Incompatible Coulomb hamiltonian extensions

We revisit the resolution of the one-dimensional Schrödinger hamiltonian with a Coulomb λ/|x| potential. We examine among its self-adjoint extensions those which are compatible with physical conservation laws. In the one-dimensional semi-infinite case, we show that they are classified on a U(1) circle in the attractive case and on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\boldsymbol{(}}{\mathbb{R}},{\boldsymbol{+}}{\boldsymbol{\infty }}{\boldsymbol{)}}$$\end{document}(R,+∞) in the repulsive one. In the one-dimensional infinite case, we find a specific and original classification by studying the continuity of eigenfunctions. In all cases, different extensions are incompatible one with the other. For an actual experiment with an attractive potential, the bound spectrum can be used to discriminate which extension is the correct one.


incompatible coulomb hamiltonian extensions G. Abramovici
We revisit the resolution of the one-dimensional Schrödinger hamiltonian with a coulomb λ/|x| potential. We examine among its self-adjoint extensions those which are compatible with physical conservation laws. In the one-dimensional semi-infinite case, we show that they are classified on a U(1) circle in the attractive case and on  ( ) + ∞ , in the repulsive one. In the one-dimensional infinite case, we find a specific and original classification by studying the continuity of eigenfunctions. In all cases, different extensions are incompatible one with the other. For an actual experiment with an attractive potential, the bound spectrum can be used to discriminate which extension is the correct one.
The Coulomb problem addresses the non-relativistic Schrödinger equation with a 3-dimensional Coulomb potential, restricted to one dimension; it has inspired a vast corpus of scientific literature for the last seventy years [1][2][3][4][5][6][7][8][9][10][11] . Some results have been much debated. Mathematical aspects are now fully understood, but physical ones want for more elaborated and robust interpretation, which we provide in details here.
In this article, we study the Coulomb potential, either restricted to a semi-infinite line, or else to a full infinite line. We will formally write the corresponding hamiltonian H = −d 2 /dx 2 + V in dimensionless units and  will represent the domain on which wavefunctions are defined, so the first case corresponds to = + * D R , while the second to = D R. When necessary, we will write  H( ) instead of H. One may note that the Schrödinger equation for D R = + * is equivalent, through a simple mapping, to the radial one for 3 = D R in 3-dimension with zero orbital momentum, L = 0.
This work lies at the frontier between physics and mathematics, because Coulomb hamiltonians + *  H( ) and H( )  , although defined on a physical basis, reveal non self-adjoint. In such a case, one usually needs to study the self-adjoint extensions K of the hamiltonian. But, in this very case, the situation is even worse, because H is not even symmetric 6,11 (that is, one can find two states ϕ and χ such that ϕ χ χ ϕ ≠ ⟨ | | ⟩ ⟨ | | ⟩ H H ). In such a situation, one must restrict the Hilbert space on which eigenstates are defined, in order to get a symmetric operator, the self-adjoint extensions K of which are well-defined. We call L this restricted Hilbert space.
When the self-adjoint extension of an operator is unique, these mathematical manipulations are transparent because the spectral theorem applies, so the action of the operator is defined unambiguously on any function of L. This is the case for almost all standard hamiltonians found in scientific literature, which are moreover generally well defined without any restriction (that is L L ( ) 2 =  ), so one does not need to care about all these mathematical subtleties.
However,  + * H ( ) and  H( ) belong to the class of operators, which admit several self-adjoint extensions. Each extension is incompatible with the other, so one must choose only one extension at a time, where to define a complete set of eigenstates. From a physical point of view, the interpretation of the operator action on a wavefunction is ambiguous, since its definition depends on the extension which is chosen. Deficiency coefficients are defined, which indicate the number of degrees of freedom, for this choice. For  +  , since the continuity of eigenfunctions at x = 0 is not guaranteed. This has been very debated and we propose an original connection process, which is founded on physical conservation laws and gives new, although compatible, results.
Altogether, we prove a new classification of the self-adjoint extensions of H( )  , excluding those which are not compatible with physical conservation laws. Accordingly, this classification maps on a space of extension parameter, which is reduced compared to that of previous classifications 15 , but the deficiency coefficient remains equal to 2. The parameter space of our classification is the product of a one-dimensional closed line by a phase similar to a gauge degree of freedom.
In what concerns the 3-dimension space, in spite of the mapping between its Schrödinger equation with that of + *  H( ), the corresponding classifications of self-adjoint extensions are different (see however Appendix in Supplementary Information), since the deficiency coefficient of H( ) 3  is zero 12 , that is  H( ) 3 is self-adjoint, when defined in L ( ) 2 3  . The present article is organized as follows: we will first focus on the D R = + * case and classify all self-adjoint extensions of  + * H( ), both for an attractive potential or a repulsive one. In particular, we define and exhibit the Dirichlet or Neumann extensions. Then, we study in details the continuation problem in the = D R case. Next, we study physical applications of D R 3 = , D R = and = + * D R cases. In a fifth part, we examine the spectral theorem. In the next one, we exhibit the extension parameter spaces. Finally, we will review the highlights of this work on the Coulomb problem. Some notations and terms are given afterwards in Table 1.

