For non-Hermitian equilateral q-pointed star-shaped quantum graphs of paper I [Can. J. Phys. 90, 1287 (2012), arXiv 1205.5211] we show that due to certain dynamical aspects of the model as controlled by the external, rotation-symmetric complex Robin boundary conditions, the spectrum is obtainable in a closed asymptotic-expansion form, in principle at least. Explicit formulae up to the second order are derived for illustration, and a few comments on their consequences are added.

http://arxiv.org/abs/1411.3828
Quantum Physics (quant-ph); Spectral Theory (math.SP)

Among quantum systems with finite Hilbert space a specific role is played by systems controlled by non-Hermitian Hamiltonian matrices \(H\neq H^\dagger\) for which one has to upgrade the Hilbert-space metric by replacing the conventional form \(\Theta^{(Dirac)}=I\) of this metric by a suitable upgrade \(\Theta^{(non−Dirac)}\neq I\) such that the same Hamiltonian becomes self-adjoint in the new, upgraded Hilbert space, \(H=H\ddagger=\Theta^{−1}H^\dagger\Theta\). The problems only emerge in the context of scattering where the requirement of the unitarity was found to imply the necessity of a non-locality in the interaction, compensated by important technical benefits in the short-range-nonlocality cases. In the present paper we show that an why these technical benefits (i.e., basically, the recurrent-construction availability of closed-form Hermitizing metrics \(\Theta^{(non−Dirac)}\) can survive also in certain specific long-range-interaction models.

http://arxiv.org/abs/1410.3583
Quantum Physics (quant-ph)

A non-Hermitian N−level quantum model with two free real parameters is proposed in which the bound-state energies are given as roots of an elementary trigonometric expression and in which they are, in a physical domain of parameters, all real. The wave function components are expressed as closed-form superpositions of two Chebyshev polynomials. In any eligible physical Hilbert space of finite dimension \(N<\infty\) our model is constructed as unitary with respect to an underlying Hilbert-space metric \(\Theta\neq I\). The simplest version of the latter metric is finally constructed, at any dimension N=2,3,…, in closed form. This version of the model may be perceived as an exactly solvable N−site lattice analogue of the \(N=\infty\) square well with complex Robin-type boundary conditions. At any \(N<\infty\) our closed-form metric becomes trivial (i.e., equal to the most common Dirac’s metric \(\Theta(Dirac)=I\)) at the special, Hermitian-Hamiltonian-limit parameters.

http://arxiv.org/abs/1409.3788
Mathematical Physics (math-ph); Quantum Physics (quant-ph)

It is shown that the toy-model-based considerations of loc. cit. (see also arXiv:1312.3395) are based on an incorrect, manifestly unphysical choice of the Hilbert space of admissible quantum states. A two-parametric family of all of the eligible correct and potentially physical Hilbert spaces of the model is then constructed. The implications of this construction are discussed. In particular, it is emphasized that contrary to the conclusions of loc. cit. there is no reason to believe that the current form of the PT-symmetric quantum theory should be false as a fundamental theory.

http://arxiv.org/abs/1404.1555
Quantum Physics (quant-ph); High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph)

Three classes of finite-dimensional models of quantum systems exhibiting spectral degeneracies called quantum catastrophes are described in detail. Computer-assisted symbolic manipulation techniques are shown unexpectedly efficient for the purpose.

http://arxiv.org/abs/1403.0723
Quantum Physics (quant-ph); Mathematical Physics (math-ph)

We propose giving the mathematical concept of the pseudospectrum a central role in quantum mechanics with non-Hermitian operators. We relate pseudospectral properties to quasi-Hermiticity, similarity to self-adjoint operators, and basis properties of eigenfunctions. The abstract results are illustrated by unexpected wild properties of operators familiar from PT-symmetric quantum mechanics.

http://arxiv.org/abs/1402.1082

Spectral Theory (math.SP); Mathematical Physics (math-ph); Quantum Physics (quant-ph)

The elementary quadratic plus inverse sextic interaction containing a strongly singular repulsive core in the origin is made regular by a complex shift of coordinate \(x=s−i\epsilon\). The shift \(\epsilon>0\) is fixed while the value of s is kept real and potentially observable, \(s∈(−\infty,\infty)\). The low-lying energies of bound states are found in closed form for the large couplings g. Within the asymptotically vanishing \(\mathcal{O}(g^{−1/4})\) error bars these energies are real so that the time-evolution of the system may be expected unitary in an {\em ad hoc} physical Hilbert space.

