Abstract
We discuss some mathematical aspects of two particular deformed versions of the Dirac Hamiltonian for graphene close to the Dirac points, one involving \(\mathscr {D}\)-pseudo bosons and the other supersymmetric quantum mechanics. In particular, in connection with \(\mathscr {D}\)-pseudo bosons, we show how biorthogonal sets arise, and we discuss when these sets are bases for the Hilbert space where the model is defined, and when they are not. For the SUSY extension of the model we show how this can be achieved and which results can be obtained.
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- 1.
We remind that, for a non o.n. set, being complete is less than being a basis. In the mathematical and physical literature it is easy to find examples of sets which are complete in a certain Hilbert space, but which are not bases. Examples are given in [6].
- 2.
In order to connect the intertwining technique with the supersymmetric quantum mechanics introduced by Witten [14], we should define the standard SUSY algebra
where \(\left[ .\,,\,.\right] \) and \(\left\{ .\,,\,.\right\} \) represent the commutator and the anticommutator, in the following way
.
- 3.
Note that (36) defining the superpotential \(\alpha (X)\) can be expressed as a partial differential equation in the original variables (x, y):
$$ x^2+2\,i\,x\,y-y^2=\sqrt{2}\,\partial _x\,\alpha (x,y)-\sqrt{2}\,i\,\partial _y\,\alpha (x,y)+2\,\alpha (x,y)^2+4\gamma . $$.
- 4.
Examples are: \(\mathscr {P}_{1}(x) = 1\), \(\mathscr {P}_{2}(x) = 2x\), \(\mathscr {P}_{3}(x) = 4(1+x^{2})\), \(\mathscr {P}_{4}(x) = 4(5x+2x^{3})\), \(\mathscr {P}_{5}(x) = 8(4+9x^{2}+2x^{4})\), and so on.
- 5.
The spectral problem \(\left( -\frac{1}{2}\frac{d^{2}}{dx^{2}}+\frac{x^{2}}{2}+\frac{1}{x^{2}}\right) \phi ^{(n)}=\left( 2n+\frac{5}{2}\right) \phi ^{(n)}\) has solutions \(\phi ^{(n)}=\left( \frac{2\varGamma (n+1)}{\varGamma \left( \frac{5}{2}+n\right) }\right) ^{\frac{1}{2}}x^{2}e^{-\frac{x^{2}}{2}}L_{n}^{\frac{3}{2}}(x^{2})\), where \(L_{n}^{a}(x)\) are the generalized Laguerre polynomials.
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Acknowledgments
F.B. acknowledges partial support by the University of Palermo and by G.N.F.N. The authors also wish to thank Prof. N. Hatano for many useful discussions.
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Appendices
Appendix 1: \(\mathscr {D}\) -pseudo-bosons
We briefly review here a few facts and definitions about \(\mathscr {D}\)-PBs. More details can be found in [5, 6].
Let \(\mathscr {H}\) be a given Hilbert space with scalar product \(\left\langle .,.\right\rangle \) and related norm \(\Vert .\Vert \). Let further a and b be two operators on \(\mathscr {H}\), with domains D(a) and D(b) respectively, \(a^\dagger \) and \(b^\dagger \) their adjoints, and let \(\mathscr {D}\) be a dense subspace of \(\mathscr {H}\) such that \(a^\sharp \mathscr {D}\subseteq \mathscr {D}\) and \(b^\sharp \mathscr {D}\subseteq \mathscr {D}\), where \(x^\sharp \) is x or \(x^\dagger \). Of course, \(\mathscr {D}\subseteq D(a^\sharp )\) and \(\mathscr {D}\subseteq D(b^\sharp )\).
Definition 1
The operators (a, b) are called \(\mathscr {D}\)-pseudo bosonic (\(\mathscr {D}\)-pb) if, for all \(f\in \mathscr {D}\), we have
Our working assumptions are the following:
Assumption \(\mathscr {D}\)-pb 1. There exists a non-zero \(\varphi _{ 0}\in \mathscr {D}\) such that \(a\,\varphi _{ 0}=0\).
Assumption \(\mathscr {D}\)-pb 2. There exists a non-zero \(\varPsi _{ 0}\in \mathscr {D}\) such that \(b^\dagger \,\varPsi _{ 0}=0\).
