Modes transformation for a Schroedinger type equation: avoided and unavoidable level crossings

Modes transformation for a Schroedinger type equation: avoided and unavoidable level crossings

Ignat Fialkovsky ifialk@gmail.com CMCC-Universidade Federal do ABC, Santo André, S.P., Brazil    Maria Perel St. Petersburg State University, 7/9 Universitetskaya nab., St. Petersburg 199034, Russia
July 20, 2019
Abstract

An asymptotic approach for a Schroedinger type equation with a non selfadjoint slowly varying Hamiltonian of a special type is developed. The Hamiltonian is assumed to be the result of a small perturbation of an operator with a twofold degeneracy (turning) point, which can be diagonalized at this point. The non-adiabatic transformation of modes is studied in the case where two small parameters are dependent: the parameter characterizing an order of the perturbation is a square root of the adiabatic parameter. The perturbation of the Hamiltonian produces a close pair of simple degeneracy points. Two regimes of mode transformation for the Schroedinder type equation are identified: avoided crossing of eigenvalues, corresponding to complex degeneracy points, and an explicit unavoidable crossing (with real degeneracy points).

Both cases are treated by a method of matched asymptotic expansions in the context of a unifying approach. An asymptotic expansion of the solution near a crossing point containing the parabolic cylinder functions is constructed, and the transition matrix connecting the coefficients of adiabatic modes to the left and to the right of the degeneracy point is derived.

Results are illustrated by an example: fermion scattering governed by the Dirac equation.

thanks: Corresponding author, electronic mail: m.perel@spbu.ru

I Introduction

The present paper is dedicated to investigation of the Schroedinger type equation:

(1)

where is a small parameter and the Hamiltonian may be non selfadjoint. Born and Fock born () showed in 1928 that under certain conditions on the Hamiltonian exact solution can be approximated by a so called adiabatic mode with an error . The latter is defined as

(2)

where is an eigenvalue of of multiplicity one,

(3)

is a corresponding eigenfunction with appropriate choice of the phase. This fact became known as the Quantum Adiabatic theorem. Numerous papers have been devoted to generalizations of this theorem, see reviews in teufel2003adiabatic () and hagedorn2007born (). The study of adiabatic approximation of solutions of (1) was primarily stimulated by problems of molecular dynamics.

Special attention was paid to cases, where adiabatic approximation fails because of local degeneration (or almost degeneration) of two eigenvalues

(4)

which induces non-adiabatic transitions between adiabatic modes taking place near . Two small independent parameters rule the problem: , which is an adiabatic parameter and the parameter characterizing the perturbation, which determines the discrepancy between almost degenerating eigenvalues. The subject of study in such cases is the transition matrix which connects adiabatic modes at different sides of the degeneracy point. It was first studied by Landau landau1965collected (), Zener zener_32 (). The refined results were obtained by Stueckelberg stueckelberg_32 () shortly afterwards. However, mathematically rigorous results came only much later. The difficulties arose, in particular, for such a relation between small parameters that enables exponentially small transitions between modes. The account of exponentially small transitions between modes, which are given by expansions in powers of small parameter, is a complicated problem. The first proof of the Landau-Zener formula in the simplest case of dependent small parameters providing the entries of the transition matrix of order unity was given in hagedorn1991proof (). In the case of arbitrary relation between small parameters the justification of formulas were given in joye1994proof (). We assume in the present paper that the small parameters are chosen dependent of each other to avoid difficulties of the exponentially small transitions.

The asymptotic results for Eq. 1 with a selfadjoint Hamiltonian are termed time-adiabatic theory, or simply adiabatics, e.g. teufel2003adiabatic (). However, a small parameter may also exist in stationary Schroedinger equation and the relevant asymptotic methods are then named semiclassical or space-adiabatic methods. Generally speaking, the stationary equation is a multidimensional one and is not of the form (1). However some semiclassical problems, which can be reduced to (1) with the Hamiltonian satisfying conditions listed below, can be studied in the framework of our formalism. Differential equations describing irregular waveguide problems encountered in Mechanics and Electrodynamics have one variable selected, it is the variable along the axis of a waveguide, it should be . The differential equations are not normally represented in the form (1) and contain the second derivatives in , but can be reduced to the form (1). For slow enough inhomogeneity, they have solutions in the form of (2), named adiabatic. As examples of such problems, we can mention the problems of wave propagation in elastic or electromagnetic waveguides and in the Timoshenko beam. Adiabatic solutions fail if the phase velocities of two modes have a local point of the degeneration. In the field of wave propagation, the problems of the description of interaction of modes or transformation of modes or coupling of modes caused by the degeneracy point are analogs of the problems of non-adiabatic transitions in Quantum Mechanics. Our aim here, in particular, is to generalize results elaborated in Quantum Mechanics to aforementioned problems, i.e. we suggest to study physical problems with a slow variation of parameters in one variable but without dissipation as a particular case of (1) but with a non selfadjoint Hamiltonian operator.