http://arxiv.org/abs/1401.1435
Quantum Physics (quant-ph)

The practical use of non-Hermitian (i.e., typically, PT-symmetric) phenomenological quantum Hamiltonians is discussed as requiring an explicit reconstruction of the ad hoc Hilbert-space metrics which would render the time-evolution unitary. Just the N-dimensional matrix toy models Hamiltonians are considered, therefore. For them, the matrix elements of alternative metrics are constructed via solution of a coupled set of polynomial equations, using the computer-assisted symbolic manipulations for the purpose. The feasibility and some consequences of such a model-construction strategy are illustrated via a discrete square well model endowed with multi-parametric close-to-the-boundary real bidiagonal-matrix interaction. The degenerate exceptional points marking the phase transitions are then studied numerically. A way towards classification of their unfoldings in topologically non-equivalent dynamical scenarios is outlined.

http://arxiv.org/abs/1305.4822
Quantum Physics (quant-ph)

The answer is yes. Via an elementary, exactly solvable crypto-Hermitian example it is shown that inside the interval of admissible couplings the innocent-looking point of a smooth unavoided crossing of the eigenvalues of Hamiltonian $H$ may carry a dynamically nontrivial meaning of a phase-transition boundary or “quantum horizon”. Passing this point requires a change of the physical Hilbert-space metric $\Theta$, i.e., a thorough modification of the form and of the interpretation of the operators of all observables.

http://arxiv.org/abs/1303.4876

Quantum Physics (quant-ph); Mathematical Physics (math-ph)

A compact review is given, and a few new numerical results are added to the recent studies of the q-pointed one-dimensional star-shaped quantum graphs. These graphs are assumed endowed with certain specific, manifestly non-Hermitian point interactions, localized either in the inner vertex or in all of the outer vertices and carrying, in the latter case, an interesting zero-inflow interpretation.

http://arxiv.org/abs/1303.4331
Quantum Physics (quant-ph)

In a pre-selected Hilbert space of quantum states the unitarity of the evolution is usually guaranteed via a pre-selection of the generator (i.e., of the Hamiltonian operator) in self-adjoint form. In fact, the simultaneous use of both of these pre-selections is overrestrictive. One should be allowed to make a given Hamiltonian self-adjoint only after an {\em ad hoc} generalization of Hermitian conjugation. We argue that in the generalized, hidden-Hermiticity scenario with nontrivial metric, the current concept of solvability (meaning, most often, the feasibility of a non-numerical diagonalization of Hamiltonian) requires a generalization allowing for a non-numerical form of metric. A few illustrative solvable quantum models of this type are presented.

http://arxiv.org/abs/1302.2035
Quantum Physics (quant-ph)

N-site-lattice Hamiltonians H are introduced and perceived as a set of systematic discrete approximants of a certain PT-symmetric square-well-potential model with the real spectrum and with a non-Hermiticity which is localized near the boundaries of the interval. Its strength is controlled by one, two or three parameters. The problem of the explicit construction of a nontrivial metric which makes the theory unitary is then addressed. It is proposed and demonstrated that due to the not too complicated tridiagonal-matrix form of our input Hamiltonians the computation of the metric is straightforward and that its matrix elements prove obtainable, non-numerically, in elementary polynomial forms.

http://arxiv.org/abs/1302.1662
Quantum Physics (quant-ph); Mathematical Physics (math-ph)

The quantum-catastrophe (QC) benchmark Hamiltonians of paper I (M. Znojil, J. Phys. A: Math. Theor. 45 (2012) 444036) are reconsidered, with the infinitesimal QC distance \(\lambda\) replaced by the total time $\tau$ of the fall into the singularity. Our amended model becomes unique, describing the complete QC history as initiated by a Hermitian and diagonalized N-level oscillator Hamiltonian at \(\tau=0\). In the limit \(\tau \to 1\) the system finally collapses into the completely (i.e., N-times) degenerate QC state. The closed and compact Hilbert-space metrics are then calculated and displayed up to N=7. The phenomenon of the QC collapse is finally attributed to the manifest time-dependence of the Hilbert space and, in particular, to the emergence and to the growth of its anisotropy. A quantitative measure of such a time-dependent anisotropy is found in the spread of the N-plet of the eigenvalues of the metric. Unexpectedly, the model appears exactly solvable — at any multiplicity N, the N-plet of these eigenvalues is obtained in closed form.

