Pseudospectra of the Schrödinger operator with a discontinuous complex potential

Pseudospectra of the Schroedinger operator with a discontinuous complex potential

Abstract

We study spectral properties of the Schrödinger operator with an imaginary sign potential on the real line. By constructing the resolvent kernel, we show that the pseudospectra of this operator are highly non-trivial, because of a blow-up of the resolvent at infinity. Furthermore, we derive estimates on the location of eigenvalues of the operator perturbed by complex potentials. The overall analysis demonstrates striking differences with respect to the weak-coupling behaviour of the Laplacian.

  • pseudospectra, non-self-adjointness, Schrödinger operators, discontinuous potential, weak coupling, Birman-Schwinger principle

  • 34L15, 47A10, 47B44, 81Q12

1 Introduction

Extensive work has been done recently in understanding the spectral properties of non-self-adjoint operators through the concept of pseudospectrum. Referring to by now classical monographs by Trefethen and Embree [33] and Davies [8], we define the pseudospectrum of an operator  in a Hilbert space  to be the collection of sets

(1.1)

parametrised by , where is the operator norm of . If  is self-adjoint (or more generally normal), then  is just an -tubular neighbourhood of the spectrum . Universally, however, the pseudospectrum is a much more reliable spectral description of  than the spectrum itself. For instance, it is the pseudospectrum that measures the instability of the spectrum under small perturbations by virtue of the formula

(1.2)

Leaving aside a lot of other interesting situations, let us recall the recent results when  is a differential operator. As a starting point we take the harmonic-oscillator Hamiltonian with complex frequency, which is also known as the rotated or Davies’ oscillator (see [8, Sec. 14.5] for a review and references). Although the complexification has a little effect on the spectrum (the eigenvalues are just rotated in the complex plane), a careful spectral analysis reveals drastic changes in basis and other more delicate spectral properties of the operator, in particular, the spectrum is highly unstable against small perturbations, as a consequence of the pseudospectrum containing regions very far from the spectrum. Similar peculiar spectral properties have been established for complex anharmonic oscillators (to the references quoted in [8, Sec. 14.5], we add [15, 24] for the most recent results), quadratic elliptic operators [27, 17, 34], complex cubic oscillators [30, 16, 21, 26], and other models (see the recent survey [21] and references therein).

A distinctive property of the complexified harmonic oscillator is that the associated spectral problem is explicitly solvable in terms of special functions. A powerful tool to study the pseudospectrum in the situations where explicit solutions are not available is provided by microlocal analysis [7, 39, 11]. The weak point of the semiclassical methods is the usual hypothesis that the coefficients of the differential operator are smooth enough (e.g. the potential of the Schrödinger operator must be at least continuous), and it is indeed the case of all the models above. Another common feature of the differential operators whose pseudospectrum has been analysed so far is that their spectrum consists of discrete eigenvalues only.

The objective of the present work is to enter an unexplored area of the pseudospectral world by studying the pseudospectrum of a non-self-adjoint Schrödinger operator whose potential is discontinuous and, at the same time, such that the essential spectrum is not empty. Among various results described below, we prove that the pseudospectrum is non-trivial, despite the boundedness of the potential. Namely, we show that the norm of the resolvent can become arbitrarily large outside a fixed neighbourhood of its spectrum. We hope that our results will stimulate further analysis of non-self-adjoint differential operators with singular coefficients.

2 Main results

In this section we introduce our model and collect the main results of the paper. The rest of the paper is primarily devoted to proofs, but additional results can be found there, too.

2.1 The model

Motivated by the role of step-like potentials as toy models in quantum mechanics, in this paper we consider the Schrödinger operator in defined by

(2.1)

In fact,  can be considered as an infinite version of the -symmetric square well introduced in [37] and further investigated in [38, 29].

Note that  is obtained as a bounded perturbation of the (self-adjoint) Hamiltonian of a free particle in quantum mechanics, which we shall simply denote here by . Consequently,  is well defined (i.e. closed and densely defined). In fact,  is m-sectorial with the numerical range (defined, as usual, by the set of all complex numbers such that and ) coinciding with the closed half-strip

(2.2)

The adjoint of , denoted here by , is simply obtained by changing  to  in (2.1). Consequently,  is neither self-adjoint nor normal. However, it is -self-adjoint (i.e. ), where  is the antilinear operator of complex conjugation (i.e. ). At the same time,  is -self-adjoint, where  is the parity operator defined by . Finally,  is -symmetric in the sense of the validity of the commutation relation .

Due to the analogy of the time-dependent Schrödinger equation for a quantum particle subject to an external electromagnetic field and the paraxial approximation for a monochromatic light propagation in optical media [23], the dynamics generated by (2.1) can experimentally be realised using optical systems. The physical significance of -symmetry is a balance between gain and loss [5].