Self-adjoint extensions in the
Here, Γ is the gamma function, M the regular confluent hypergeometric function and U the logarithmic confluent hypergeometric function 16 . Both F η and G η are continuous and bounded, see ref. 11 for asymptotic behavior and other properties. The case e = 0 extends this case when the potential is attractive, see section 'Solutions of zero energies' . For e < 0 (bound states of negative energy), the solutions of (1) read and Here, ψ dig is the digamma function. One finds For λ < 0 so ′ < qq 0 and the potential is attractive, the spectrum of any self-adjoint extension will reveal infinite and discrete. As we shall find, all solutions corresponding to η = −n, with n ∈ *  , belong to the same extension and read f u u L u n ( ) e (2 ) 2 , the standard Rydberg solution, with L n the Laguerre polynomial. They obey Dirichlet condition f −n (0) = 0. On the other hand, for η − ∉ *  , f η (0) ≠ 0, see ref. 11 for more details. We will call Rydberg states, those following η = −n with ∈ * n  , and non Rydberg states the others. Note that the definition of g η must be changed into since, in that very case η = −n, u e −u M(1 −n, 2, 2u) is proportional to u e −u U(1 −n, 2, 2u). For λ > 0 so qq 0 ′ > and the potential is repulsive, the spectrum of any self-adjoint extension will reveal discrete, with a unique bound state of strictly negative energy, but in a specific case that we will explain further on. existence of self-adjoint extension. The existence of self-adjoint extensions for the Coulomb potential has been fully established in several references [12][13][14][15] and needs not to be discussed here again. Indeed, the deficiency coefficients m ± are found equal to 1, although not explicitly calculated in ref. 13 . We will construct all self-adjoint extensions as follows.
We will write ω + *  H ( ) the self-adjoint extensions of + *  H( ), parametrized by ω, a symbolic index, the meaning of which will be explained later on. The boundary triples theorem implies that Description of a self-adjoint extension. In this part, we consider the attractive case. Let e ω < 0 be in the is an eigenfunction of ω + *  H ( ) and belongs to ω D . There is such e ω , otherwise the spectrum of ω + *  H ( ) would be included in +  , which case we exclude later on.
, so does k ϕ ω by definition, while η ω g diverges, letting 0 The other factor reads then e k μ = θ ω ω , a constant phase factor which can be fixed arbitrarily.
One observes that not all functions ϕ k belong to ω D , because the scalar product η η ⟨ | ⟩ F F 1 2 ,which we calculate in Appendix (see Supplementary Information), with arbitrary momenta k i = λ/(2η i ), is not always zero. Let us establish this result: we note γ E the Euler constant and define function g b : then, the scalar products reads (this expression is valid when η 1 → η 2 and the limit is 1); therefore an operator admitting all such eigenfunctions would not be symmetric 6,11 .
. We will prove that the set of bound states of  ω + * H ( ) corresponds to functions generated by . We study the zeros of g b (η) −g b (η ω ) further on. (5) implies that any function ϕ k orthogonal to k ϕ ω obeys g b (η) = g b (η ω ) so all functions in ω B are either proportional or orthogonal to each other. By construction, ω B is maximal, because any function orthogonal to ϕ ω k belongs to it; there cannot be any other eigenfunction in D ω corresponding to a bound state, so φ ∈ ∈ ⊆ ω ω ω However, we cannot claim yet that this inclusion is an equality, because the scalar product of a bound state with a free one could be different from zero.
Let us discard this possibility and thus prove e Let us examine free states. Let F ω be the set of functions φ e = Ψ k , with e > 0 and momentum = k e, such that using (2). Let us define g f :  2 so one must choose β = 0 k 2 and gets k 2 ζ = ∞. (8) extends in this case, since g b (η ω ) →∞ when η ω → −n with n  ∈ * . (8) implies that Ψ k is orthogonal to any function k 1 ϕ ∈ ω B as soon as it is orthogonal to ϕ ω k . All free eigenfunctions of ω + *  H ( ) must belong to F ω , so they respect (8); thus, they are all orthogonal to any ( ) ; this ends our demonstration.
■ Conversely, all elements in ω F are eigenfunctions of ω + * H ( )  . In that purpose, let us establish the generalized orthonormality of all elements in F ω . Let e 1 φ and φ e 2 be in ω F , with e 1 ≠ e 2 . The scalar products ⟨ | ⟩ (9) We have proved that all bound eigenfunctions of ω + *  H ( ) are in B ω while all free ones are in ω F . Therefore, we Let (e 1 , e 2 ) be such that ψ φ = e 1 and e 2 ϕ φ = (depending on whether ψ belongs to the free or the bound spectrum, either e 1 ∈ +  or S ∈ ω e 1 , and idem for ϕ with e 2 ). One writes then The last equality is proved by discussing whether e 1 ≠ e 2 , so ψ φ φ = ⊥ e e 1 2 = ϕ, following all previous discussions, or else e 1 = e 2 .
