˇˇ sup ˇˇ φ ˇˇ ` sup ˇˇ V p φ q ˇˇˇˇ ď sup ˇˇ } φ } ` | a |} φ } ˇˇ ď ˇˇ φ ˇˇ ` | a | ˇˇ φ ˇˇ “ p ` | a |q ˇˇ φ ˇˇ “ p ` | a |q ¨ “ } φ } BV L,R . Then we have that the operator norm of the Hamiltonian is given by(71) } ˆ H } op “ } ˆ Hφ } L L,R } φ } BV L,R “ p ` | a |q | φ |} φ } BV L,R “ p ` | a |q ¨ p ` | a |q“ . Moreover, if φ is Lipschitz continuous such that for a real number M ě with } φ p n q }} φ } ď M for all n ą , with } φ } “ and | a | ď M , the previous norm-bounded result can bestrengthened to } ˆ Hφ } L L,R ď M p ` | a |q “ M } φ } BV L,R . Inserting the Lipschitz result in theset of Eqs. (71), the same result is obtained under a stronger form of continuity.3.5.
The Hilbert space of C L,R p X L,R q . In the last section we saw that the }¨} BV L,R -normis the norm completion of C L,R p X L,R q with } ¨ } sup -norm, and that D ˆ H “ C L,R p X L,R q . Itfollows that the wave function | ψ y must also belong to this Sobolev-type space, or belongto a completely continuous function space after two derivatives, which could be describedas the measurable (integrable) functions of C L p X L q X C R p X R q “ C λ p X λ q on all compactsubsets of R .In order to have a Hilbert space, any wave function | ψ y which satisfies these conditionsshould be self-dual, and the functional equation x ˆ H, ψ ˚ ψ y „ x ˆ H, ψ y should also satisfy theHölder inequality with p ` q “ . It follows that quantum mechanics requires we identify(72) x ˆ H, ψ ˚ ψ y “ x ˆ H, ψ y „ c x ψ ˇˇˇ ˆ H ˇˇˇ ψ y . With this in mind, it follows that we must restrict | ψ y to be those C L,R p X L,R q functionswith Euclidean norm. Therefore we need a norm defined by } ¨ } BV L,R
X } ¨ } L L,R . This impliesthat the Hilbert space norm of | ψ y is given by b } ¨ } BV L,R . Therefore we define the Hilbertspace to be the following.
Definition 3.5. : The semicontinuous Hilbert space of C L,R p X L,R q Let the Hamiltonian operator be given by Eq. (63) , and ψ P C L,R p X L,R q . The Hilbert EMICONTINUOUS BANACH SPACES FOR SCHRÖDINGER’S EQ. WITH DIRAC- δ POTENTIAL 53 space for ψ is defined by the twice differentiable semicontinuous space of left/right Lebesgue-Stieltjes measurable (integrable) functions with BV L,R H : “ t ψ ˇˇ ψ P C L,R p X L,R q , and c (cid:13)(cid:13)(cid:13) p ψ ˚ , ˆ Hψ q (cid:13)(cid:13)(cid:13) BV L,R ă8u . Since we have a Hilbert space which is dependent upon the measure of an operator on C L,R p X L,R q functions, this is becomes equivalent to the Schatten norm over these functionspaces. In light of Ex. 2.1.1 and as a result of the semicontinuous quotient space construction,we have the following result, which by now may be obvious. Theorem 3.6. : Decomposition of H The Hilbert space of H of C L,R p X L,R q -functions has the orthogonal decomposition of left(resp. right) measurable functions such that H “ H L ‘ H R , where H L and H R are separableorthogonal subspaces and submanifolds of left/right µ L { µ R -measurable (integrable) functionson the measure spaces X L and X R , respectively. Moreover, for any non-atomic completelydiscontinuous function f (neither left, nor right semicontinuous) over an interval I “ p¨ , a qYp a, b q Y p b, ¨q Ă R which is Lebesgue measurable for f over the subinterval I “ p a ` , b ´ q suchthat λ p f q p a ` ,b ´ q ‰ , finite and with Lebesgue-Stieltjes measure µ L p f q I “ µ R p f q I “ , isseparately left and right µ -extendable to the