Limiting absorption principle and virtual levels of operators in Banach spaces

The idea of introducing a small absorption into the wave equation for selecting particular solutions goes back to Ignatowsky Ign (05) and is closely related to the Sommerfeld radiation condition Som (12). We start with the Helmholtz equation

For $z\in\mathbb{C}\setminus\overline{\mathbb{R}_{+}}$, equation (1.1) has a unique $L^{2}$-solution $(-\Delta-zI)^{-1}f$, with $(-\Delta-zI)^{-1}$ represented by the convolution with ${e^{-|x|\sqrt{-z}}}/{(4\pi|x|)}$, $\mathop{\rm{R\hskip-1.0pte}}\nolimits\sqrt{-z}>0$. For $z\geq 0$, there may be no $L^{2}$-solution; moreover, when $z>0$, there are different solutions of similar norm, and one faces the question of choosing an appropriate one. The way to select a unique solution is known as the radiation principle. V.I. Smirnov, in his widely renowned “Course of higher mathematics” Smi (41), credits the radiation principle to V.S. Ignatowsky Ign (05) and to A. Sommerfeld Som (12); the work of Ignatowsky was also publicized by A.N. Tikhonov, both in his lectures at mechmat at Moscow State University and in the textbook written jointly with A.A. Samarskii TS (51) (and even in their ’1950 Russian translation of A. Sommerfeld’s textbook Som (48)). In Ign (05), Ignatowsky considered the electromagnetic field scattered by a wire using the expression $Z(t,x)=Ae^{\mathrm{i}(\omega t-\varkappa x)}$ for the electric field, with $\omega$ and $\varkappa$ certain parameters. The absorption in the medium corresponded to $\varkappa$ having nonzero imaginary part; its sign was taken into account when choosing an appropriate solution to the Helmholtz equation. Following this idea, A.G. Sveshnikov, a student of Tikhonov, specifies in Sve (50) a solution to (1.1) by

calling this approach the limiting absorption principle (LAP) and attributing it to Ignatowsky.¹¹1We suppose that in the twenties and thirties, between Ign (05) and Smi (41), the idea of the limiting absorption principle was being refined when V.S. Ignatowsky worked at St. Petersburg University, where in particular he taught mathematical theory of diffraction and likely was in contact with V.I. Smirnov. Let us mention that, besides his work on diffraction, Ignatowsky is known for his contributions to the theory of relativity (see VG (87)) and for developing optical devices while heading the theoretical division at GOMZ, the State Association for Optics and Mechanics (which later became known as “LOMO”). On 6 November 1941, during the blockade of St. Petersburg, Ignatowsky was arrested by NKVD (an earlier name of KGB), as a part of the “process of scientists”, and shot on 30 January 1942. (During this process, V.I. Smirnov was “credited” by NKVD the role of a Prime Minister in the government after the purportedly planned coup; Smirnov avoided the arrest because he was evacuated from St. Petersburg in August 1941, shortly before the blockade began.) As a result, Ignatowsky’s name has been unknown to many: the reference to his article disappeared from Smirnov’s “Course of higher mathematics” until post-1953 editions (see e.g. the English translation (Smi, 64, §230)).   Russians are used to such rewrites of the history, joking about the “History of the history of the Communist Party”, a reference to a mandatory and ever-changing Soviet-era ideological course in the first year of college. As the matter of fact, the very “Course of higher mathematics” mentioned above was started by V.I. Smirnov together with J.D. Tamarkin, with the first two volumes (published in 1924 and 1926) bearing both names, but after Tamarkin’s persecution by GPU (another earlier name of KGB) and his escape from the Soviet Union with smugglers over frozen lake Chudskoe in December 1924 Hil (47), Tamarkin’s authorship eventually had to disappear. His coauthor Smirnov spent the next year pleading with the authorities (and succeeding!) for Tamarkin’s wife Helene Weichardt – who tried to follow her husband’s route with the smugglers over the icy lake but was intercepted at the border and jailed – to be released from prison and allowed to leave the Soviet Union to join her husband AN (18). We note that (1.2) leads to

where the choice of the branch of the square root in the exponent is dictated by the need to avoid the exponential growth at infinity. Sveshnikov points out that Ignatowsky’s approach does not depend on the geometry of the domain and hence is of more universal nature than that of A. Sommerfeld Som (12), which is the selection of the solution to (1.1) satisfying the Sommerfeld radiation condition

in agreement with (1.3).