■ Let us prove that ∼ ω H is maximal ad absurdum. Since it is symmetric, it admits a self-adjoint extension K, which is defined on C D ⋃ ω , where C is some non empty space, by hypothesis. Let us write C K the restriction of K on C.
be a basis of D ω , and ψ ∈ J j { , } j a basis of C. One writes Eventually, we have established that C K is symmetric. From standard algebra 17 , there exists at least an eigenfunction 0 C φ ∈ , and its eigenvalue e 0 is real. Applying the boundary triples theorem, the function φ 0 is a solution of the differential equation H is symmetric and maximal, that is, it is a self-adjoint extension. Furthermore, ∼ ω H is simple. Concerning bound states, this results from the elimination of functions proportional to g η . Concerning free states, it follows (8). Now, let φ e be any eigenfunction included in the domain of H ( )  ω + * . This domain includes φ ω e , so φ e must be either orthogonal to e φ ω or have eigenvalue e ω . In the first case, φ e belongs to D ω . In the second case, it is proportional to φ ω e (still resulting from the elimination of functions proportional to g η ω ). This proves that the domain of which is therefore completely determinate. ■ Classification in the attractive case. Set S ω contains the zeros of , which we represent for several values of ω in Fig. 1. To characterize each set B ω , we follow the results in ref. 12 and define ϕ ϕ λ ωϕ . This condition differs from the more usual one . Another possible characterization is given in ref. 13 . For a given number  ω ∈ , we define η ω to be any solution of g b (η) = −ω (one can chose the highest η, as we will prove further on that this set has a maximum).
Let us prove = ∼ ω ω B B . First, we will show that two functions in ∼ ω B are either proportional or orthogonal. One finds, for non Rydberg Thus, it comes that all elements in ∼ ω B verify ω = −g b (η), so, using (5), the proposition is proved, except for Rydberg states such that η − ∈ *  . For these, the last limit gives ∞. However, these eigenfunctions are well known and indeed orthogonal (see section 'Dirichlet solutions'), so the result extends to this case immediately. Conversely, any index η corresponding to ϕ ∈ ω www.nature.com/scientificreports www.nature.com/scientificreports/ , which corresponds to Rydberg states, one gets This proves exactly that φ e belongs to ∼ ω F . Reversely, let us show that any element e F φ ∈ ∼ ω belongs to ω F . Using (2), one writes φ e = α k F η + β k G η . Then, from the definition of ∼ ω F , one gets where ∈ ω e S , with ω = −g b (n), as explained further on. The curves seem to form pairs corresponding to (n, n + 1), in particular, one could believe that each pair intersects on the η-axis (abscissa), but this is wrong, except for (n, n + 1) = ( −2, −1) which correspond to the same Rydberg set S ∞ . All the other intersections are only close to zero, so that, indeed, The topology of the parameter space is studied in section 'Structure for = + * D R in the repulsive case' .
Classification in the repulsive case. We now consider the repulsive case. The physical situation is very different to the previous one, for instance, one observes that there is no Rydberg state, that is no eigenfunction obeying φ e (0) = 0, however many steps of the calculations are similar, so we will only point out the differences.
Keeping the definition of g b with η > 0, one finds (4) with the opposite sign. Then, (5) has no solutions, but the existence of a bound state will hold in the repulsive case, which means that it is a unique bound state. This is true for all ω, see for instance In the free spectrum, a similar sign difference occurs: where the definition of g f is unchanged. The scalar product expression ⟨ | ⟩ G F 1 2 η η is unchanged but mind that its real sign is also changed after that of η. Eventually, the demonstration that all functions in F ω respect ⟨ | ⟩ Φ Φ e e 1 2 =0 holds, and, consequently, the determination of  (note the sign difference). With this new definition, index ω(η) has the same expression than in the attractive case. The demonstration is straight forward for the bound states; for free ones, one finds the expression obtained for F η are unchanged, but mind that the real sign is changed after that of η. Eventually, there is no sign change for index ω(η) in all cases. We plot this function in Fig. 3 and observe another major difference: it maps + * . As a consequence, ω is bounded from above. The particular value ω = 2γ E brings a very peculiar situation and must be studied elsewhere.  We suppose ad absurdum that the spectrum is included in  + . We consider two eigenfunctions k 1 Ψ and Ψ k 2 . We can choose momenta k 1 ≠ k 2 , otherwise ω + *  H ( ) would only act on functions F 1 η and G 1 η , which norm are infinite; no integrable function could be constructed and this extension would not be physical. The same argument holds if there is only one eigenfunction.