intervals I L “ p a, b sYp b, ¨s and I R “ r¨ , a qYr a, b q ,where µ L p f q I L ‰ implies f I L P H L , and µ R p f q I L “ and µ R p f q I L R H L . Similarly, µ L p f q I R “ and µ R p f q I R ‰ implies µ R p f q I R P H R and µ R p f q I L R H R . Therefore eachleft/right extension is separately unique in H L and H R .Proof. : The proof of this can most easily be seen by first referring to Ex. 2.1.1. Thisshows for a simple step function (Heaviside function) that there is a unique left and rightextension of the Heaviside function such that µ L p H L q K µ R p H R q . Separability of H L and H R is inherited by the countable measure topology basis of X L,R . The left/right extensionis chosen such that the function is piecewise extended to the left/right by a set of Lebesguemeasure zero, and the left/right function value over the interval extension is continuous (i.e.with no jump). Given any µ L,R extendable function f in the spaces C L,R p X L,R q (which byconstruction may not consist only of atoms), f may be approximated by a sequence of stepfunctions semicontinuous step functions χ L n P C L p X L q or χ R n P C R p X R q . There are twopossible cases at each boundary point, and a third separate case which we will explain afterthe boundary point cases. Case i) Take a sequence of left continuous step functions. Each left sequence is, bedefinition zero over any boundary point which is discontinuous from the left and lies outsidethe interval extension (i.e. if the extended interval χ p a,b q “ χ p a,b s , this by definition implies χ p a q “ unless there is a different step function defined over the interval χ p¨ ,a s ). Theanalogous holds for a right continuous sequence of step functions.Case ii) Take a completely discontinuous function f over the joined intervals I “ p¨ , a q Yp a, ¨q , where a jump occurs at f p a q . Choose to extend f such that it becomes left continuous, p f “ f L over the interval I L “ p¨ , a s Y p a, b s , such that f p a q ‰ f p a ` q . In the topologicalmeasure space X L , µ L p f p x “ a qq is well defined, however µ R p f “ a q “ H , and µ R pHq “ .The analogous holds for f right extended, such that p f “ f R on I R “ r¨ , a q Y r a, ¨q . Sincein either case, there is a jump discontinuity at f p a q , and p f is chosen to be continuous fromeither the left or the right, we have that f L p a q ‰ f R p a q , otherwise there would be no jump,and thus each left/right extension is unique.In either Case i) or ii), we have that either µ L p f q “ H and µ R p f q “ H , which implies µ L p f q “ µ R p f q “ , or p f “ f L , which implies µ L p f q ‰ and µ R p f q “ (and similarly for p f “ f R ). Again, the left/right extension is unique.Case iii) There is a function g over some interval I , for which µ L p g q I “ µ R p g q I “ and g is not µ L,R -extendable. In this case g „ r s (the equivalence class of the zero function forboth X L and X R ). However r s is the only function equivalence class which may be commonto both X L and X R .The remaining aspects of the proof follow straightforwardly. (cid:3) Indefiniteness of H and Krein Spaces In this section we only wish to make some cursory comments regarding the formalismdeveloped above and the theory of Krein spaces (and Krein space operators). For a conciseoverview of Krein spaces see [36]. For a more comprehensive introduction see [14].4.1.
Krein spaces and Krein space operators.