2.2 The spectrum

As a consequence of (2.2), the spectrum of  is contained in . Moreover, the -symmetry implies that the spectrum is symmetric with respect to the real axis. By constructing the resolvent of  and employing suitable singular sequences for , we shall establish the following result.

Proposition 2.1.

We have

(2.3)

The fact that the two rays form the essential spectrum of  is expectable, because they coincide with the spectrum of the shifted Laplacian in and the essential spectrum of differential operators is known to depend on the behaviour of their coefficients at infinity only (cf. [12, Sec. X]). The absence of spectrum outside the rays is less obvious.

In fact, the spectrum in (2.3) is purely continuous, i.e. , for it can be easily checked that no point from the set on the right hand side of (2.3) can be an eigenvalue of  (as well as ). An alternative way how to a priori show the absence of the residual spectrum of , , is to employ the -self-adjointness of  (cf. [20, Sec. 5.2.5.4]).

2.3 The pseudospectrum

Before stating the main results of this paper, let us recall that a closed operator  is said to have trivial pseudospectra if, for some positive constant , we have

or equivalently,

(2.4)

Normal operators have trivial pseudospectra, because for them the equality holds in (2.4) with .

In view of (2.2), in our case (2.4) holds with if the resolvent set is replaced by . However, the following statement implies that (2.4) cannot hold inside the half-strip .

Theorem 2.2.

For all , there exists a positive constant  such that, for all with ,

(2.5)

Although the estimates give a rather good description of the qualitative shape of the pseudospectra, the constants and dependence on for are presumably not sharp.

In view of Theorem 2.2,  represents another example of a -symmetric operator with non-trivial pseudospectra. The present study can be thus considered as a natural continuation of the recent works [30, 16, 21]. However, let us stress that the complex perturbation in the present model is bounded. Moreover, comparing the present setting with the situation when (2.1) is subject to an extra Dirichlet condition at zero (cf. Section 7.3), the difference between these two realisations is indeed seen on the pseudospectral level only.

Even though the step-like shape of the potential in (2.1) is a feature of the present study, we stress that the discontinuity by itself is not the source of the non-trivial pseudospectra, see Remark 4.1 below.

The pseudospectrum of  computed numerically using Eigtool [36] by Mark Embree is presented in Figure 1.

Figure 1: The curves in the complex -plane computed for several values of ; the different colours correspond to , while the thick black lines are the essential spectrum of . (Courtesy of Mark Embree.)

2.4 Weak coupling

Inspired by (1.2), we eventually consider the perturbed operator

(2.6)

in the limit as . Here  is the operator of multiplication by a function that we denote by the same letter. Since  is not necessarily relatively bounded with respect to , the dotted sum in (2.6) is understood in the sense of forms. We remark that the perturbation does not change the essential spectrum, i.e., , and recall Proposition 2.1.

If  were the free Hamiltonian  and  were real-valued, the problem (2.6) with is known as the regime of weak coupling in quantum mechanics. In that case, it is well known that (under some extra assumptions on ) the perturbed operator possesses a unique discrete eigenvalue for all small positive  if, and only if, the integral of  is non-positive (see [32] for the original work). This robust existence of “weakly coupled bound states” is of course related to the singularity of the resolvent kernel of the free Hamiltonian at the bottom of the essential spectrum. Indeed, these bound states do not exist in three and higher dimensions, which is in turn related to the validity of the Hardy inequality for the free Hamiltonian (see, e.g., [35]).

Complex-valued perturbations of the free Hamiltonian have been intensively studied in recent years [1, 14, 6, 22, 9, 13, 10]. In [4, 25] the authors consider perturbations of an operator which is by itself non-self-adjoint. In all of these papers, however, the results are inherited from properties of the resolvent of the free Hamiltonian.

In the present setting, the unperturbed operator  is non-self-adjoint. Moreover, its resolvent kernel has no local singularity, but it blows up as when , see Section 3. Consequently, discrete eigenvalues of  can only “emerge from the infinity”, but not from any finite point of (2.3). The statement is made precise by virtue of the following result.

Theorem 2.3.

Let . There exists a positive constant  (independent of  and ) such that, whenever

we have

(2.7)

It is interesting to compare this estimate on the location of possible eigenvalues of  with the celebrated result of [1]

(2.8)

Our bound (2.7) can be indeed read as an inverse of (2.8). It demonstrates how much the present situation differs from the study of weakly coupled eigenvalues of the free Hamiltonian.

Under some additional assumptions on , the claim of Theorem 2.3 can be improved in the following way.

Theorem 2.4.

Let and . There exist positive constants  and  such that, for all , we have

(2.9)

In particular, if for instance belongs to the Schwartz space , then every eigenvalue of  must “escape to infinity” faster than any power of  as , namely .