Using (2) and (9), one gets . So, either both β k are zero, or both are different from zero. In the first case, this property extends to all free states, which are therefore all Rydberg free ones; thus,  ω + * H ( ) can extend on all standard Rydberg solutions, including bound ones, which contradicts our hypothesis.
The remaining case leads to β ≠ 0 k ∀i = 1, 2, which means that momenta k i correspond to non Rydberg states. Multiplying by One can assume k i β real, without loss of generality. Let us define the real and purely imaginary parts of eigen- (1) cf. section 'Classification in the attractive case' , in which this item holds both for repulsive or attractive case) and is contradictory, unless Ψ = 0 k i i . Altogether, this implies that k i ζ is real ∀i = 1, 2. Eventually, one gets is a real constant, which we write  ω. From the classifications above, one observes that all functions Ψ k are eigenfunctions of ω + * ∼  H ( ), which proves an extension of ω + *  H ( ) and therefore contains bound eigenstates. We have reached a contradiction. In all cases, we have shown that there is at least one bound state. ■ In the repulsive case, it is the only one. In the attractive case, they are infinitely many; let us study that of highest energy.

Maximum of ω
S S . For the attractive case, η < 0, so one is interested in the maximal value η ω max corresponding to the maximum of ω S . There exists such a maximum, this is visible on Fig. 4, which is a close focus of Fig. 1  , are flat, while the sign of the slope of the other curves is positive for [n] + even and negative for [n] + odd. Therefore, the maximal η < 0, related to an energy S ∈ ω e is the first zero from the right. The only difficult case would be that of the flat curves; these however correspond to the standard Rydberg solutions ∈ − *  n , the maximal value of which is indeed −1. ■ www.nature.com/scientificreports www.nature.com/scientificreports/

Infinite energy state.
In what precedes we exclude value η = 0. Limit η → 0 of eigenfunctions corresponding to bound states reads f 0 (u) = e −u , but using rescaled ϕ k (x) = f η (kx) and renormalizing by D η , one gets for all  x ∈ + * but not for x = 0 ( ± = + in the repulsive case, ± = − in the attractive one). The so called infinite energy −∞ would correspond to a singular distribution with {0} support. Looking for such a solution, one substitutes a n n n 0 0 ( ) ϕ δ = ∑ = ∞ in (1). In the η = 0 limit, all coefficients a n are found zero, which definitely discards such solution.
The limit η → 0 of eigenfunctions corresponding to free states reads . Using rescaled Φ 0 (x) = α 0 F 0 (kx) + β 0 G 0 (kx) (but no renormalization is needed, since the limit of C η is 1), one gets The first is zero so the limit of eigenfunctions when e → +∞ is the constant function Ψ(x) = 1. Eventually, we should compare these limits to the solutions of (1), where η is replaced by 0. They read φ ∞ (x) = ax + b, but a ≠ 0 gives divergent non physical functions, so, up to an arbitrary phase, one finds b = 1, which is the e = +∞ limit.
■ Incidentally, we are in position to discuss the long-standing claim 1 of a solution | ⟩ φ −∞ with energy −∞: we see that this solution does not exist, putting an end to this old story.

Discussion of some particular cases. Dirichlet solutions.
We consider the attractive case. When ω → ±∞, one gets the Dirichlet condition φ e (0) = 0. For bound states, this can be shown by examining the limit ϕ k (0 + ) = D η /Γ(1 + η), which we give in section 'Classification in the attractive case' and which is also valid in the repulsive case. For free states, this follows, firstly, from the fact that ζ k →∞, as shown in the same section, which implies β k → 0 so φ e ∝ F η , secondly from the limit F η (0 + ) = 0, still proved in that section. Then, the corresponding values of ∞ S are exactly −λ 2 /(4n 2 ), for all  ∈ * n , which is the standard Rydberg spectrum (in dimensionless unit). Moreover, the function η ↦ ω(η) = −g b (η) respects ω(η + 1) = ω(η) for all η = −n with n ∈ *  and only for these values.
In the repulsive case, one must recall that there is no Rydberg state, even in the limit ω → −∞, so this discussion is not relevant for this case.
Neumann solutions. The case ω = 0 will be called the Neumann solutions, because the finite part 18 where the essential divergent function ln k x ( ) is left aside, is exactly zero at x = 0. These functions are very close to the anomalous solutions of ref. 11 , however those do not belong to a single extension: they are proportional to G η in the free spectrum and correspond to ζ k = 0. We have shown previously that ζ , which zeros are not exactly periodic, on the contrary, each one belongs to a different extension. The very small difference between any such anomalous state and the closest Neumann one explains the small violation of orthogonality that www.nature.com/scientificreports www.nature.com/scientificreports/ was calculated 11 (when η →∞, the difference between Neumann and anomalous solutions tends to zero, as well as the scalar products between anomalous solutions).