We summarize some basic definitionsof Krein spaces and Krein space operators given by [36, Sec. 3]. Let H , K denote Kreinspaces on R . A Krein space (which may also be a Pontryagin space) is an indefinite innerproduce space which is representable as the orthogonal direct sum H “ H ` ‘ H ´ , where H ` “ t p H , x¨ | ¨yq u is a positive-definite Hilbert space and H ´ “ t p H , ´ x¨ | ¨yq u is the EMICONTINUOUS BANACH SPACES FOR SCHRÖDINGER’S EQ. WITH DIRAC- δ POTENTIAL 55 antispace of a Hilbert space with a negative inner product. A fundamental symmetry on H are symmetries expressible an orthogonal direct sum. The Hilbert space topology is thestrong topology induced on H , and the dim H ˘ are the indices of H . A Pontryagin space isa Krein space with finite H ´ index.The spaces of continuous linear functionals (operators) and adjoint operators are denotedby L p H q and L p f , K q . For some operator A P L p H , K q then A ˚ P L p K , H q , with x Af, g y “x f, A ˚ g y for some f P H and g P K . Definition 4.1. : Properties of Krein space functionsLet A P L p H q . Then A is:self adjoint if A ˚ “ A ,a projection if A ˚ “ A and A “ A ,nonnegative if x Af, f y ě , @ f P H . Definition 4.2. : Properties of Krein space operatorsLet A P H be self adjoint, and denote the supremum of all r for which there exists an r -dimensional subspace of H that is a (anti)Hilbert space by ind ` A , respectively ind ´ A , in theinner product given by x f, g y A “ x Af, g y for f, g P H . Let B be an operator in L p H , K q , then B is:isometric if B ˚ B “ H ,partially isometric if BB ˚ B “ B ,unitary if both B and B ˚ are isometric,a contraction if B ˚ B ď H ,a bicontraction if both B and B ˚ are contractions. We also note a difference between Krein spaces ( H ) and Hilbert spaces ( H ) regardingorthogonality. Let M Ă H be a closed subspace in H . It is generally not true that H “ M ‘ M K . However, if in addition to being a linear subspace of H , M is also a regularsubspace (a Krein subspace) if it is closed and a Krein space in the inner product of H . Ifthese conditions hold, then we have the following: Definition 4.3. : Krein subspacesLet M be a regular subspace of H , then the following are equivalent: M is a Krein subspace, H “ M ‘ M K ,For a projection operator ˆ P P L p H q such that ˆ P : H Ñ M , then M “ ran ˆ P . H as a Krein space. From what we have seen in Section 4.1, the Hilbert spacecertainly has the properties of a Krein space. The Hamiltonian functional operator Eq. (63)on the orthogonal measure spaces X L,R provides a decomposition of H . Let ψ L , φ L P H L and ψ R , φ R P H R . Since Eq. (63) was found explicitly on the measure space X L , we denotethe left/right Hamiltonian by ˆ H L “ Eq. (63), and H R “ Eq. (63) with sgn L p x q Ñ sgn R p x q ,and denote the initial Hamiltonian function ˆ H “ Eq (58). Then by repeating the stepsfrom moving from Eq. (58) to Eq. (63) in the case of X R and denoting the eigenvalues of ˆ H L,R | ψ L,R y “ E L,R | ψ L,R y , we have the functional result(73) x ψ | ˆ H | ψ y “ x ψ L | ˆ H | ψ L y ‘ x ψ R | ˆ H | ψ R y ‘ x ψ L | ˆ H | ψ R y ‘ x ψ R | ˆ H | ψ L y“ x ψ L | ˆ H L ψ L y ‘ x ψ R | ˆ H R ψ R y ‘ x ψ L | ˆ H R ψ R y ‘ x ψ R | ˆ H L ψ L y“ E L x ψ L | ψ L y ‘ E R x ψ R | ψ R y ‘ ¨ x ψ L | ψ R y ‘ ¨ x ψ R | ˆ ψ L y“ E L x ψ L | ψ L y ‘ E R x ψ R | ψ R y where orthogonality of | ψ L y , | ψ R y is used from the second to the the third lines.Eq. (73) shows that we have the Hilbert space H “ p H L , E L x ψ L | ψ L yq‘p H R , E R x ψ R | ψ R yq .However H being expressible as an orthogonal decomposition does not ensure that H is itselfis Krein. A necessary condition is that H L,R are each a Krein subspace. Therefore we mustbe able to show that H L,R “ H L,R, ` ‘ H L,R, ´ , where H L,R, ` is a Hilbert space and H L,R, ´ is the associated anti-Hilbert space. We already have the indefinite structure built into ourleft/right states via the coupling constant α . The Hamiltonian functional was defined suchthat the sign of α was unspecified.Let H be a Krein space associated with the Hilbert space H , A a definitizable operator in H , and E the spectral function of A . If A is positive, its spectrum is σ p A q P R , where 0 isthe only non-negative semi-simple eigenvalue of σ p A q . Also t , may be the only criticalpoints of the spectrum. If t , are regular critical points (non-singular), and 0 is not aneigenvalue, then A ˚ “ A in the Hilbert space H “ p H , xp E p R ` q ´ E p R ´ qq¨ , ¨yq . EMICONTINUOUS BANACH SPACES FOR SCHRÖDINGER’S EQ. WITH DIRAC- δ POTENTIAL 57
It was shown in [11] the if H “ L p R q with y P D A “ W , , where W , is the L Sobolevspace, then the Julian operator
