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We prove the simplicity and analyticity of the eigenvalues of the cubic oscillator Hamiltonian,$H(\beta)=-d^2/dx^2+x^2+i\sqrt{\beta}x^3$,for $\beta$ in the cut plane $\C_c:=\C\backslash (-\infty, 0)$. Moreover, we prove that the spectrum consists of the perturbative eigenvalues $\{E_n(\beta)\}_{n\geq 0}$ labeled by the constant number $n$ of nodes of the corresponding eigenfunctions. In addition, for all $\beta\in\C_c$, $E_n(\beta)$ can be computed as the Stieltjes-Pad\'e sum of its perturbation series at $\beta=0$. This also gives an alternative proof of the fact that the spectrum of $H(\beta)$ is real when $\beta $ is a positive number. This way, the main results on the repulsive PT-symmetric and on the attractive quartic oscillators are extended to the cubic case., Comment: 23 pages, 3 figures

We consider a \textit{PT}-symmetric cubic oscillator with an imaginary double well. We prove the existence of an infinite number of level crossings with a definite selection rule. Decreasing the positive parameter $\hbar$ from large values, at a value $\hbar_n$ we find the crossing of the pair of levels $(E_{2n+1}(\hbar),E_{2n}(\hbar))$ becoming the pair of levels $(E_n^+(\hbar),E_n^-(\hbar))$. For large parameters, a level is a holomorphic function $E_m(\hbar)$ with different semiclassical behaviors, $E_j^\pm(\hbar),$ along different paths. The corresponding $m$-nodes delocalized state $\psi_m(\hbar)$ behaves along the same paths as the semiclassical $j$-nodes states $\psi_j^\pm(\hbar),$ localized at one of the wells $x_\pm$ respectively. In particular, if the crossing parameter $\hbar_n$ is by-passed from above, the levels $E_{2n+(1/2)\pm(1/2)}(\hbar)$ have respectively the semiclassical behaviors of the levels $E_n^\mp(\hbar)$ along the real axis. These results are obtained by the control of the nodes. There is evidence that the parameters $\hbar_n$ accumulate at zero and the accumulation point of the corresponding energies is aninstability point of a subset of the Stokes complex called the monochord, consisting of the vibrating string and the sound board., Comment: 38 pages, 10 fugures

We make a nodal analysis of the processes of level crossings in a model of quantum mechanics with a PT-symmetric double well. We prove the existence of infinite crossings with their selection rules. At the crossing, before the PT-symmetry breaking and the localization, we have a total P-symmetry breaking of the states.

Here, we show how the molecular structure appears and becomes stable for supercritical physical conditions. In particular we consider, for the ammonia molecule in a gas, a simple model based on a standard non-linear double-well Schroedinger equation with a dissipative term and a term representing weak collisions., Comment: 4 pages, 2 figures

We consider the properties of the family of double well quantum Hamiltonians Hħ = − ħ2 (d2/dx2) + i(x3 − x), x ∈ ℝ, ħ > 0, starting from the resonances of the cubic oscillator Hϵ = − (d2/dx2) + x2 + ϵx3, ϵ > 0, and studying their analytic continuations obtained by generalized changes of representation. We prove the existence of infinite crossings of the eigenvalues of Hħ together with the selection rules of the pairs of eigenvalues taking part in a crossing. This is a semiclassical localization effect. The eigenvalues at the crossings accumulate at a critical energy for some of the Stokes lines.

We study a periodically driven double well model. As in the case of autonomous models, previously treated in a joint paper with A. Martinez,(7) we have the destruction of the splitting for critical metastability. The relevance of the model for the understanding of the red shift in the inversion line of the molecule of ammonia is shortly discussed. We show that, in order to have a reasonable behavior of the metastability as a function of the frequency, a non-monochromatic perturbation is needed.

We consider a non-linear Stark effect as a model for localization and symmetry breaking of a molecule in a gas. By a comparison method with respect to the linear Stark effect, we prove the existence of level bifurcation and symmetry breaking at a critical value of the gas pressure exponentially small for large nuclear masses. Extending the Davies results, we confirm the predictions of Claverie-Jona Lasinio for pyramidal molecules as the ammonia one.

We consider a Zener double-well problem related to the magnetic bands in a superlattice Bloch operator. We give the precise asymptotic behaviour of the level splittings. This way we extend the Peierls substitution rule to an exponentially small term and furthermore, for the first time, we rigorously compute an exponentially small term in a Zener problem.

We prove that the Stark effect perturbation theory of a class of bound states uniquely determines the position and the width of the resonances by Distributional Borel Sum. In particular the small field asymptotics of the width is uniquely related to the large order asymptotics of the perturbation coefficients. Similar results apply to all the “resonances” of the anharmonic and double well oscillators.