Remark 2.5.

The reader will notice that statement (2.7) differs from (2.9) in that the latter does not highlight the dependence of the right hand side on the potential  but only on its amplitude . The reason is that it is the behaviour of  on diminishing  that primarily interests us. Moreover, the proofs of the theorems are different and it would be cumbersome (but doable in principle) to gather the dependence of the right hand side in (2.9) on (different) norms of .

2.5 The content of the paper

The organisation of this paper is as follows.

In Section 3, we find the integral kernel of the resolvent , cf. Proposition 3.1, and use it to prove Proposition 2.1.

In Section 4, the explicit formula of the resolvent kernel is further exploited in order to prove Theorem 2.2.

The definition of the perturbed operator (2.6) and its general properties are established in Section 5. In particular, we locate its essential spectrum (Proposition 5.5) and prove the Birman-Schwinger principle (Theorem 5.3).

Section 6 is divided into two respective subsections, in which we prove Theorems 2.3 and 2.4 with help of the Birman-Schwinger principle and, again, using the explicit formula of the resolvent kernel.

Finally, in Section 7, we present two concrete examples of the perturbed operator (2.6). Moreover, we make a comparison of the present study with a decoupled model due to an extra Dirichlet condition.

3 The resolvent and spectrum

Our goal in this section is to obtain an integral representation of the resolvent of . Using that result, we give a proof of Proposition 2.1.

In the following, we set

where we choose the principal value of the square root, i.e., is holomorphic on and positive on .

Proposition 3.1.

For all , is invertible and, for every ,

(3.1)

where

(3.2)
Remark 3.2.

The kernel is clearly bounded for every and fixed . Moreover, using (4.1) below, it can be shown that it remains bounded for as well. Hence, contrary to the case of the resolvent kernel of the free Hamiltonian  in one or two dimensions, the resolvent kernel of  has no local singularity. On the other hand, and again contrary to the case of the Laplacian, for all fixed , as , . Hence, the kernel exhibits a blow-up at infinity. The absence of singularity will play a fundamental role in the analysis of weakly coupled eigenvalues in Section 6. Moreover, we shall see in Section 4 that the singular behaviour at infinity is responsible for the spectral instability of .

Proof of Proposition 3.1.

Let and . We look for the solution of the resolvent equation .

The general solutions  of the individual equations

(3.3)

where and , are given by

where , are functions to be yet determined. Variation of parameters leads to the following system:

Hence, we can choose

where are arbitray complex constants. The desired general solutions of (3.3) are then given by

(3.4)

with .

Among these solutions, we are interested in those which satisfy the regularity conditions

(3.5)

These conditions are equivalent to the system

whence we obtain the following relations:

(3.6)

Summing up, assuming (3.6), the function

(3.7)

belongs to and solves the differential equation (3.3) in the whole . It remains to check some decay conditions as in addition to (3.6). This can be done by setting

(3.8)
(3.9)

Indeed, then

goes to as , and similarly for .

By gathering relations (3.6), (3.8) and (3.9), we obtain the following values for  and :

(3.10)
(3.11)

Replacing the constants by their values (3.8), (3.10), (3.11) and (3.9), respectively, expression (3.7) with (3.4) gives the desired integral representation

(3.12)

for a decaying solution of the differential equation (3.3) in .

To complete the proof, it remains to check that  given by (3.12) is indeed in the operator domain . Using for instance the Schur test (cf. (4.5) below), it is straightforward to check that  is in provided that . Therefore , whence and . ∎

This representation of the resolvent will be used in Sections 5 and 6 to study the location of weakly coupled eigenvalues. It will also enable us to prove the existence of non-trivial pseudospectra in Section 4. In this section we use it to prove Proposition 2.1.

Proof of Proposition 2.1.

According to Proposition 3.1, we have

It remains to prove the inverse inclusion. This can be achieved by a standard singular sequence construction.

Let be a real increasing sequence such that, for all , . Let be such that , for all , and

for some .

Then, for all , the sequence

where is chosen so that , is a singular sequence for corresponding to in the sense of [12, Def. IX.1.2]. Hence, according to [12, Thm. IX.1.3], we have

This completes the proof of the proposition. ∎

4 Pseudospectral estimates

The main purpose of this section is to give a proof of Theorem 2.2.

Proof of Theorem 2.2.

Let , where and . Recall our convention for the square root we fixed at the beginning of Section 3. The following expansions hold

(4.1)

as . As a consequence, we have the asymptotics

(4.2)
(4.3)
(4.4)

as .