http://arxiv.org/abs/1212.0734
Quantum Physics (quant-ph); Mathematical Physics (math-ph)

We show that the eigenvectors of the PT -symmetric imaginary cubic oscillator are complete, but do not form a Riesz basis. This results in the existence of a bounded metric operator having intrinsic singularity reflected in the inevitable unboundedness of the inverse. Moreover, the existence of nontrivial pseudospectrum is observed. In other words, there is no quantum-mechanical Hamiltonian associated with it via bounded and boundedly invertible similarity transformations. These results open new directions in physical interpretation of PT -symmetric models with intrinsically singular metric, since their properties are essentially different with respect to self-adjoint Hamiltonians, for instance, due to spectral instabilities.

http://arxiv.org/abs/1208.1866
Quantum Physics (quant-ph); Mathematical Physics (math-ph)
Physical Review D 86, 121702 (2012)

The bound-state spectrum of a Hamiltonian H is assumed real in a non-empty domain D of physical values of parameters. This means that for these parameters, H may be called crypto-Hermitian, i.e., made Hermitian via an ad hoc choice of the inner product in the physical Hilbert space of quantum bound states (i.e., via an ad hoc construction of the so called metric). The name of quantum catastrophe is then assigned to the N-tuple-exceptional-point crossing, i.e., to the scenario in which we leave domain D along such a path that at the boundary of D, an N-plet of bound state energies degenerates and, subsequently, complexifies. At any fixed \(N \geq 2\), this process is simulated via an N by N benchmark effective matrix Hamiltonian H. Finally, it is being assigned such a closed-form metric which is made unique via an N-extrapolation-friendliness requirement.

http://arxiv.org/abs/1206.6000
Quantum Physics (quant-ph); Mathematical Physics (math-ph)

A new exactly solvable model of a quantum system is proposed, living on an equilateral q-pointed star graph (q is arbitrary). The model exhibits a weak and spontaneously broken form of \({\cal PT}-\)symmetry, offering a straightforward generalization of one of the standard solvable square wells with \(q=2\) and unbroken \({\cal PT}-\)symmetry. The kinematics is trivial, Kirchhoff in the central vertex. The dynamics is one-parametric (viz., \(\alpha-\)dependent), prescribed via complex Robin boundary conditions (i.e., the interactions are non-Hermitian and localized at the outer vertices of the star). The (complicated, trigonometric) secular equation is shown reducible to an elementary and compact form. This renders the model (partially) exactly solvable at any \(q \geq 2\) — an infinite subset of the real roots of the secular equation proves q-independent and known (i.e., inherited from the square-well \(q=2\) special case). The systems with \(q=4m-2\) are found anomalous, supporting infinitely many (or, at m=1, one) additional real m-dependent and \(\alpha-\)dependent roots.

http://arxiv.org/abs/1205.5211
Quantum Physics (quant-ph)

For the description of quantum evolution, the use of a manifestly time-dependent quantum Hamiltonian \(\mathfrak{h}(t) =\mathfrak{h}^\dagger(t)\) is shown equivalent to the work with its simplified, time-independent alternative \(G\neq G^\dagger\). A tradeoff analysis is performed recommending the latter option. The physical unitarity requirement is shown fulfilled in a suitable ad hocrepresentation of Hilbert space.

http://arxiv.org/abs/1204.5989
Quantum Physics (quant-ph); General Relativity and Quantum Cosmology (gr-qc)

Besides the standard quantum version of the Coulomb/Kepler problem, an alternative quantum model with not too dissimilar phenomenological (i.e., spectral and scattering) as well as mathematical (i.e., exact-solvability) properties may be formulated and solved. Several aspects of this model are described. The paper is made self-contained by explaining the underlying innovative quantization strategy which assigns an entirely new role to symmetries.

http://arxiv.org/abs/1204.1257
Quantum Physics (quant-ph)