We prove the Pad?e (Stieltjes) summability of the perturbation series of any energy level En,1(?), n ? N, of the cubic anharmonic oscillator, H1(?) = p2 +x2 +i??x3, as suggested by the numerical studies of Bender and Weniger. At the same time, we give a simple proof of the positivity of every level of the PT -symmetric Hamiltonian H1(?) for positive ? (Bessis–Zinn Justin conjecture). The n zeros, of a state ?n,1(?), stable at ? = 0, are confined for ? on the cut complex plane, and are related to the level En,1(?) by the Bohr–Sommerfeld quantization rule (semiclassical phase-integral condition). We also prove the absence of non-perturbative eigenvalues and the simplicity of the spectrum of our Hamiltonians.

The spectral problem of the Dirac equation in an external quadratic vector potential is considered using the methods of the perturbation theory. The problem is singular and the perturbation series is asymptotic, so that the methods for dealing with divergent series must be used. Among these, the Distributional Borel Sum appears to be the most well suited tool to give answers and to describe the spectral properties of the system. A detailed investigation is made in one and in three space dimensions with a central potential. We present numerical results for the Dirac equation in one space dimension: these are obtained by determining the perturbation expansion and using the Pad\'e approximants for calculating the distributional Borel transform. A complete agreement is found with previous non-perturbative results obtained by the numerical solution of the singular boundary value problem and the determination of the density of the states from the continuous spectrum., Comment: 10 pages, 1 figure

We study the Schrodinger equation for an electron in a one dimensional crystal submitted to a constant electric field. We prove the existence of ladders of resonances, the imaginary part of which is exponentially small with the field.

The authors prove that the small field asymptotic behaviour of the Stark-Wannier ladders near the real direction is generically highly singular. This result is in agreement with the conjecture of a chaotic behaviour of the lifetime of the states because of infinitely many crossings.

We consider a non-linear perturbation of a symmetric double-well potential as a model for molecular localization. In the semiclassical limit, we prove the existence of a critical value of the perturbation parameter giving the destruction of the beating effect. This value is twice the one corresponding to the first bifurcation of the fundamental state. Here we make use of a particular projection operator introduced by G. Nenciu in order to extend to an infinite dimensional space some known results for a two-level system.

In this paper, we give the Heisenberg position operator in the crystal momentum representation and we prove the acceleration theorem for Bloch oscillators. As an application, we discuss the motion of well localized states.

We consider a Wannier–Stark problem with only one ladder for weak field. We prove that a generic first-band state is a metastable state (Wannier–Bloch oscillator) oscillating because of a beating effect and decaying at the rate given by the imaginary part of the Wannier-Stark resonances. By this result we have at the same time the realization of the ideas of Bloch about the oscillations, of Wannier about the approximate quantization and of Zener about the metastability. Such oscillators, which generically perform a breathing mode motion in a large spatial region, have been experimentally observed.

We consider the small field asymptotics of the lifetime of metastable states in Wannier-Stark problems. Assuming that at zero field we have Bloch operators with only the first gap open and using the regular perturbation theory, we prove that the behavior of the lifetime computed by means of the Fermi Golden Rule is proportional to the correct one with the factor \(\). The connection with adiabatic problems is briefly discussed.

We give in a rigorous way the time behavior of the metastable Bloch oscillators for weak electric field. The validity of the Fermi golden rule, with the change of the numerical prefactor suggested by Kane and Blount, is definitely proved. Moreover, we give a new version of the acceleration theorem and the behavior of the Bloch oscillators in the adiabatic limit.

We consider the energy levels of a Stark family, in the parameterj, of quartic double wells with perturbation parameterg: $$H(g,j) = p^2 + x^2 (1 - gx)^2 - j\left( {gx - \frac{1}{2}} \right).$$ For non-evenj (and for the symmetric casej=0) we prove analyticity in the full Nevanlinna disk ℜg−2 >R−1 of theg2-plane, as predicted by Crutchfield. By means of an approximation we give a heuristic estimate of the asymptotic smallg behaviour, showing the relation between the avoided crossings and the failure of the usual perturbation series.

In this paper we perform the semiclassical analysis of a pair of resonances in the case of a quasi-symmetrical unstable double well. We consider two kinds of asymmetric perturbations: one supported in the infinite external well, the other one of the Stark kind. We prove that the first perturbation is able to localize each state inside one of the internal wells so that we have linear Stark effect and vanishing of the splitting at the crossing point of the two resonances. This phenomenon is critical in the ratio between the internal and external barrier lengths, and the critical value of the ratio is close to two. Possible applications to the molecular structure and to the vanishing of the inversion frequency are briefly discussed.