To be specific, we assume that the non-selfadjoint Hamiltonian may be factorized

(5)

with selfadjoint operators and (all the conditions on are scrutinized in Section II.1). Then Eq. 1 is reduced to the form

(6)

In this case, for construction of adiabatic modes (2) we can use the same , , which now can be considered as eigenvalues and eigenfunctions of a linear selfadjoint operator pencil

(7)

Basing on (6) we can study problems of mechanics or electrodynamics by analogy with time-adiabatic methods of Quantum mechanics. The degeneracy points (often named the turning points) in the semiclassical theory and points of (avoided) crossing of energy levels are then all treated on the same footing.

At first time, the equation (6) was applied to an anisotropic electromagnetic waveguide in the monography felsen1994radiation (), where the operator was a matrix differential operator with respect to variables transverse to . Equation (6) has been already used by the some of us in several applied problems dealing with asymptotics near degeneracy points of different types: for Maxwell equations in the Earth-ionosphere waveguide perel1990overexcitation () (with a non-invertible ), Maxwell equations in curvilinear coordinates near the smooth boundary of the convex body And-Zai-Per (), the Timoshenko beam equations perel2000resonance (), and elastic wave equations perel2005asymptotic (). It was also applied to investigation of liquid crystals Aksenova (). The Dirac equation for quasiparticles in graphene is given in the form of (6) in Section III. Moreover, equation (6) may be used in the framework of -symmetric quantum mechanics with playing the role of -symmetry operator, see Bender2007 () and reference therein.

One of the advantages of the formulation of all these problems in form (6) is that it allows for a clear and intuitive physical interpretation of processes of non-adiabatic transitions. Indeed, (6) possesses a conservation law

(8)

where, generally speaking, the quadratic form is not positively defined. For problems of wave propagation, it has a meaning of conservation of the time-averaged flux of energy, see the literature cited above. Then all modes can be divided into two groups depending on the sign of this constant, ones carrying energy to a degeneracy point, and others away from it. Such interpretation of the direction of mode propagation often does not coincide with that based on the sign of the phase velocity of modes.

Figure 1: Eigenvalues vs. (in arbitrary units.) The non perturbed eigenvalues are shown by thin lines, while the exact ones by thick lines. Dashed/solid lines (also blue/green on-line) distinguish modes number 1 and 2. \subrefcompl_TP Avoided crossing: exact eigenvalues have two complex turning points. \subrefreal_TP Unavoidable crossing: exact eigenvalues have two real turning points.

The problems of constructing of asymptotic solutions as in the presence of degeneracy points for the ODEs of second order and their generalizations to systems of ODEs has been studied in many papers; see the books and reviews berry72 (), fedoryuk2012asymptotic (), olver2014asymptotics (), slavyanov1996asymptotic (), wasow2012linear () and references therein. All the methods applied to investigation of degeneracy points can be divided into three groups. First ones are the uniform methods, which work both in the vicinity of degeneracy points and away from them; see Buldyrev-Slavyanov (), Cherry (), langer1937connection (). The second group comprises methods based on the Fourier representation of the unknown function, such as the Maslov method kucherenko1974asymptotics (), maslov2001semi () and the microlocal analysis verdier_99 (). Methods based on local considerations in the vicinity of the degeneracy point with further matching of local solutions with adiabatic ones constitute the third group. There is the method of matched asymptotic expansions bender_book (); wasow2012linear (), also called the boundary layer method boundl_babich_79 (). In the field of nonadiabatic transitions it was applied in hagedorn1991proof (). This is the method we apply in the present paper. The same method was used in our previous papers perel1990overexcitation (), perel2000resonance (), perel2005asymptotic (), And-Zai-Per () (although it was not always written explicitly), where different asymptotic situations for waveguide problems were considered.