We consider one-dimensional Pauli Hamiltonians in a bounded interval with possibly non-self-adjoint Robin-type boundary conditions. We study the influence of the spin-magnetic interaction on the interplay between the type of boundary conditions and the spectrum. A special attention is paid to PT-symmetric boundary conditions with the physical choice of the time-reversal operator T.

http://arxiv.org/abs/1203.5011
Quantum Physics (quant-ph); High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph)

A new version of PT-symmetric quantum theory is proposed and illustrated by an N-site-lattice Legendre oscillator. The essence of the innovation lies in the replacement of parity P (serving as an indefinite metric in an auxiliary Krein space) by its non-involutory alternative P(positive)=Q>0 playing the role of a positive-definite nontrivial metric in an auxiliary, redundant, unphysical Hilbert space. It is shown that the QT-symmetry of this form remains appealing and technically useful.

http://arxiv.org/abs/1201.5058
Quantum Physics (quant-ph)

A given Hamiltonian matrix H with real spectrum is assumed tridiagonal and non-Hermitian. Its possible Hermitizations via an amended, ad hoc inner-product metric are studied. Under certain reasonable assumptions, all of these metrics are shown obtainable as recurrent solutions of the hidden Hermiticity constraint called Dieudonne equation. In this framework even the two-parametric Jacobi-polynomial real- and asymmetric-matrix N-site lattice Hamiltonian is found tractable non-numerically at all N.

http://arxiv.org/abs/1201.2263
Mathematical Physics (math-ph); Quantum Physics (quant-ph)

A non-unitary version of quantum scattering is studied via an exactly solvable toy model. The model is merely asymptotically local since the smooth path of the coordinate is admitted complex in the non-asymptotic domain. At any real angular-momentum-like parameter the reflection R and transmission T are shown to change with the winding number (i.e., topology) of the path. The points of unitarity appear related to the points of existence of quantum-knot bound states.

http://arxiv.org/abs/1112.2644
Mathematical Physics (math-ph); Quantum Physics (quant-ph)

Two discrete N-level alternatives to the popular imaginary cubic oscillator are proposed and studied. In a certain domain \({\cal D}\) of parameters \(a\) and \(z\) of the model, the spectrum of energies is shown real (i.e., potentially, observable) and the unitarity of the evolution is shown mediated by the construction of a (non-unique) physical, ad hoc Hilbert space endowed with a nontrivial, Hamiltonian-dependent inner-product metric \(\Theta\). Beyond \({\cal D}\) the complex-energy curves are shown to form a “Fibonacci-numbered” geometric pattern and/or a “topologically complete” set of spectral loci. The dynamics-determining construction of the set of the eligible metrics is shown tractable by a combination of the computer-assisted algebra with the perturbation and extrapolation techniques. Confirming the expectation that for the local potentials the effect of the metric cannot be short-ranged.

http://arxiv.org/abs/1111.0484
Quantum Physics (quant-ph); Mathematical Physics (math-ph)

An extension of the scope of quantum theory is proposed in a way inspired by the recent heuristic as well as phenomenological success of the use of non-Hermitian Hamiltonians which are merely required self-adjoint in a Krein space with an indefinite metric (chosen, usually, as the operator of parity). In nuce, the parity-like operators are admitted to represent the mere indefinite metric in a Pontryagin space. A constructive version of such a generalized quantization strategy is outlined and, via a non-numerical illustrative example, found feasible.

http://arxiv.org/abs/1110.1218
Mathematical Physics (math-ph)

The increasingly popular concept of a hidden Hermiticity of operators (i.e., of their Hermiticity with respect to an {\it ad hoc} inner product in Hilbert space) is compared with the recently introduced notion of {\em non-linear pseudo-bosons}. The formal equivalence between these two notions is deduced under very general assumptions. Examples of their applicability in quantum mechanics are discussed.

http://arxiv.org/abs/1109.0605
Mathematical Physics (math-ph); Functional Analysis (math.FA); Quantum Physics (quant-ph)

We consider one-dimensional Schroedinger-type operators in a bounded interval with non-self-adjoint Robin-type boundary conditions. It is well known that such operators are generically conjugate to normal operators via a similarity transformation. Motivated by recent interests in quasi-Hermitian Hamiltonians in quantum mechanics, we study properties of the transformations in detail. We show that they can be expressed as the sum of the identity and an integral Hilbert-Schmidt operator. In the case of parity and time reversal boundary conditions, we establish closed integral-type formulae for the similarity transformations, derive the similar self-adjoint operator and also find the associated “charge conjugation” operator, which plays the role of fundamental symmetry in a Krein-space reformulation of the problem.