Then, if (a, b) satisfy Definition 1, it is obvious that \(\varphi _0\in D^\infty (b):=\cap _{k\ge 0}D(b^k)\) and that \(\varPsi _0\in D^\infty (a^\dagger )\), so that the vectors
\(n\ge 0\), can be defined and they all belong to \(\mathscr {D}\) and, as a consequence, to the domains of \(a^\sharp \), \(b^\sharp \) and \(N^\sharp \), where \(N=ba\). We further introduce \({\mathscr {F}}_\varPsi =\{\varPsi _{ n}, \,n\ge 0\}\) and \({\mathscr {F}}_\varphi =\{\varphi _{ n}, \,n\ge 0\}\).
It is now simple to deduce the following lowering and raising relations:
as well as the following eigenvalue equations: \(N\varphi _n=n\varphi _n\) and \(N^\dagger \varPsi _n=n\varPsi _n\), \(n\ge 0\). In particular, as a consequence of these last equations, choosing the normalization of \(\varphi _0\) and \(\varPsi _0\) in such a way \(\left\langle \varphi _0,\varPsi _0\right\rangle =1\), we deduce that
for all \(n, m\ge 0\). The third assumption we introduced in [5] is the following:
Assumption \(\mathscr {D}\)-pb 3. \({\mathscr {F}}_\varphi \) is a basis for \(\mathscr {H}\).
This is equivalent to the request that \({\mathscr {F}}_\varPsi \) is a basis for \(\mathscr {H}\) [6]. In particular, if \({\mathscr {F}}_\varphi \) and \({\mathscr {F}}_\varPsi \) are Riesz bases for \(\mathscr {H}\), we have called our \(\mathscr {D}\)-PBs regular. Since this assumption is rarely satisfied in concrete models, we introduced in [5] a weaker version of Assumption \(\mathscr {D}\)-pb 3: for that, let \({\mathscr {G}}\) be a suitable dense subspace of \(\mathscr {H}\). Two biorthogonal sets \({\mathscr {F}}_\eta =\{\eta _n\in {\mathscr {G}},\,g\ge 0\}\) and \({\mathscr {F}}_\varPhi =\{\varPhi _n\in {\mathscr {G}},\,g\ge 0\}\) have been called \({\mathscr {G}}\)-quasi bases if, for all \(f, g\in {\mathscr {G}}\), the following holds:
Is is clear that, while Assumption \(\mathscr {D}\)-pb 3 implies (A.71), the reverse is false. However, if \({\mathscr {F}}_\eta \) and \({\mathscr {F}}_\varPhi \) satisfy (A.71), we still have some (weak) form of resolution of the identity. Now Assumption \(\mathscr {D}\)-pb 3 is replaced by the following:
Assumption \(\mathscr {D}\)-pbw 3. For some subspace \({\mathscr {G}}\) dense in \(\mathscr {H}\), \({\mathscr {F}}_\varphi \) and \({\mathscr {F}}_\varPsi \) are \({\mathscr {G}}\)-quasi bases.
To refine further the structure, let us assume there exists a self-adjoint, invertible, operator \(\varTheta \), which, together with \(\varTheta ^{-1}\), leaves \(\mathscr {D}\) invariant: \(\varTheta \mathscr {D}\subseteq \mathscr {D}\), \(\varTheta ^{-1}\mathscr {D}\subseteq \mathscr {D}\). Then we say that \((a,b^\dagger )\) are \(\varTheta -\)conjugate if \(af=\varTheta ^{-1}b^\dagger \,\varTheta \,f\), for all \(f\in \mathscr {D}\). One can prove that, if \({\mathscr {F}}_\varphi \) and \({\mathscr {F}}_\varPsi \) are \(\mathscr {D}\)-quasi bases for \(\mathscr {H}\), then the operators \((a,b^\dagger )\) are \(\varTheta -\)conjugate if and only if \(\varPsi _n=\varTheta \varphi _n\), for all \(n\ge 0\). Moreover, if \((a,b^\dagger )\) are \(\varTheta -\)conjugate, then \(\left\langle f,\varTheta f\right\rangle \rangle 0\) for all non zero \(f\in \mathscr {D}\).
We refer to [6] for more results on \(\mathscr {D}\)-PBs.
Appendix 2: A Class of Potentials Isospectral to \(H_0\)
In this Appendix we briefly discuss a particular class of isospectral potential to the Harmonic oscillator Hamiltonian, given by the choice of the parameter \(\gamma \) in the confluent hypergeometric equation (39). Its solution will be expressed in terms of combinations of Hermite and Pseudo-Hermite polynomials [20].