As is well understood now, the correct choice is to consider functions in B 0 . On the contrary, it is not physical to consider any two anomalous states together 19 , because they do not belong to the same self-adjoint extension.
Physical interpretation of ω. We did not give any physical interpretation of ω yet. It is the limit of the ratio φ between the derivative of the wavefunction and the wavefunction itself when x → 0, after subtracting the divergent term ln x ( ) λ ± ( ± = + when the potential is attractive, ± = − when it is repulsive). This ratio relates to the initial condition that one fixes at x = 0 when solving Schrödinger equation Hφ = Eφ. An infinite ratio corresponds to choosing Dirichlet conditions, a zero ratio to Neumann ones, and any finite value in-between means fixing an intermediate condition, that mixes φ and φ′.
Solutions of zero energy. Writing  + , we have indicated that 0 must be included in the free spectrum. This is worth giving some details.
The solutions of (1) for e = 0 and λ < 0 read where J 1 and Y 1 are Bessel functions of, respectively, the first and second kind. That for λ > 0 read where I 1 and K 1 are modified Bessel functions of, respectively, the first and second kind.
We have extended the notations we use for free states, because these solutions are indeed the limit of those ones, j ∝ F −∞ , y ∝ G −∞ , ι ∝ F ∞ and κ ∝ G ∞ . The attractive case η < 0 brings nothing special, solutions j and y have the standard properties of the eigenfunctions corresponding to free states; one may say that this limit is regular.
On the contrary, the repulsive case η > 0 is extraordinary. Instead of heavy mathematical considerations, let us explain the situation by hand. When one looks at the curves of functions x ↦ F η (x) and x ↦ G η (x), for increasing η, one observes that there are two regions x ∈ [0, x η ] and ∈ ∞ η x x [ , [, where x η is a separating parameter which we do not care to define properly here. In region [0, x η ], F η resembles eigenfunction g η (in other words, it grows considerably, as if it were diverging) and G η resembles eigenfunction f η (in other words, it becomes exponentially small). But, as these functions reach x η , they rapidly change shape and behave like those corresponding to standard free states (bounded and oscillating).
This peculiar behavior, resembling bound states in a first region then free ones afterwards, reaches its climax when η →∞, where x η →∞: indeed, solution ι is diverging, while ⋂ κ ∈ + * + *   L L ( ) ( ) 1 2 . In this very case, F ∞ must be discarded and the scalar products between G ∞ and eigenfunctions f η reads and is non zero, as observed on Fig. 5. The orthogonal combination of eigenfunctions F η and G η is governed by ratio Our guess is that, in the repulsive case, a singular contribution δ(E) appears in the density of states, contrary to the situation of the attractive case. This belief is founded by the existence of a bound eigenstate, to which corresponds an integrable function, with eigenvalue e = 0.
Eventually, one is interested in the corresponding value of index ω(∞ ). One finds ω(∞ ) = 2γ E . Moreover, the limit of regular bound eigenfunction ϕ k , when η →∞, does not exist, so there is exactly one bound eigenstate of energy e = 0 corresponding to ω(∞ ) = 2γ E , which is exactly that proportional to κ.

the real line problem
We discuss here the attractive case for D R = . We should point out that there was no need to use of any physical constraint in the previous cases, except when we have discarded the hypothesis of a unique energy e > 0 or that with only two energies e 1 > e 2 > 0. On the contrary, our determination of self-adjoint extensions for = D R is much more involved with physical laws. Our aim is to classify self-adjoint extensions that are compatible with physical constraints.
We note φ e eigenfunctions defined on , e φ > their restriction on  + * and e φ < that on − * The continuity of all functions φ e as well as their derivatives is easily verified for all x ≠ 0 from (1) and (12). The only difficulty lies at x = 0. Let us define the self-adjoint extensions of  H( ).
D R, has already been done 12 but no effort has been made yet to interpret these from a physical point of view. We want to select, among all extensions, only those, the eigenfunctions of which describe physical states.
Usually, authors impose continuous boundary conditions for all wavefunctions and their derivative 20-23 but these conditions reveal often too restrictive and other boundary conditions have been suggested 24,25 . So, we choose weaker and universal constraints, which are compatible with any of these conditions and fit with all experimental observations: the density of probability cannot vary discontinuously, therefore ρ = |φ| 2 must be continuous. ρ also obeys the conservation of probability law (14). This implies eventually that dj/dx be defined at all ∈  x . We introduce boundary condition C( ) θ : lim ( ( ) e ( )) 0; we will find that physical states do respect conditions C( ) θ . We will therefore construct self-adjoint extensions, with these boundary conditions. More precisely, we will show that there are at maximum two values θ 1 and θ 2 , such that eigenfunctions obey C θ ( ) i , with i = 1, 2. As for = + * D  , we will admit the existence of self-adjoint extensions and construct them as maximal symmetric operators. We write them H ( )  ϖ , where ϖ is a symbolic parameter, the meaning of which we will clarify further on. We write B ϖ the set of eigenfunctions in the bound spectrum, F ϖ that of eigenfunctions in the free spectrum, ⋃ = ϖ ϖ ϖ D B F and ϖ S the corresponding bound spectrum.
continuity of probability. Let φ e be an eigenfunction of self-adjoint extension  H ( ) ϖ . We will first use the continuity of ρ(x) = |φ e (x)| 2 .