Let us also mention the limiting amplitude principle TS (48) (the terminology also introduced in Sve (50)) which specifies a solution to (1.1) by $u(x)=\!\!\lim\limits_{t\to+\infty}\!\!\psi(x,t)e^{\mathrm{i}kt}$, where $\psi(x,t)$ is a solution to the wave equation

Thus, $u$ is the limiting amplitude of the periodic vibration building up under the action of a periodic force for a long time. This corresponds to using the retarded Green function, represented by the convolution with $G_{\mathrm{ret}}(x,t)=\frac{\delta(t-|x|)}{4\pi|x|}$, yielding the solution to (1.5) in the form

in agreement both with the limiting absorption principle (1.2) (cf. (1.3)) and with the Sommerfeld radiation condition (1.4).

Presently, a common meaning of the LAP is the existence of a limit of the resolvent at a given point of the essential spectrum. While the resolvent of $A:\,\mathbf{X}\to\mathbf{X}$ cannot have a limit at the essential spectrum as an operator in $\mathbf{X}$, it can have a limit as a mapping

where $\mathbf{E}$ and $\mathbf{F}$ are some Banach spaces such that the embeddings $\mathbf{E}\hookrightarrow\mathbf{X}\hookrightarrow\mathbf{F}$ are dense and continuous. Historically, this idea could be traced back to eigenfunction expansions Wey (10); Car (34); Tit (46) and Krein’s method of directing functionals Kre (46, 48) (see (AG, 81, Appendix II.7)). This was developed in Pov (50, 53); GK (55); Ber (57); Bir (61) (see also rigged spaces in (GV, 61, I§4), also known as equipped spaces and related to Gelfand’s triples from GK (55)). Gradually the theory takes the form of estimates on the limit of the resolvent at the essential spectrum in certain spaces; this further development becomes clearer in Eid (62); Vai (66); Eid (69) (the convergence of the resolvent is in the sense of distributions), then in (Rej, 69, Lemma 6.1) (where certain spaces are introduced), and finally in (Agm, 70, Theorem 2.2), (Yam, 73, Theorem 4.1) (for Dirac operators), and (Agm, 75, Appendix A), where the convergence of the resolvent is specified with respect to weighted $L^{2}$ spaces. See also Kur (78) and BAD (87). Let us also mention that in Agm (98) this same approach – to consider the resolvent as a mapping from $\mathbf{E}$ to $\mathbf{F}$, with the embeddings $\mathbf{E}\hookrightarrow\mathbf{X}\hookrightarrow\mathbf{F}$ being dense and continuous – is used to define resonances of an operator as poles of the analytic continuation of its resolvent.

Perhaps the simplest example of LAP is covered by S. Agmon in (Agm, 75, Lemma A.1): by that lemma, the operator $(\partial_{x}-zI)^{-1}$, $z\in\mathbb{C}$, $\mathop{\rm{R\hskip-1.0pte}}\nolimits z\neq 0$, is uniformly bounded as an operator from $L^{2}_{s}(\mathbb{R})$ to $L^{2}_{-s}(\mathbb{R})$, $s>1/2$, and has a limit (in the uniform operator topology) as $\mathop{\rm{R\hskip-1.0pte}}\nolimits z\to\pm 0$. For example, for $\mathop{\rm{R\hskip-1.0pte}}\nolimits z<0$, the solution to the equation $(\partial_{x}-z)u=f$ is given by the operator $f\mapsto u(x)=\int_{-\infty}^{x}e^{z(x-y)}f(y)\,dy$, which is bounded from $L^{1}(\mathbb{R})$ to $L^{\infty}(\mathbb{R})$ and hence from $L^{2}_{s}(\mathbb{R})$ to $L^{2}_{-s}(\mathbb{R})$, $s>1/2$, uniformly in $z\in\mathbb{C}$, $\mathop{\rm{R\hskip-1.0pte}}\nolimits z<0$. Here we use the standard notation

for any $p\in[1,+\infty]$, $s\in\mathbb{R}$, $d\in\mathbb{N}$, with $\langle x\rangle=(1+x^{2})^{1/2}$. Agmon then shows that the LAP is available for the Laplacian when the spectral parameter approaches the bulk of the essential spectrum: by (Agm, 75, Theorem 4.1), for $s,\,s^{\prime}>1/2$, the resolvent

is bounded uniformly for $z\in\varOmega\setminus\overline{\mathbb{R}_{+}}$, for any open neighborhood $\varOmega\subset\mathbb{C}$ such that $\overline{\varOmega}\not\ni\{0\}$, and has limits as $z\to z_{0}\pm\mathrm{i}0$, $z_{0}>0$. For the sharp version (the $\mathbf{B}\to\mathbf{B}^{*}$ continuity of the resolvent in the Agmon–Hörmander spaces), see (Yaf, 10, Proposition 6.3.6).