Methods of distinguishing different types of asymptotic solutions also vary greatly. For simplest differential equation of the second order, the asymptotic solutions near the degeneracy point depend on the type of the -dependence of eigenvalues. For a system of ordinary differential equations ( is a matrix) and for more general cases, where is a differential operator, some information about eigenfunctions is also necessary. To build the classification of asymptotic cases, different approaches have been suggested in the literature. For equation in the form of (1) with a non selfadjoint Hamiltonian, the most typical cases were analyzed in buslaev_grigis_01 (); grinina2000solution () in assumption that a so called model equation is known. However, a recipe of reducing a concrete problem to a model equation was not suggested. The type of singularity of the adiabatic coupling function should be prescribed in berry1993universal (); betz2005precise () for classification of asymptotic cases. A detailed classification of points of level crossing in terms accepted in molecular dynamics is given in hagedorn1994molecular (). In our opinion, the reduction of a problem to (6) enables an easier classification of asymptotic cases in terms of the matrix elements of and . One of the typical cases is considered in the present paper.

We assume that the operator is the result of a perturbation of an operator with a single degeneracy point, which is a point of crossing of two eigenvalues. It is well-known that if is an identity matrix, the crossing of eigenvalues turns under perturbation to an avoided crossing. It is a typical case for non-adiabatic transitions described by Landau-Zener formula. However introducing we expanded the class of problems. The connection formulas should be revisited, and transition matrix should be generalized. A perturbation of operator with two eigenvalues crossing may cause now not only avoided crossing but unavoidable crossing too. Moreover the knowledge of eigenvalues behavior (for example, crossing or avoided crossing) is not sufficient for unique determination of the transition matrix. It is crucial whether there are two independent eigenfunctions at (in other words if the restriction of the operator to invariant subspace corresponding to is diagonalizable) or an eigenfunction and its associated. Particular cases of the treatment of the degeneracy points with linear independent eigenfunctions were presented in hagedorn1989adiabatic (); perel2000resonance (); And-Zai-Per (), and ones with linear dependent eigenfunctions in perel2005asymptotic (). Here we consider the former case.

We construct an inner and an outer formal asymptotic expansions, show that the domains of their validity intersect, and match them to find the transition matrix. The obtained expansions are asymptotic solutions, see subsection IV.1.2, IV.2.1. The outer expansion is the adiabatic one, which is well known for the selfadjoint Hamiltonians and was discussed for the non selfadjoint ones in fedoryuk2012asymptotic () for linear systems of ordinary equations, and in nenciu1992adiabatic () for systems with large dissipation. The equations for the leading term of the inner expansion can be interpreted as the model equation of buslaev_grigis_01 (); grinina2000solution (). We show also that under the assumptions listed in Section II.1 these expansions are formal solutions of the equation (6).

Our main aim is to suggest studying asymptotics of (6) for problems of mode transformation in waveguide problems, some of which were already studied by us. We give here one new example of the problem for the Dirac equation in the simplest case, where transverse to direction of propagation momentum is dependent on the adiabatic parameter. We intend to give example of elastic waveguides in the further paper.

Ii Statement of the problem and main result

ii.1 Statement of the problem

We study asymptotic solutions of the following equation:

(9)

in the Hilbert space. We assume that , , and its inverse, are operators that are selfadjoint for real . The latter two of them are supposed to be bounded , as well as . For simplicity, we consider independent on . (This condition may be removed but it is satisfied in all practical cases known to the authors.)

We introduce two crucial assumptions concerning eigenvalues and eigenfunctions of the spectral problem

(10)

and an additional third assumption, which can be relaxed. Note that eigenvalues of (10) can be both real and complex, while eigenfunctions are -orthogonal, see Appendix B for more details..

  1. Two eigenvalues of (10), and , are degenerating at a single point and they stay real on the whole interval. The point is a degeneracy point of a crossing type, i.e.,

    (11)

    Here does not depend on , as , and both and are separated from the rest of the spectrum of (10) (if any) with a gap independent on .