We consider the semiclassical Stark effect for a family of asymmetric unstable double well models and we study the crossing and anticrossing of the field dependent resonances in the complex field plal}e. We prove that a Bender-Wu type singularity crosses the real axis when the internal barrier is nearly twice "larger" than the external one and the beating period is close to the shorter life-time of the resonances. At this critical point we have the anticrossing-crossing transition and for larger instability we have the single well localization.

In the framework of regular perturbation theory we discuss the weak field crossing behavior of the resonances in double ladder (and double well) Stark problems. We get a precise condition for the anticrossing in terms of the Agmon length of the Zener barriers. This condition has a simple physical meaning: as a general rule we have anticrossing and beating effect if the lifetime of the system is larger than the beating period. Of course, we have full delocalization in the anticrossing case only.

LetHB be any fixed one-dimensional Bloch Hamiltonian with only the firstm gaps open andHF=HB+Fx be the corresponding Stark Hamiltonian. For any positiveF small enoughHF has onlym ladders of sharp resonances given by the analytic translation method, the decoupled band approximation and the regular perturbation theroy. This way, the Wannier conjecture becomes a definite regular perturbation theory for the Stark ladders as eigenvalues of the translated Hamiltonian.

We consider the Bloch problems with a finite number of open gaps and we prove, for any external weak enough electric field, the existence of a finite number of Stark ladders, given by complex translation, decoupled band approximation and regular perturbation theory.

We consider the one-dimensional Dirac equation for the harmonic oscillator and the associated second order separated operators giving the resonances of the problem by complex dilation. The same operators have unique extensions as closed PT-symmetric operators defining infinite positive energy levels converging to the Schroedinger ones as c tends to infinity. Such energy levels and their eigenfunctions give directly a definite choice of metastable states of the problem. Precise numerical computations shows that these levels coincide with the positions of the resonances up to the order of the width. Similar results are found for the Klein-Gordon oscillators, and in this case there is an infinite number of dynamics and the eigenvalues and eigenvectors of the PT-symmetric operators give metastable states for each dynamics., Comment: 13 pages, 2 figures

Following Avron (1982) the authors consider the Stark effect for Bloch electrons in the case of a finite number of gaps. They prove that the ladders of resonances are given by the Wannier decoupled-band approximation and the perturbation theory. The Fermi golden rule yields the width behaviour of Buslaev and Dmitirieva (1990).

We study the top resonance states of the cubic anharmonic oscillator H(β) = p2 + x2 + i√βx3 for β on the complex plane cut on the negative semiaxis. In particular, by the semiclassical scaling and semiclassical methods, we prove that the top resonance states do not belong to L2(R).

By slightly restricting the conditions given by Herbst and Howland, we prove the existence of resonances in the Stark effect of disordered systems (and atomic crystals) for large atomic mean distance. In the crystal case the ladders of resonances have the Wannier behavior for small complex field.

The 1/R perturbation series for H 2 + has a complex Borel sum whose imaginary part determines the asymptotics of the perturbed energy coefficients E ( N ) . The full asymptotic expansion for the energy includes complex, exponentially small terms: E ( R ) ˜ Σ E ( N ) ( 2 R ) − N + e − R / n Σ a ( N ) ( 2 R ) - N + e - 2 R / n [ Σ d ( N ) ( 2 R ) - N + log R terms ] ± i e - 2 R / n Σ c ( N ) ( 2 R ) - N + … . The explicit imaginary terms cancel the implicit imaginary part of the Borel sum. An exact relation between the double-well gap series, . exp ( - R / n ) Σ a ( N ) ( 2 R ) - N and the i exp(−2 R / n ) series is derived. PACS numbers: 31.15. +q, 03.65.−w, 31.10, +z

We discuss a close connection between the formula of Banks, Bender, and Wu for the asymptotics of the Rayleigh-Schrodinger coefficients of the two-dimensional rotationally symmetric anharmonic oscillator and the behavior of resonances of the hydrogen Stark problem in two regimes: small field (Oppenheimer's formula) and large field (where we obtain the new results arg E → −π/3, ∣E∣ ∼α[F(lnF)^(2/3) for F, the electric field strength, going to infinity). We also announce a rigorous proof of Bender-Wu-type formulas.

Existence of resonances in hydrogen Stark effect is proved. It is also proved that the divergent time-independent perturbation expansions are Borel summable to the resonances, and a simple application of the Borel-Pade method for computing their position and width is indicated.