One put apart the case when φ e (0 + ) = 0 or φ e (0 − ) = 0. Indeed, the only eigenfunctions which have such limit are the Rydberg ones. In such case, the continuity of ρ gives φ e (0 + ) = φ e (0 − ) = 0 and φ e is eventually continuous on .
We recall that non Rydberg functions do not cancel at x = 0. For such functions, the continuity of ρ implies  ρ(x, t) = |ψ(x, t)| 2 represents a density of probability and must be continuous with respect to x at all times. One finds x The continuity of x ↦ ρ(x, t), valid for all α, β, ζ and t, implies that of ( ) e e 1 2 R φ φ and I( ) e e 1 2 φ φ ; so one gets www.nature.com/scientificreports www.nature.com/scientificreports/ Thus, φ e is said to be θ ϖ -symmetrical, where θ-symmetry is also written  θ ( ) and defined by We assume now that there are two or more non Rydberg eigenfunctions in the bound spectrum, let us write them . Their scalar product reads When they are not proportional, k 1 ϕ and k 2 ϕ can be eigenfunctions of the same ϖ  H ( ) only if ϕ > k 1 and ϕ > k 2 , their restriction on + *  , are orthogonal each other. From part 'Self-adjoint extensions in the  + * case' , we get ω(η 1 ) = ω(η 2 ). Let us call ω ϖ this constant. Altogether, we have established the existence of parameters ω ϖ and θ ϖ , such that all non Rydberg eigenfunctions φ e , in the bound spectrum, obey ( ) ■ We will examine now the situation, where there is also a Rydberg eigenstate in the domain of ϖ  H ( ), and prove that this Rydberg states has the opposite symmetry to the non Rydberg one, in the following sense.
One can expand φ e 2 into a θ ϖ -symmetrical and a θ ϖ + π-symmetrical parts, e e e 2 2 , as demonstrated in Appendix (see Supplementary Information). Then, one finds φ = (the second term is zero by symmetry, cf. appendix) so e2 . This proves that 0 ) . ■ Let us examine now free states. We consider a non Rydberg eigenfunction φ e with e > 0. We will find that φ e obeys ( ) θ ϖ and that φ = Ψ ∈ ω > > ϖ e k F , but the demonstration is more involved and relies also on the current continuity. To begin with, following (2), one can write φ α β = + η η conservation of current. We still consider H ( ) ϖ  and two independent eigenfunctions e 1 φ and e 2 φ in the domain of H ( )  ϖ and calculate the current associated to the mixed state | ⟩ ψ t ( ) defined in section 'Continuity of probability' . It becomes, after some calculation, where j 1 and j 2 are constant. The conservation of probability law applies independently on the sinus and cosine terms, so it eventually reads Let us continue the proof concerning non Rydberg free states, which was sketched in the previous section. We choose | ⟩ φ e 1 a non Rydberg bound state and | ⟩ e 2 φ a non Rydberg free one (we assume their existence; one observes that they are independent). So We have proved that all non Rydberg obey  θ ϖ ( ), although we have not determined the set to which belongs φ > e when e > 0. ■ Before taking advantage of this result, let us conclude on the current of probability. For e 1 φ and e 2 φ non Rydberg, j is odd and the limit of j(x)/x when x → 0 becomes This calculation is valid in both attractive or repulsive cases. For Rydberg states, the same three limits give zero (the case e 1 < 0 and e 2 < 0 extends exactly; the case e 1 > 0 and e 2 < 0 also extends, because the wrong normalisation vanishes in the zero limit; the case e 1 > 0 and e 2 > 0 is apart). Altogether, (14) is respected at all cases. ■ Self-adjoint extensions. We still consider self-adjoint extension ϖ  H ( ). We assume first that there exists a non Rydberg bound eigenfunction e 1 φ . We have shown that there are two parameters ω ϖ and θ ϖ such that it reads . In other words, φ e 1 is a θ-symmetrical eigen- Let us achieve the proof concerning non Rydberg free states; so we assume there is such an eigenfunction e 2 φ , with e 2 > 0. We know φ e 2 obeys ( ) ■ Let us eventually consider any Rydberg eigenfunction φ e 3 of the same operator ϖ H ( )  . We know that this function is θ ϖ + π-symmetrical. Reversely, all θ ϖ + π-symmetrical Rydberg eigenfunctions are orthogonal to any (here non Rydberg) θ ϖ -symmetrical function (cf. Appendix in Supplementary Information), so H ( ) ϖ  can be extended into a symmetric operator (one can choose a trivial action , acting on all eigenstates φ e obeying R( ) and on all eigenstates φ e obeying  θ π + . This extension is maximal by construction and reads where π θ is the projector on θ-symmetrical functions. Since H ( )  ϖ is maximal by definition, it is equal to this extension, and our classification is complete, within the assumption that there are non Rydberg eigenfunctions (2020) 10:7280 | https://doi.org/10.1038/s41598-020-62144-2 www.nature.com/scientificreports www.nature.com/scientificreports/ (one at least in the bound spectrum, one at least in the free spectrum). In such case, we define ϖ = (ω ϖ , θ ϖ ) and our results prove that and ω →∞, because one can expand any eigenfunction as the sum of its θ-symmetrical and θ + π-symmetrical parts. existence of a non Rydberg bound state. We consider a self-adjoint extension  ϖ H ( ). We assume there is at least a non Rydberg eigenstate, otherwise ϖ =∞, which situation exists and has been studied above.