  2. The operator is diagonalizable in the invariant subspace corresponding to

  3. Operator families and are holomorphic in some domain including an interval containing .

    We will use the following consequences of the third assumption (see, for example, kato2013perturbation ()):

    1. Operators and can be expanded in the Taylor series with bounded operator coefficients.

    2. The eigenvalues are holomorphic and can be expanded in the Taylor series.

    3. The eigenfunctions are holomorphic and can be defined at by continuity, the obtained eigenfunctions , are -orthogonal:

      (12)

Our aim is to construct asymptotic outer (inner) expansions outside (inside) the neighborhood of the point . The outer expansion we also call adiabatic one or adiabatic mode. After that we seek the transition matrix connecting two sets of adiabatic solutions at both sides of the point .

We estimate the terms of obtained expansions and find conditions for them to have an asymptotic nature. The straightforward consequence of these estimates is the fact that obtained expansions are asymptotic solutions. We call an asymptotic solution of order in powers of a small parameter , on an interval of if it satisfies (9) up to the terms of order , i.e.,

(13)

uniformly with respect to . If a partial sum of an asymptotic series is an asymptotic solution of order for any , then we call the series an asymptotic solution.

ii.2 Main result

Our mains result is based on construction of the adiabatic solution of (9), which is performed in close analogy to the case of the self-adjoint Schroedinger equation (1), see Section IV.1. Our construction differs from the standard one by two facts. First, is itself a function of the small parameter , i.e., , see (9). Eigenfunctions and eigenvalues of denoted and , respectively, also depend on and are found by the perturbation method from the eigenfunctions and eigenvalues of in Sections C.1, C.2. Therefore we find asymptotic solution as expansion in terms of but not . Second feature is the presence of in the right-hand side of (9), which permits to express the eigenelements of the adjoint problem as , .

The leading order term (the asymptotic solution of order ) reads

(14)

where , are arbitrary constants. Adiabatic modes, generally speaking, differ on both sides of the point and the subscript indicates the side, where the mode is constructed. We note that the leading term (14) depends on non-perturbed eigenfunctions , see (10), and perturbed eigenvalues . The - normalization factor and the diagonal conversion coefficient are defined as

(15)
(16)

where the scalar product is that inherent from the Hilbert space, in which acts. Coefficients are connected with the Berry phase.

The choice of the lower limits of integration in (14) influences the arbitrary constants. We take these limits equal to real parts of the degeneracy points for the perturbed operator , which are as follows:

(17)

where

(18)

See details about the degeneracy points in Section C.2. Such choice of the lower limit is convenient although adiabatic solutions themselves are not applicable in the vicinity of , as we will see later. However, calculating the integral of the eigenvalue over this region is possible, and to this end we find the expansion of eigenvalues in Appendix C. The sum of the first terms of expansions of eigenvalues contributing the leading order term of the adiabatic mode is denoted in the further text.

After these preliminary remarks and definitions we are able to formulate our main result, which concerns the matching of adiabatic modes.

Theorem 1

Let , , and eigenvalues and eigenvectors of meet the conditions stated in Section II.1. Then the equation (9) has an asymptotic solution of order 0 that has the following representations in terms of adiabatic modes to the left and to the right of the turning point, ,

(19)

, and the complex constants , are connected by the transition matrix

(20)

The dimensionless parameter governing the result is as follows

(21)

If , (20) coincides with the Landau-Zener formula . However for , (20) gives a matrix, which does not arise in considerations of self-adjoint Hamiltonians. These two cases describe two qualitatively distinct cases of eigenvalues asymptotics near : avoided crossing (with complex turning points, Fig. 1) and unavoidable crossing (with classical turning points, Fig. 1). Two distinct physical processes correspond to these cases.

The transition matrix is dependent on arbitrary constants . The form (20) of corresponds to the following choice of :

(22)
(23)

We call the adiabatic modes with such choice of the canonical modes.

It is important to note that although, generally speaking, , , may vanish in degeneracy points, the conditions 1)-2) from Section II.1 ensure that it does not happen. It is shown in Appendix B.

Iii Example. Dirac equation in dimensions

The massless Dirac equation in - dimensional space with the external potential for a stationary wave function reads

(24)

where is a pair of the Pauli matrices, ; for simplicity we put the Fermi velocity equal to one, .