While the mapping (1.7) has a limit as $z\to z_{0}\pm\mathrm{i}0$ with $z_{0}>0$, for any $d\geq 1$, the behaviour at $z_{0}=0$ depends on $d$. For example, in three dimensions, as long as $s,\,s^{\prime}>1/2$ and $s+s^{\prime}\geq 2$, the mapping (1.7), represented by the convolution with $(4\pi|x|)^{-1}e^{-|x|\sqrt{-z}}$, $\ z\in\mathbb{C}\setminus\overline{\mathbb{R}_{+}}$, $\ \mathop{\rm{R\hskip-1.0pte}}\nolimits\sqrt{-z}>0$, remains uniformly bounded and has a limit as $z\to z_{0}=0$. A similar boundedness of the resolvent in an open neighborhood of the threshold $z_{0}=0$ persists in higher dimensions, but breaks down in dimensions $d\leq 2$. In particular, for $d=1$, the resolvent is represented by the convolution with $e^{-|x|\sqrt{-z}}/(2\sqrt{-z})$, $z\in\mathbb{C}\setminus\overline{\mathbb{R}_{+}}$, $\mathop{\rm{R\hskip-1.0pte}}\nolimits\sqrt{-z}>0$, and cannot have a limit as $z\to 0$ as a mapping $\mathbf{E}\to\mathbf{F}$ as long as $\mathbf{E},\,\mathbf{F}$ are weighted Lebesgue spaces (at the same time, see Example 1.0 below). There is a similar situation in two dimensions. We say that the threshold $z_{0}=0$ is a regular point of the essential spectrum for $d\geq 3$ and that it is a virtual level if $d\leq 2$.

Mathematically, virtual levels correspond to particular singularities of the resolvent at the essential spectrum. This idea goes back to J. Schwinger Sch60b and was further addressed by M. Birman Bir (61), L. Faddeev Fad (63), B. Simon Sim (73, 76), B. Vainberg Vai (68, 75), D. Yafaev Yaf (74, 75), J. Rauch Rau (78), and A. Jensen and T. Kato JK (79), with the focus on Schrödinger operators in three dimensions. Higher dimensions were considered in Jen (80); Yaf (83); Jen (84). An approach to more general symmetric differential operators was developed in Wei (99). The virtual levels of nonselfadjoint Schrödinger operators in three dimensions appeared in CP (05). Dimensions $d\leq 2$ require special attention since the free Laplace operator has a virtual level at zero (see Sim (76)). The one-dimensional case is covered in BGW (85); BGK (87). The approach from the latter article was further developed in BGD (88) to two dimensions (if $\int_{\mathbb{R}^{2}}V(x)\,dx\neq 0$) and then in JN (01) (with this condition dropped) who give a general approach in all dimensions, with the regularity of the resolvent formulated via the weights which are square roots of the potential (and consequently not optimal). There is an interest in the subject due to dependence of dispersive estimates on the presence of virtual levels at the threshold point, see e.g. JK (79); Yaf (83); ES (04); Yaj (05) in the context of Schrödinger operators; the Dirac operators are treated in Bou (06, 08); EG (17); EGT (19). Let us mention the dichotomy between a virtual level and an eigenvalue manifested in the large-time behavior of the heat kernel and the behavior of the Green function near criticality; see Pin (92, 04). We also mention recent articles BBV (20) on properties of virtual states of selfadjoint Schrödinger operators and GN (20) proving the absence of genuine (non-$L^{2}$) virtual states of selfadjoint Schrödinger operators and massive and massless Dirac operators, as well as giving classification of virtual levels and deriving properties of eigenstates and virtual states.

On the contrary, properties (P1) – (P3) are not satisfied for $H=-\Delta$ in $L^{2}(\mathbb{R}^{3})$, with $\mathfrak{D}(H)=H^{2}(\mathbb{R}^{3})$. Regarding (P1), we notice that nonzero solutions to $(-\Delta+V)\psi=0$ (with certain compactly supported potentials) can behave like the Green function, $\sim|x|^{-1}$ as $|x|\to\infty$, and one expects that this is what virtual states should look like, while nonzero solutions to $\Delta\psi=0$ cannot have uniform decay as $|x|\to\infty$, so should not qualify as virtual states; the integral kernel of $R_{0}^{(3)}(z)=(-\Delta-zI)^{-1}$,

remains pointwise bounded as $z\to 0$ and has a limit in the space of mappings $L^{2}_{s}(\mathbb{R}^{3})\to L^{2}_{-s^{\prime}}(\mathbb{R}^{3})$, $s,\,s^{\prime}>1/2$, $s+s^{\prime}>2$ (see e.g. JK (79)), failing (P2); finally, small perturbations cannot produce negative eigenvalues (this follows from the Hardy inequality), so (P3) also fails. In this case, we say that $z_{0}=0$ is a regular point of the essential spectrum.