We can rapidly exclude the situation, where there are no non Rydberg free eigenstates. Indeed, one knows that all non Rydberg bound states' energies belong to some set ω ϖ S and that their eigenfunctions obey  θ ϖ ( ) , with θ π ∈ ϖ [0, 2 [; so they belong to and there are indeed non Rydberg free eigenstates. On the contrary, the situation with non Rydberg free eigenstates and no bound ones can not be discarded so easily. The demonstration is close to that of section 'Existence of a bound state' .
We first study the case of a unique non Rydberg free eigenstate | ⟩ φ e 1 . There must be a Rydberg free one | ⟩ gives β α β α Since the eigenspace associated to e 2 is of dimension 1 (because of the Dirichlet condition, since e 2 φ is Rydberg), one deduces that φ e 2 obeys ( ) θ π + ϖ . Thus, from the relation above, φ e 1 obeys  θ ϖ ( ). Therefore, α β α β ζ = ≡ ω θ ϖ  , which admits non Rydberg bound eigenstates. This is indeed contradictory and the case can be discarded. ■ Let us assume now there are two independent non Rydberg free states | ⟩ φ e 1 and e 2 | ⟩ φ .
(2) reads We have already found that there exist θ ϖ such that We use the continuity of j the same way as before, constructing a state φ α φ β φ = +   Adding (15) and (16) proves that ζ ζ + +

( , )
k k 1 2 obeys (10); thus, ζ + k i are real and obey (11). Subtracting (15) and (16)  are real and obey (11). Then, (16) proves that the same ω can be associated to all functions k 1 φ > , k 1 φ < , φ > k 2 and k 2 φ < . Therefore, notwithstanding we did not establish ζ ζ = + − i i , one can introduce any bound state associated to ϕ η with g b (η) = −ω, and extend the action of ϖ H ( )  on these bound states, keeping the operator symmetric. This is contradictory, so the result is proved. ■ (2020) 10 the dirichlet case in one dimension. We focus on the case ϖ =∞ and study eigenfunctions φ e . Both attractive and repulsive case can be considered, but we will focus on the first one.
Let us consider the bound spectrum. From what precedes, μ in section 'Description of a self-adjoint extension' is entirely free. Therefore, φ φ is a basis of the eigenspace E e corresponding to energy e. This is an exceptional violation of the general result, which asserts that an energy in the bound spectrum is non degenerated in one dimensional systems. Here, the eigenspace E e has dimension 2. However, examining the standard demonstration 26 , on observes that it is based on a Wronskian theorem, which can not apply here. Another

physical applications
We study different possible extensions of this work to real physical situations.
the hydrogenate case in three dimension. Let us focus on the case = 3 D R , using the mapping Φ(r) = φ(r)/r, where φ is the one-dimensional solution and Φ the radial part of the three-dimensional wavefunction. We will only consider the attractive case here.
Let us connect our parametrization ω with that of ref. 14 , which parameter is written α. We will show the connection for bound states only, but this can be done for all states. The first order expansion of any state φ e with e < 0 reads this expression holds both in attractive and repulsive cases. ω can be expressed in terms of b/a, which reads In ref. 14 , where λ reads γ, one finds parameters φ 0 = a and φ 1 = b, so one gets As it is well known 27 , for L > 0, the solutions of the Schrödinger equation which do not cancel at r = 0 do not belong to L ( ) 2 3  and must therefore be discarded. On the contrary, that, corresponding to the case L = 0, belong to  L ( ) 2 3 (all g η solutions, which diverge at r →∞ are excluded from this discussion). This is the reason why the L ≠ 0 subspaces appearing in (2.1.13) of ref. 14 have no parametrization, contrary to the L = 0 one. This helps us interpreting what these authors mean by ≪H γ,α,y describes the Coulomb interaction plus an additional point interaction ≫ : the eigenfunctions for α <∞ are divergent eigenfunctions and not physical, although they belong to L ( ) 2 3  , so they do not describe the physical Coulomb interaction. Most authors have similarly assumed that the only admissible Coulomb bound states are the Rydberg ones, given by the Laguerre polynomial r n 3/2 with a specific normalization (assuming that the spherical function reads π 1/ 4 for kinetic momentum L = 0). This solution exactly corresponds to the ω =∞ Dirichlet case, which is also the α =∞ one.