One of the physical problems described by (24) is the electron scattering on an electrostatic potential barrier inside graphene katsnelson2007graphene (). If the external potential depends on one variable only, , we can separate the -derivative in the equation (24), and use the Fourier transform for the -dependence

Then, the case, where the electrons are incident almost perpendicularly to the potential barrier can be studied reijnders2013semiclassical (); zalipPRB.91 (). The interest in this case was invoked by the so-called Klein paradox klein1929reflexion () present in graphene, which is characterized by the unit probability of tunneling through the barrier, for applications in graphene, see katsnelson2006chiral ().

Close to normal scattering of electrons on a potential is characterized by small values of . In particular, if we assume that is of order , , , then we write (24) in the form (9)

(25)

with the Hermitian operators

(26)

The operators , , and act in the Hilbert space The conservation law (8) in this case corresponds to the conservation of the -component of the electron current Bogoliubov (), due to the fact that ,

(27)

and can be both positive and negative.

The generalized eigenvalue problem is solved by

(28)

and the normalization of modes reflects the electric charge current transferred by these modes, having opposite signs. So, in this case we have

(29)

The degeneracy points are those where , i.e.,

(30)

Supposing that on the interval of interest there is only one such point , we conclude that all the assumptions of the Section II are satisfied if the potential has a non vanishing first derivative at , .

For constructing the canonical modes, we calculate the matrix entries

(31)

and other necessary ingredients

(32)
(33)
(34)

We also note that in this case the eigenvalues are symmetric, .

The canonical modes are

(35)

The eigenvalues of are

(36)

Outside the vicinity of the crossing point , we have the expansion:

(37)

If is in the vicinity of the turning points of order inclusively, we get

(38)

. The transition matrix is given by (123) with from above.

Thus, our method can be applied readily to the description of the nearly normal scattering of electrons on an external potential. This case was investigated successfully in reijnders2013semiclassical (); zalipPRB.91 () by a different method, and it is a straightforward task to verify that our result for the transition matrix is in agreement with theirs. We have an assumption that is of order one, but result is valid for any .

Iv Asymptotic derivation

iv.1 Adiabatic (outer) expansion

To proceed with resolving the connection problem via the method of matched asymptotic expansions, in this section we construct adiabatic, or outer, expansions away from the degeneracy points of the original equation

(39)

iv.1.1 Construction of adiabatic expansion

We search for an adiabatic expansion of a solution of (39) in the form of

(40)

both and are given by formal series in powers of , respecting the order of magnitude of the perturbation

(41)
(42)

The standard adiabatic approximation contains only an expansion of , while we also introduce the second expansion in the phase factor. We are entitled to impose an additional condition

(43)

It fixes the arbitrariness of possible multiplication of the whole ansatz (40) by an arbitrary series in . This condition makes the representation (40)–(42) unique. As we will see in the sequel, it also guarantees that the amplitude factor depends on the local properties of the medium only, while all the integral (nonlocal) ones are contained in the phase factor.

Following the perturbation method, we insert (41), (42) into (39). Equating the coefficients at equal powers of , we obtain a sequence of equations

(44)
(45)
(46)
(47)

Aiming at constructing the ‘first’ mode, we choose as the solution of the principal order equation (44) the first eigenfunction of and corresponding eigenvalue

(48)

We solve equations (45), (46) and (47) step by step. All these equations are solvable if their right-hand sides are orthogonal to the solution of the homogeneous equation (44). This condition with account of (43) and (48) yields

(49)
(50)

We recall here that the eigenvalues , , are assumed real.

Taking into account (43), we write the higher order approximations in the form

(51)

where is a scalar function of , and is -orthogonal to and

(52)

In the following sections we consider the degeneracy between and , so we separate the term proportional to , because it contains the main singularity where is close to , see Section IV.1.2. To find , we substitute (51) into (47), calculate its scalar product with , take into account (52) and the orthogonality properties of eigenfunctions, see Appendix B. We find

(53)
(54)

To complete the construction of the adiabatic solution, we need to deduce the -orthogonal component , . However, as we shall see in the next section, it does not influence the transformation of the modes, at least in the principal order. Thus, we only need to show the possibility of its determination to prove that the recurrent system (44)-(47) can indeed be solved step by step. We rewrite equation (47) as follows

(55)

where is the right-hand side of (47). In view of the assumption of the presence of a finite gap between and the rest of the spectrum and the fact that is -orthogonal to , , equation (55) has a single solution , see Appendix A, (property 5). Thus, all the terms of the formal series can be constructed.