J y “ sgn p x q d dx y is a fundamental symmetry on H . Then Ay “ ´ sgn p x q d dx y is congruent to a self adjoint operator on L p R q . Moreover A has noeigenvalues, σ p A q “ R , and t , are regular and the only critical critical points of σ p A q .Given the work of [11], we can directly infer the following about the Hamiltonian functionaloperator Eq. (63) and the Hilbert (anti)spaces. Theorem 4.4. : Properties of ˆ H as a Krein space H Let ˆ H be the Hamiltonian functional operator given by Eq. (63) on the Hilbert space H “ H L ‘ H R , and H “ C L,R p X L,R q Ą L L,R p X L,R q with an indefinite inner product. Then H isdirect sum decomposable as H “ p H L ‘ H R q ` ‘ p H L ‘ H R q ´ , with H L, ` “ ` H , x E L p R p¨ , ¨s q¨ , ¨y ˘ being the left continuous Hilbert space, H R, ` “ ` H , x E R p R r¨ , ¨q q¨ , ¨y ˘ being the right continuous Hilbert space, H L, ´ “ ` H , ´ x E L p R p¨ , ¨s q¨ , ¨y ˘ being the left continuous Hilbert antispace,and H R, ´ “ ` H , ´ x E R p R r¨ , ¨q q¨ , ¨y ˘ being the right continuous Hilbert antispace.Moreover ˆ H is self adjoint on H L,R, ` and anti-self adjoint on H L,R, ´ .Proof. : Each Hilbert space and antispace H L,R, ˘ are mutually orthogonal and constructedfrom the quotient spaces of half-open topologies on R . The Hilbert space topology is thestrong topology inherited by H , therefore H inherits the left/right continuity of X L,R . Anal-ogous arguments to [11] show that ˆ H is self adjoint on H L,R, ` separately when α ą , whichcorresponds to the operator A “ ´ sgn p x q d dx , defined in the previous paragraph. Since theJulian operator defined above by J “ sgn p x q d dx is a fundamental symmetry of H (againshown in [11]), which corresponds to the case of α ă and ´ A “ J , it follows that H isanti-self adjoint on H L,R, ´ . (cid:3) Summary and Concluding Remarks
We have investigated the quantum mechanics Hamiltonian with a Dirac- δ potential asa continuous linear functional operator. The particular aspect of this equation is that thekinetic energy operator is a diffeomorphism mapping from the space of weakly continuouslinear functions L to another function space L by p D ˝ D q : L Ñ L Ñ L . However δ may be considered (after one integration by parts) as a measure or distribution on the linear transformation from L Ñ L , or rather T : L Ñ L Ñ C (or R ). Moreover the space oftest functions of δ and δ is equivalent to any L p space due to L p containing discontinuousfunctions.In order to resolve the domain incompatibilities for distributional potentials in quantummechanics, we constructed the spaces of semicontinuous functions. The spaces L PL,R areprojective subspaces of the standard L p spaces, which single point extensions/restrictionson each disconnected open set for which they are defined on L p . This effectively allows all L p (aside from functions of atomic sets) functions to be identified as left/right semicontinuousfunction, which is topologically continuous when defined on their corresponding topologicallysemicontinuous measure spaces, X µ L or X µ R . This continuity is reduced to semicontinuityon each measure space. The }¨} sup bounds all }¨} L pL,R including the finitely additive measuresof bounded variation, } ¨ } BV L,R . Under these conditions, we have that L pL,R ã Ñ C L,R is apartial embedding of semicontinuous L p functions into semicontinuous spaces C L,R , withRiemann-Stieltjes integral. We may view Riemann-Stieltjes measures as an extension ofthe Riemann measure to include half-open intervals, or the Lebesgue-Stieltjes integral as arestriction of the Lebesgue measure to half-open intervals. In this way, they are equivalenton C L,R spaces. The C L,R spaces provide two advantages over the standard C p R q and L p spaces. The first advantage is that C L,R allows us define a common space of test functions for δ and δ functionals which includes as subspaces semicontinuous L p functions. The secondadvantage is that we may include Banach spaces of regulated distributions, which invertregulated distributions in terms of their primitive functions.In Section 3 we analyzed the functional Hamiltonian Eq. (4) on the semicontinuous spacesas differentiable manifolds, complete with the tangent (and cotangent) fiber bundle struc-tures. We then obtained a connection form transformation of Eq. (4), which was shown tobe canonical on the cotangent bundle. This permitted an equivalence class identification,which was a foliation of the of the cotangent bundle in terms of the cohomology classes oflinear functionals with derivative of their primitive functions. The fact that inverses of regu-lated distributions is possible in the semicontinuous function spaces is needed to make