The energy of any bound state of the hydrogen molecule ion ${\mathrm{H}}_{2}$${\mathrm{}}^{+}$ has an expansion in inverse powers of the internuclear distance R of the form Rayleigh-Schr\"odinger perturbation theory (RSPT) gives the coefficients ${E}^{(N)}$ but is otherwise unable to treat the exponentially small series, which in part are characteristic of the double-well aspect of ${\mathrm{H}}_{2}$${\mathrm{}}^{+}$. (Here n denotes the hydrogenic principal quantum number.) We develop a quasisemiclassical method for solving the Schr\"odinger equation that gives all the exponentially small subseries.The RSPT series diverges: for the ground state ${E}^{(N)}$\ensuremath{\sim}-(N+1)!/${e}^{2}$ for large N. The ${E}^{(N)}$ asymptotics are governed via a dispersion relation by the imaginary ${e}^{\mathrm{\ensuremath{-}}2R/n}$ series, which itself is given by the square of the ${e}^{\mathrm{\ensuremath{-}}R/n}$ series times a ``normalization integral.'' That the expansion itself contains imaginary terms might seem inconsistent with the reality of the ${\mathrm{H}}_{2}$${\mathrm{}}^{+}$ eigenvalues. In fact, the RSPT series is Borel summable for R complex. The Borel sum has a cut on the real R axis, and its limit from above or below the positive R axis is complex. The imaginary ${e}^{\mathrm{\ensuremath{-}}2R/n}$ (and higher) series consist of just the counterterms to cancel the imaginary part of the Borel sum.Extensive numerical examples are given. Of interest is a weak (down by a factor ${N}^{\mathrm{\ensuremath{-}}6}$) alternating-sign contribution to ${E}^{(N)}$, which is uncovered both theoretically and numerically. Also of interest is the identification of the Borel sum of the RSPT series with nonphysical boundary conditions. This too is illustrated both theoretically and numerically.

We give a rigorous proof of the analyticity of the eigenvalues of the double-well Schrodinger operators and of the associated resonances. We specialize the Rayleigh-Schrodinger perturbation theory to such problems, obtaining an expression for the complex perturbation series uniquely related to the eigenvalues through a summation method. By an approximation we obtain new series expansions directly computable, still summable, which, in the case of the Herbst-Simon model, can be given in an explicit form.

Existence of resonances is proved for the time‐independent Klein–Gordon equation describing the interaction of a charged particle with an external uniform field of small strength F in addition to the Coulomb attraction. It is further shown that the resonances reduce to the exactly known bound states of the problem as F→0, and to the resonances of the nonrelativistic Stark effect as c→∞.

An indeterminacy criterion is proven for the moment problem associated with the coefficients of a Borel summable power series of Stieltjes type which diverge faster than (2n) !. As an application we show that the Stieltjes type continued fraction corresponding to the Rayleigh–Schrodinger perturbation expansions for the energy eigenvalues of the anharmonic oscillators (x2(m+1) and in any finite number of dimensions) does not converge to the eigenvalues if m≳2. In particular, this implies the nonconvergence of the Pade approximants to the eigenvalues of p2+x2+λx2(m+1) if m≳2.

The authors prove that every fixed m resonance eigenfunction of hydrogen in a constant electric field is a finite sum of functions which are products in squared parabolic coordinates. This shows that the standard ansatz yields all resonances.

We discuss a close connection between the formula of Banks, Bender, and Wu for the asymptotics of the Rayleigh-Schr\"odinger coefficients of the two-dimensional rotationally symmetric anharmonic oscillator and the behavior of resonances of the hydrogen Stark problem in two regimes: small field (Oppenheimer's formula) and large field (where we obtain the new results $\mathrm{arg}E\ensuremath{\rightarrow}\ensuremath{-}\frac{\ensuremath{\pi}}{3}$, $|E|\ensuremath{\sim}\ensuremath{\alpha}{[F(\mathrm{ln}F)]}^{\frac{2}{3}}$ for $F$, the electric field strength, going to infinity). We also announce a rigorous proof of Bender-Wu-type formulas.

The Stieltjes type continued fraction (i.e., any diagonal Pade approximants sequence) of the perturbation series for the resonances of the so‐called one‐dimensional Stark effect converges to the resonances.

We complete and correct some proofs of an earlier paper on distributional Borel summability and we add an application which can be useful in the discussion of semiclassical problems.

The resonance energies for the hydrogen atom in an electric field, both the real and imaginary parts, have been calculated together from the real Rayleigh-Schr\"odinger perturbation series by Borel summation. Pad\'e approximants were used to evaluate the Borel transform. The numerical results compare well with values obtained by the complex-coordinate variational method and by sequential use of Pad\'e approximants.