In the case of a purely discrete spectrum of , the first order approximation has the form of

(56)

where the summation ranges all the modes except for the first two.

In the approximation, we have

(57)

where is a constant phase factor, reads

(58)

it comprises the first terms of expansion of the eigenvalue in powers of , which is valid in the area of applicability of adiabatic modes and which is found in Appendix C in (165). The coefficients were defined in (16). The imaginary part of is the Berry phase Berry-phase (), while its real part fixes the normalization of the whole solution. Indeed, for smooth we find

(59)

and thus upon integration and exponentiation in (57), at the upper limit of integration it gives exactly and a constant at the lower one, so that the principal order of the adiabatic mode (40) is

(60)

Thus, it can be made -normalized, , assuming the eigenfunctions are normalized in . The overall sign of the normalization factor , however, cannot be fixed and represents intrinsic properties of solution. It is shown in Appendix B(property 4) that is not equal to zero.

The structure of the amplitude guarantees that under the transformation , , it maps in the same way: . And at the same time, the Berry phase develops an opposite contribution under the same phase shift if is not constant, ,

(61)

So, the only ambiguity left in definition of is an overall constant phase factor. It can be interpreted purely in terms of the lower limit of integration , but for simplicity of further analysis we introduced an additional parameter .

We call the formal series constructed here the adiabatic expansion or adiabatics. The principal term of the expansion is named the adiabatic approximation or adiabatic mode. The other solution, is obtained by simply interchanging the indices .

iv.1.2 The validity region of adiabatic solutions and the slow variable

All the approximations , , are of order one if is of order one. It follows from conditions in Section II.1. Operators , are bounded and norms of eigenfunctions , and their derivatives are bounded. Norms of are also bounded as it follows from Appendix B.

According to (53), (54) the higher order terms, , , in expansion of , contain the singularity near the degeneracy point , where . We study here how rapidly the degree of singularity increases with the growth of the order of approximation .

The first important fact is that main singularity of is contained in the term The second necessary fact is that , , and are of order unity, as follows from our assumptions. By induction on it is easy to show that for it holds

(62)
(63)

Therefore, the higher terms of the expansions (41), (42) can be estimated near the degeneracy point as

(64)

They stay small and thus guarantee the asymptotic nature of expansions (57) for

(65)

for any . This suggests to seek the resonant or inner expansion of (39) in the vicinity of in terms of the slow, or stretched, variable

(66)

We note that the constructed asymptotic expansion is an asymptotic solution. To obtain an asymptotic solution of -th order, we need to take partial sums of terms of expansions (41), (42) and insert these sums into (40). Then we should substitute the result into the equation (39) and estimate the discrepancy. The terms of order up to are cancelled because of (44)-(47), and the discrepancy comprises terms from the right-hand side of equations of order from till , which can be estimated as , as it follows from the reasonings given above.

We can now analyze the structure of the outer asymptotic expansion in terms of for further constructing ansatz of inner expansion. Expanding (50) and (54) for small we deduce

(67)
(68)
(69)
(70)

where and , are constant scalars and vectors correspondingly. After inserting (67)-(70) into (41), (42), we can collect the terms with the same powers of . Principal singularities of every term of the adiabatic expansion, i.e. the highest order terms in , contribute to the first term of the inner expansion. The same is correct for the second terms and so on. We will find thus

(71)
(72)

Upon integration over (as required in (40)) the first term and the first parenthesis of (71) are non small for any values of , while the third and all consecutive are negligible if

(73)

This defines the area of asymptotic nature of the expansion in term of , and thus also the applicability area of the inner solution (since such expansion is unique!) Under this condition we can expand the exponent containing the third and higher terms of (71), and obtain the following ansatz of the inner expansion:

(74)

The functions , and , will be found in Section IV.2.

iv.1.3 Canonical modes

For the simplification of the further matching procedure and a simpler form of the transition matrix, we shall now introduce canonical modes by fixing arbitrary factors in the principal term (60). First of all, we note that in its phase

(75)

we can actually choose the lower limit of integration differently for each of terms, for it amounts only to a overall constant phase factor in . As it will be evident from Section IV.3.2, the most simple form of the transition matrix is obtained for the canonical modes defined as

(76)