http://arxiv.org/abs/1108.4946
Spectral Theory (math.SP); Mathematical Physics (math-ph); Functional Analysis (math.FA); Quantum Physics (quant-ph)

Non-Hermitian ring-shaped discrete lattices share the appeal with their more popular linear predecessors. Their dynamics controlled by the nearest-neighbor interaction is equally phenomenologically interesting. In comparison, the innovative nontriviality of their topology may be expected to lead to new spectral effects. Some of them are studied here via solvable examples. Main attention is paid to the perturbation-caused removals of spectral degeneracy at exceptional points.

http://arxiv.org/abs/1108.0620
Quantum Physics (quant-ph); Mathematical Physics (math-ph)

The one-dimensional real line of coordinates is replaced, for simplification or approximation purposes, by an N-plet of the so called Gauss-Hermite grid points. These grid points are interpreted as the eigenvalues of a tridiagonal matrix \(\mathfrak{q}_0\) which proves rather complicated. Via the “zeroth” Dyson-map \(\Omega_0\) the “operator of position” \(\mathfrak{q}_0\) is then further simplified into an isospectral matrix \(Q_0\) which is found optimal for the purpose. As long as the latter matrix appears non-Hermitian it is not an observable in the manifestly “false” Hilbert space \({\cal H}^{(F)}:=\mathbb{R}^N\). For this reason the optimal operator \(Q_0\) is assigned the family of its isospectral avatars \(\mathfrak{h}_\alpha\), \(\alpha=(0,)\,1,2,…\). They are, by construction, selfadjoint in the respective \(\alpha-\)dependent image Hilbert spaces \({\cal H}^{(P)}_\alpha\) obtained from \({\cal H}^{(F)}\) by the respective “new” Dyson maps \(\Omega_\alpha\). In the ultimate step of simplification, the inner product in the F-superscripted space is redefined in an {\it ad hoc}, $\alpha-$dependent manner. The resulting “simplest”, S-superscripted representations \({\cal H}^{(S)}_\alpha\) of the eligible physical Hilbert spaces of states (offering different dynamics) then emerge as, by construction, unitary equivalent to the (i.e., indistinguishable from the) respective awkward, P-superscripted and \(\alpha-\)subscripted physical Hilbert spaces.

http://arxiv.org/abs/1107.1770
Quantum Physics (quant-ph); Mathematical Physics (math-ph)

A toy-model quantum system is proposed. At a given integer \(N\) it is defined by the pair of \(N\) by \(N\) real matrices \((H,\Theta)\) of which the first item \(H\) specifies an elementary, diagonalizable non-Hermitian Hamiltonian \(H \neq H^\dagger\) with the real and explicit spectrum given by the zeros of the \(N-\)th Chebyshev polynomial of the first kind. The second item \(\Theta\neq I\) must be (and is being) constructed as the related Hilbert-space metric which specifies the (in general, non-unique) physical inner product and which renders our toy-model Hamiltonian selfadjoint, i.e., compatible with the Dieudonne equation \(H^\dagger \Theta= \Theta\,H\). The elements of the (in principle, complete) set of the eligible metrics are then constructed in closed band-matrix form. They vary with our choice of the \(N-\)plet of optional parameters, \(\Theta=\Theta(\vec{\kappa})>0\) which must be (and are being) selected as lying in the positivity domain of the metric, \(\vec{\kappa} \in {\cal D}^{(physical)}\).

http://arxiv.org/abs/1105.1863
Quantum Physics (quant-ph); High Energy Physics – Lattice (hep-lat); Mathematical Physics (math-ph)

The “Hermitizability” problem of quantum theory is explained, discussed and illustrated via the discrete-lattice cryptohermitian quantum graphs. In detail, the description of the domain ${\cal D}$ of admissible parameters is provided for the “circular-model” three-parametric quantum Hamiltonian $H$ using periodic boundary conditions. It is emphasized that even in such an elementary system the weak- and strong-coupling subdomains of ${\cal D}$ become, unexpectedly, non-empty and disconnected.

http://arxiv.org/abs/1103.4001
Quantum Physics (quant-ph); High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph)