Actually, no fundamental principle of quantum mechanics justifies discarding solutions that diverge for r → 0, since the probability ∫|Φ(r)| 2 r 2 dr is finite (in the basic meaning "not infinite"). However, experimental evidences, from the original Rydberg spectrum, are in excellent agreement with this assumption. We find that experimental data 28  , for several (m, n) couples, as determined from these data, with that calculated from the exact values of ω S . Actually, (m, n) = (5, 3) gives the highest (best) limit of possible values for ω.
Based on these physical grounds, we will follow the common choice and, dealing with the case  = D 3 , discard all divergent wavefunctions, therefore reducing the parameter range to ω =∞, the self-adjoint extension corresponding to Dirichlet solutions. We can justify this choice, from a mathematical point of view, by reminding that the deficiency coefficient of  H( ) 3 is zero. We will discuss this point further on.
Explicit spectra for a semi-infinite line. The calculated spectra ω + *  S ( ) vary significantly, for different values of ω. We show three of them in Fig. 6, corresponding to ω 1 =∞ (Rydberg spectrum), ω 2 = ω( −1/4) ≈ 2.3 and ω 3 = ω( −1/2) ≈ −0.27 (close to the Neumann case). As already pointed out, in any one-dimensional system endeqnarray* For ω =∞ and λ < 0 (attractive case), this formula is equivalent to Eq. 19.171, in ref. 26 with a different normalization (we preferred to use k parameter, rather than E). We have checked this formula numerically on several examples, x e x 2  − ,  x x e x 2 − , etc. One can, in particular, expand a function ϕ k , with ω ω ≠ λ ( )  On the left, we show the absolute values, on the right, we normalize energies so that the lowest energy is −1. The variation of E n+1 −E n , when n is increased, is steeper for ω 3 .
 H ( ) acts on ψ e as  H ( ) ω + * with ω = ω(e), the index of energy e. However, + *  H ( ) is not a good operator, because it does not correspond to the same self-adjoint extension, for each state.
Technically, the last result can be understood as follows: d/dx does not commute with ∫dk in the former development. Indeed, when the derivation is performed inside the integral, it produces a factor η ∝ 1/k which makes it improper.
This analysis is common with that, which can be made for H = −d 2 /dx 2 ; the divergence of the Coulomb potential is not entirely responsible of the loss of self-adjointness.
Spectral theorem in . The spectral theorem in  can be formulated after that in  + * . Each θ-symmetrical and θ + π-symmetrical part of any function can be expanded separately. Considering ϖ  H ( ), with ϖ = (ω, θ), any function  L ( ) 2 φ ∈ expands into φ = φ θ + φ θ+π . It applies also in the particular case ϖ =∞, choosing any arbitrary θ. In this case, one can also write f = f > + f < (where f > extends in − *  as zero and f < extends in + *  as zero). f > expands in   F . This is the right place to observe that μ, defined in in section 'Description of a self-adjoint extension' , is not determinate in this particular case. One can indeed choose μ = 0 (i.e. f = f > ) or μ =∞ (i.e. f = f < ). We discuss this supplementary degree of freedom further.

Topological classification of the extension parameter space
Structure for * D R = + in the repulsive case. The structure of the order parameter seems to be equivalent to the interval γ −∞ [ , 2 ] E in the repulsive case, which is topologically equivalent to interval [0,1] . This is notwithstanding the special case  H ( ) 2 E γ + * , which we found for the zero energy. This case corresponds to ω = 2γ E , but, what should now be pointed out is that the regular limit ω → 2γ E , which can be constructed, using g η , does not exist. One finds indeed that eigenfunction g η tends to a singular distribution with {0} support. Looking for and e 3 ɛ from bottom to top) versus ɛ ln (1/ ) in dimensionless y-scale. The asymptotic limit is indicated by an arrow on the right, for each curve and by the horizontal straight lines. Figure 9. Representation of the order parameter space in the repulsive case (left) or attractive case (right). Left are represented the ω = 2γ E closing circle, the ω axis (which is supposed to vary from −∞ to 2γ E ) and the gauge parameter θ. Right is represented the ω = ±∞ point at the strangling point and a θ-circle is pointed out: all orthogonal lines to this circle vary with ω.