theseequivalence identifications well defined. In that way, semicontinuity was the key propertywhich made such constructions possible. The semicontinuity of the differentiable manifolds EMICONTINUOUS BANACH SPACES FOR SCHRÖDINGER’S EQ. WITH DIRAC- δ POTENTIAL 59 then allowed us to define a common domain of Eq. (4), and determine a 0-form wave func-tion solution exists in such a way it is in the cohomology class of harmonic 0-forms for boththe kinetic energy operator and the δ potential. The orthogonality of L pL,R function spacesextends to the Hilbert spaces of the Hamiltonian operator Eq. (63), and thus provided ansemicontinuous orthogonal decomposition of the Hilbert space H .In Section 4, we discussed the Hilbert space H within the indefinite structure of Kreinspaces, H . The indefinite structure was implicitly manifest through indefinite multiplicativecoupling α sgn p x q , defined in Eq. (63). Therefore we found that the Hilbert space and asso-ciated antispace were regular subspaces of H . The work of [11], shows that the Hamiltonianfunctional equation, Eq. (63), is self adjoint on C L,R for the case of α ą and anti-selfadjoint for the case when α ă .There remains open questions to which we leave for future work. In particular, theexistence of an antispace of H implies the existence anti-particle states inherent in QFT. Herethey are manifest in a basic quantum mechanics construction. A complete spectral analysisof the system provide insight between quantum mechanics and quantum field theory withrespect to this system. What is interesting is that our construction was based on a classicalBanach space formulation of quantum mechanics, yet it seems that notions of quantumfield theory are almost implicit. Obviously restricting to the half-line would remove thenegative definite components of the spectrum. However, that notion seems unsatisfying.There is nothing special regarding R ´ . Regardless, the coupling term α sgn p x q is almostbetter viewed as a Pfaffian-like β -function in the QFT path integral quantization. Thistouches upon work current in progress regarding a Feynman path integral formulation ofthe system for which we will investigate (among other things) anomalous bound states, andghost states resulting from the Feynman "measure", and the possibility of supersymmetricstates. In regards to the latter, a supersymmetric field is defined via fields which obey ananti-symmetric commutator algebra. An investigation of some algebraic structure, as inRmrk. 8, would be needed for such an analysis.In light of the discussion in Section 4 and the odd character of the Dirac- δ potential,supersymmetric states may be implicit in a very natural way. Preliminary calculationssuggest this to be the case. Anomalous bound (or even possibly scattering) states mayalso provide some theoretical predictions regarding low dimensional solid state systems. For example, the possible quantization of magnetically induced current flows in carbon nano-tubes and other small scale structures for which quantum interactions become dominant. AFeynman path integral formulation of the Dirac- δ system shows that the Feynman measurecan introduce non-trivial dynamics through the exponential (i.e. ghosts), which can becomedominant if the coupling constant is on the order of unity.Another intriguing component of our study here is the connection form Eq. (35) derivedfrom Eq. (4). Eq. (35) has a form similar in nature to the Dirac-Born-Infeld (DBI) operator.It would be interesting to generalize what has done hear to R n and look to see if this analogyindeed holds true. Intuitively, one would expect that 1-dimensional Pfaffians would be comecomponents of spinors in higher dimensions. Also, the possibility of defining a consistent"Lie algebra" using this canonical form of Eq. (35), and the relevant implications for analgebra representation for jets, or for pseudo-differential operators. It would be interestingto investigate the limits of our construction here in terms of these formalisms, both separatelyand in conjunction with the possible DBI operator connection. Acknowledgments.
The author would like to thank Prof. Helge Holden for helpful com-ments regarding historical developments and useful references for this work. The authoris particularly grateful to Dr. Michael Maroun for his friendship, the many helpful com-ments, uncountable enlightening discussions, and suggested references. This work would nothave been possible without his input. Finally, the author is also tremendously grateful toDr. Tuna Yildirim for his friendship and willingness to help proof read this document forgrammatical errors.
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