The Laplacian in an unbounded tubular neighbourhood of a hyperplane with non-Hermitian complex-symmetric Robin-type boundary conditions is investigated in the limit when the width of the neighbourhood diminishes. We show that the Laplacian converges in a norm resolvent sense to a self-adjoint Schroedinger operator in the hyperplane whose potential is expressed solely in terms of the boundary coupling function. As a consequence, we are able to explain some peculiar spectral properties of the non-Hermitian Laplacian by known results for Schroedinger operators.

http://arxiv.org/abs/1102.5051
Spectral Theory (math.SP); Mathematical Physics (math-ph)

An array of N subsequent Laguerre polynomials is interpreted as an eigenvector of a non-Hermitian tridiagonal Hamiltonian $H$ with real spectrum or, better said, of an exactly solvable N-site-lattice cryptohermitian Hamiltonian whose spectrum is known as equal to the set of zeros of the N-th Laguerre polynomial. The two key problems (viz., the one of the ambiguity and the one of the closed-form construction of all of the eligible inner products which make $H$ Hermitian in the respective {\em ad hoc} Hilbert spaces) are discussed. Then, for illustration, the first four simplest, $k-$parametric definitions of inner products with $k=0,k=1,k=2$ and $k=3$ are explicitly displayed. In mathematical terms these alternative inner products may be perceived as alternative Hermitian conjugations of the initial N-plet of Laguerre polynomials. In physical terms the parameter $k$ may be interpreted as a measure of the “smearing of the lattice coordinates” in the model.

http://arxiv.org/abs/1101.1183
Mathematical Physics (math-ph); Quantum Physics (quant-ph)

Non-hermitian quantum graphs possessing real (i.e., in principle, observable) spectra are studied via their discretization. The discretized Hamiltonians are assigned, constructively, an elementary pseudometric and/or a more complicated metric. Both these constructions make the Hamiltonian Hermitian, respectively, in an auxiliary (Krein or Pontryagin) vector space or in a less friendly (but more useful) Hilbert space of quantum mechanics.

http://arxiv.org/abs/1101.1015
Quantum Physics (quant-ph); Mathematical Physics (math-ph)

In an innovative inverse-problem construction the measured, experimental energies $E_1$, $E_2$, …$E_N$ of a quantum bound-state system are assumed fitted by an N-plet of zeros of a classical orthogonal polynomial $f_N(E)$. We reconstruct the underlying Hamiltonian $H$ (in the most elementary nearest-neighbor-interaction form) and the underlying Hilbert space ${\cal H}$ of states (the rich menu of non-equivalent inner products is offered). The Gegenbauer’s ultraspherical polynomials $f_n(x)=C_n^\alpha(x)$ are chosen for the detailed illustration of technicalities.

http://arxiv.org/abs/1011.4803
Quantum Physics (quant-ph); Mathematical Physics (math-ph)
Phys. Rev. A 82 (2010) 052113
DOI:10.1103/PhysRevA.82.052113

A new version of an elementary PT-symmetric square well quantum model is proposed in which a certain Hermiticity-violating end-point interaction leaves the spectrum real in a large domain of couplings $\lambda\in (-1,1)$. Within this interval we employ the usual coupling-independent operator P of parity and construct, in a systematic Runge-Kutta discrete approximation, a coupling-dependent operator of charge C which enables us to classify our P-asymmetric model as CPT-symmetric or, equivalently, hiddenly Hermitian alias cryptohermitian.

http://arxiv.org/abs/1011.4806
Quantum Physics (quant-ph); High Energy Physics – Theory (hep-th)

We establish that a perfect-transmission scattering problem can be described by a class of parity and time reversal symmetric operators and hereby we provide a scenario for understanding and implementing the corresponding quasi-Hermitian quantum mechanical framework from the physical viewpoint. One of the most interesting features of the analysis is that the complex eigenvalues of the underlying non-Hermitian problem, associated with a reflectionless scattering system, lead to the loss of perfect-transmission energies as the parameters characterizing the scattering potential are varied. On the other hand, the scattering data can serve to describe the spectrum of a large class of Schroedinger operators with complex Robin boundary conditions.

http://arxiv.org/abs/1011.4281
Mathematical Physics (math-ph); Quantum Physics (quant-ph)