For \(\gamma =-\frac{1}{2}-p\), (40) becomes
It is simple to show that
for p even, and
for p odd. Equation (A.72) thus becomes
In view of the relations
Equation (A.77) can be cast as
where \(\mathscr {H}_p(x)\) are defined as the Pseudo-Hermite polynomials [20] and \(\mathscr {P}_{p}(x)\) are linear combination of the standard Hermite polynomials H(x),
The polynomials \(\mathscr {P}_{p}(x)\) are nodeless polynomials of degree \((p-1)\), odd/even if p is even/odd.Footnote 4 Remark that \(\frac{d^{p}}{dx^{p}}\left( e^{x^{2}}\mathrm {Erf}(x)\right) \) is nodeless for odd p, but it has a singularity at \(x=0\) for even p, while \(\frac{d^{p}}{dx^{p}}\left( e^{x^{2}}\right) \) is nodeless for even p, but it has a singularity at \(x=0\) for odd p. The formulas above contain the classes of isospectral potentials discussed in the literature. For instance, the Mielink’s result of [15] is recovered for \(\epsilon =\frac{1}{2}\), \(k=1\) and \(p=0\) with the choice \(c_{2}=1\) and \(c_{1}=C_{0}^{M}\),
(that (41) with \(C_{0}^{M}=\frac{C_{1}}{C_{2}}\)). Rational nodeless solutions are obtained for \(c_{2}=0\) and even p, while they present a singularity in \(x=0\) for odd p,
the case of the class of solutions \(\phi _{p}(x)=e^{\frac{x^{2}}{2}}(-i)^{p}\mathscr {H}_{p}(i x)\) recently discussed in [20] is recovered with the choice of the constants \(c_{1}=\frac{2^{p\ }\varGamma \left( \frac{1+p}{2}\right) }{\sqrt{\pi }},\ c_{2}=0\) for even values of p and \(c_{2}=\frac{2^{p\ }p\ \varGamma \left( \frac{p}{2}\right) }{\sqrt{\pi }},\ c_{1}=0\) for odd values of p. In particular, for even p
and the isospectral Hamiltonian is
with ground state \(\psi _{+}(x)=C_{0}\frac{e^{\frac{x^{2}}{2}}}{h_{p}(x)} =C_{0}\frac{e^{-\frac{x^{2}}{2}}}{\mathscr {H}_{p}(x)}\), where \(C_{0}=\left[ \frac{(2m)!2^{m}}{\sqrt{\pi }}\right] ^{\frac{1}{2}}\) is a normalization constant. The spectrum obtained by selecting only the odd eigenstates of the harmonic oscillator is given by
with eigenvalues \((2j+\frac{3}{2}+k)\) for \(j=0,1\ldots ,\ m=0,1\ldots ,\ (k,\epsilon )\in \mathbb {R}.\)
Another example is found in [21], where the authors studied the one-dimensional quantum system described by the potential
It correspond to the choice \(\epsilon =-\frac{3}{2},\ k=1, p=2,\ c=1,\ c_{2}=0\). The ground state \(\psi _{+}=\frac{2}{\sqrt{\pi }}\frac{e^{-\frac{x^{2}}{2}}}{(1+2x^{2})}\) is well defined and the eigenstates \(\psi _{1}^{(n)}\) with eigenvalues \((n+\frac{3}{2})\) are obtained for all n
The rational (odd p) and transcendental (even p) potentials obtained for \(c_1=0\) possess a singularity in \(x=0\),
The eigenstates of the isospectral Hamiltonian \(H_{1}\) are then well defined only for odd values of the quantum number n. An example of such a rational and singular solution is given by the radial part (\(l=1\)) of the three-dimensional harmonic oscillator, recovered for \(\epsilon =-\frac{1}{2},\ k=1\,(p=1),\ c_{1}=0,\ c_{2}=1\).Footnote 5 The hamiltonian
isospectral to \(H_{0}+1\), has eigenstates
but, to in order to avoid divergences, we have to consider only the ones corresponding to odd values of n
with spectrum \(2j+5/2\).
Before concluding the section, it is in order to point out that it is simple to extend the previous results to the case \(\varGamma =\frac{1}{2}+p\), in that it merely coincides with the same problem for \(u_{\gamma }(x)\) up to the change of variable \(y=i x\). Therefore, the general solution would be simply
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Bagarello, F., Gianfreda, M. (2016). \(\mathscr {D}{-}\)Deformed and SUSY-Deformed Graphene: First Results. In: Bagarello, F., Passante, R., Trapani, C. (eds) Non-Hermitian Hamiltonians in Quantum Physics. Springer Proceedings in Physics, vol 184. Springer, Cham. https://doi.org/10.1007/978-3-319-31356-6_7
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