Generalizing earlier results on the Hydrogen case it is proved, through a dilation analyticity technique different from the canonical one, that the action of a weak electric field shifts the isolated eigenvalues of any atomic system into resonances of the Stark effect, uniquely determined by the perturbation series through the Borel summation method.

It is proved that the action of a weak electric field shifts the eigenvalues of the Hydrogen atom into resonances of the Stark effect, uniquely determined by the perturbation series through the Borel method. This is obtained by combining the Balslev-Combes technique of analytic dilatations with Simon's results on anharmonic oscillators.

The authors prove the existence and give the semiclassical magnetic asymptotics of the magnetic bands in superlattices. They use the Wannier single-band approximation which leads to a dual semiclassical Bloch model with a band function as a potential. A picture of x-dependent bands suggests exponentially small magnetic gap widths as given by the beating effect of a Zener double well.

It is proved that the Stark-Wannier states, as functions of the electric field, are analytic in a disc tangential to the real axis at the origin, with asymptotic expansion to the second order which coincides with the Wannier approximation up to the first order.

In this paper we discuss the splitting instability in a periodically driven double well. The physical motivation of this study comes from the relevance of the concept of molecular structure in chemistry, but the model could be tested directly by means of heterostructures and microwaves. Let us recall the old problem of the explanation of the molecular localization ~ML! hypothesis, successfully used in chemistry as the concept of molecular structure, in the rigorous quantummechanics ~QM! framework @1#. QM requires that the probability distribution of stationary states have the same symmetry of the Hamiltonian, in marked contrast with the ML requirement. The qualitative explanation of this apparent contradiction is simple: since the molecule is not an isolated system, its states cannot be stationary @2#. The main problem is the understanding of the quantitative aspect of the phenomenon, as it results from the following question by Woolley @1#: ‘‘Why should the general quantum theory describing energy eigenstates turn out to be of such little use in chemistry, or put in another way, why should transitions out of the time-dependent molecular quantum states which empirically appear to be an essential ingredient of any useful quantum chemistry, be so slow?’’ Although it is generally accepted that the phenomenon should be explained by means of decoherence arguments @3#, it is also clear that explicit models are needed. Thus, by means of the study of an explicit model, we want to point out the role of instability in the localization phenomenon. Indeed, we expect the existence of metastable states in perturbed systems and we want to study the smallness of the interaction between a pair of such states for large instability. Let us consider the case of the ammonia molecule NH3, where the model for the motion of the nitrogen atom N is a double well with a large internal barrier @1#. In this model we have the pyramidal shape of the molecule ~molecular structure! if the state is localized in one of the wells. The inversion line of the molecular microwave emission gives the energy splitting of the stationary states. Experiments on ammonia gas show that the localization and the inversion line are dependent on pressure. In particular, the localization probability increases and the inversion line broadens and decreases as the pressure increases, giving the so-called redshift ~RS! effect @2#. Some previous explicit models @4# which are able to explain ML, are autonomous, i.e., they make use of timeindependent potentials. In particular in a recent paper @5# ,b y using an unstable autonomous model, both ML and RS are obtained. In the present paper we use a nonautonomous model ~time-dependent potential! so that the instability caused by the molecular, collisions is represented in a more realistic way. In particular, our model consists of a double-well potential with a time-dependent perturbation simulating the dynamical influence of the environment on the ammonia molecule, i.e., the collisions with the other molecules of the gas, where the collision frequency is related to the pressure. Let us notice that the present model is more physical than the previous ones for the reasons stated above, but it is still simplified. One simplification is the choice of a perturbation periodic in time. This choice is technical and is due to the recent improvement of methods for handling periodic problems. We point out that the classical resonance effect between different frequencies are not relevant for the results. In any case the results give an a posteriori justification of the model. Since the first~but not the second! order perturbation term vanishes, we set the perturbation of the same order of the square root of the splitting. We consider the large internal barrier regime, so that both the splitting and the perturbation are exponentially small. This choice of parameters is similar to previous ones, and allows us to apply the same comparison with experiments given by Claverie and Jona Lasinio @4#, although in that case the frequency parameter was absent. If the periodic perturbation is strong enough, and the Fourier coefficients are slowly decreasing for increasing index, we have localization for a frequency larger than a critical value but smaller than a large value. This upper bound should be related to the simplification of the model given by the periodicity of the time behavior. Since the model is linear, we have no spontaneous symmetry breaking, so that the perturbation is asymmetric, but not too much, in order to have the RS. We control the asymmetry of the perturbation by varying the coefficient of the time independent perturbation.