Let us prove the upper bound in (2.5) using the Schur test:

(4.5)

After noticing the symmetry relation valid for all (which is a consequence of the -self-adjointness of ), we simply have

(4.6)

By virtue of (3.2), for all ,

(4.7)

Similarly, if ,

(4.8)

According to (4.2)–(4.4), the right hand sides in (4.7) and (4.8) are both equivalent to

whence (4.6) yields the upper bound in (2.5).

In order to get the lower bound, we set

(4.9)

where  denotes the characteristic function of a set . Then according to (3.2),

(4.10)
(4.11)
(4.12)

On the other hand, we have

(4.13)

Hence, using (4.3) and (4.4),

as , and the lower bound in (2.5) follows. ∎

Remark 4.1 (Irrelevance of discontinuity).

Although the proof above relies on the particular form of the potential , it turns out that the discontinuity at is not responsible for the spectral instability highlighted by Theorem 2.2. Indeed, consider instead of the potential a smooth potential such that, for some , the difference

is supported in the interval . In order to get a lower bound for the norm of the resolvent of the regularised operator , we shall use the pseudomode

where the function  is introduced in (4.9). Using again the asymptotic expansions (4.1), one can check that, provided that is large enough,

for some independent of . Thus, in view of (4.13), we have

as , . On the other hand, (4.12) yields

for some independent of . Consequently, is a -pseudomode for , or more specifically,

(4.14)

with independent of , as , .

Summing up, despite of the fact that the lower bound in (4.14) is not as good as that of Theorem 2.2, the presence of non-trivial pseudospectra for the operator  clearly indicates that the discontinuity of the potential does not really play any essential role in the spectral instability of .

5 General properties of the perturbed operator

In this section, we state some basic properties about the perturbed operator  introduced in (2.6). Here  is not necessarily small and positive.

5.1 Definition of the perturbed operator

The unperturbed operator  introduced in (2.1) is associated (in the sense of the representation theorem [18, Thm. VI.2.1]) with the sesquilinear form

In view of (2.2),  is sectorial with vertex  and semi-angle . In fact,  is obtained as a bounded perturbation of the non-negative form  associated with the free Hamiltonian ,

Given any function , let  be the sesquilinear form of the corresponding multiplication operator (that we also denote by ), i.e.,

As usual, we denote by the corresponding quadratic form.

Lemma 5.1.

Let . Then and, for every ,

(5.1)
Proof.

Set . For every , an integration by parts together with the Schwarz inequality yields

By density of in , the inequality extends to all and, in particular, whenever . ∎

It follows from the lemma that  is -subordinated to , which in particular implies that  is relatively bounded with respect to  with the relative bound equal to zero. Classical stability results (see, e.g., [20, Sec. 5.3.4]) then ensure that the form is sectorial and closed. Since  is a bounded perturbation of , we also know that is sectorial and closed. We define  to be the m-sectorial operator associated with the form . The representation theorem yields

(5.2)

where should be understood as a distribution. By the replacement , we introduce in the same way as above the form and the associated operator  for any . Of course, we have .

5.2 The Birman-Schwinger principle

As regards spectral theory,  represents a singular perturbation of , for we are perturbing an operator with purely essential spectrum. An efficient way to deal with such problems in self-adjoint settings is the method of the Birman-Schwinger principle, due to which a study of discrete eigenvalues of the differential operator  is transferred to a spectral analysis of an integral operator. We refer to [2, 28] for the original works and to [31, 32, 3, 19] for an extensive development of the method for Schrödinger operators. In recent years, the technique has been also applied to Schrödinger operators with complex potentials (see, e.g., [1, 22, 13]). However, our setting differs from all the previous works in that the unperturbed operator  is already non-self-adjoint and its resolvent kernel substantially differs from the resolvent of the free Hamiltonian. The objective of this subsection is to carefully establish the Birman-Schwinger principle in our unconventional situation.

In the following, given , we denote

so that .

We have introduced  as an unbounded operator with domain acting in the Hilbert space . It can be regarded as a bounded operator from to . More interestingly, using the variational formulation,  can be also viewed as a bounded operator from to , by defining for all by

where denotes the duality bracket between and .

Similarly, in addition to regarding the multiplication operators and as operators from to , we can view them as operators from to , due to the relative boundedness of  with respect to  (cf. Lemma 5.1 and the text below it).

Finally, let us notice that, for all , the resolvent can be viewed as an operator from to . Indeed, for all , there exists a unique such that

(5.3)

where denotes the inner product in . Hence the operator is bijective.

With the above identifications, for all , we introduce

(5.4)

as a bounded operator on to .  is an integral operator with kernel

(5.5)

where  is the kernel of the resolvent written down explicitly in (3.2). The following result shows that  is in fact compact.

Lemma 5.2.

Let . For all , is a Hilbert-Schmidt operator.

Proof.

By definition of the Hilbert-Schmidt norm,

(5.6)