We claim that the properties (P1) – (P3) are essentially equivalent, even in the context of the general theory BC (21). These properties are satisfied when $z_{0}$ is either an eigenvalue of $H$ or, more generally, a virtual level. To motivate the general theory, we can start from the Laplace operator in one dimension, considering the problem

For any $f\in C_{\mathrm{comp}}(\mathbb{R})$, there is a $C^{2}$-solution to (2.3). If we consider $z\in\mathbb{C}\setminus\overline{\mathbb{R}_{+}}$, then the natural choice of a solution is

where the resolvent $R_{0}^{(1)}(z)=(-\Delta-zI)^{-1}$ has the integral kernel $R_{0}^{(1)}(x,y;z)$ from (2.1). This integral kernel is built of solutions $e^{\pm x\sqrt{-z}}$; the choice of such a combination is dictated by the desire to avoid solutions exponentially growing at infinity. For $z\neq 0$, since $R_{0}^{(1)}(x,y;z)$ is bounded, the mapping $f\mapsto R_{0}^{(1)}f$ defines a bounded mapping $L^{1}(\mathbb{R})\to L^{\infty}(\mathbb{R})$. This breaks down at $z=0$, since $e^{\pm x\sqrt{-z}}$ are linearly dependent when $z=0$. To solve (2.3) at $z=0$, one can use the convolution with the fundamental solution $G(x)=|x|/2+xC$, with any $C\in\mathbb{C}$. While such fundamental solutions provide a solution $u=G\ast f$ to (2.3), this solution may no longer be from $L^{\infty}$; any of the above choices of $G$ would no longer be bounded as a mapping $L^{1}\to L^{\infty}$. This problem is resolved if a potential $V\in C_{\mathrm{comp}}(\mathbb{R},\mathbb{C})$ is introduced into (2.3),

so that the Jost solution $\theta_{-}(x)$ to $(-\partial_{x}^{2}+V)u=0$ with $\lim_{x\to-\infty}\theta_{-}(x)=1$, tends to infinity as $x\to+\infty$ and is linearly independent with the Jost solution $\theta_{+}(x)$, $\lim_{x\to+\infty}\theta_{+}(x)=1$. To construct a fundamental solution to (2.4) at $z_{0}=0$, we set

with $W[\theta_{+},\theta_{-}](y)=\theta_{+}(y)\theta_{-}^{\prime}(y)-\theta_{+}^{\prime}(y)\theta_{-}(y)$, the Wronskian. This will work if $|\theta_{-}(x)|$ grows as $x\to+\infty$ (and similarly if $|\theta_{+}(x)|$ grows as $x\to-\infty$); if, on the other hand, $\theta_{\pm}$ remain bounded, then, as the matter of fact, these functions are linearly dependent, their Wronskian is zero, and (2.5) is not defined. In this construction the space $L^{\infty}$ appears twice: it contains the range of $G(z_{0})\and{L^{2}_{s}(\mathbb{R})}$, $s>3/2$, when $\theta_{\pm}$ are linearly independent (see BC (21)), and it is the space where $\theta_{\pm}$ live when they are linearly dependent. This is not a coincidence: from $-u^{\prime\prime}=f\in C^{\infty}_{\mathrm{comp}}(\mathbb{R})$, we can write $-u^{\prime\prime}+Vu=f+Vu$, and then $u=(-\partial_{x}^{2}+V-z_{0}I)^{-1}(f+Vu)$ is in the range of $(-\partial_{x}^{2}+V-z_{0}I)^{-1}\big{(}C_{\mathrm{comp}}(\mathbb{R})\big{)}\subset L^{\infty}(\mathbb{R})$.

We point out that in the case of general exterior elliptic problems the above dichotomy – either boundedness of the truncated resolvent or existence of a nontrivial solution to a homogeneous problem with appropriate radiation conditions – was studied by B. Vainberg Vai (75).

• Let $H=-\Delta+V$ with $V\in C_{\mathrm{comp}}(\mathbb{R}^{d},\mathbb{R})$ be a Schrödinger operator in $L^{2}(\mathbb{R}^{d})$, and assume that the associated quadratic form

  $$\mathbf{a}[u]:=\int_{\mathbb{R}^{d}}(|\nabla u|^{2}+V|u|^{2})\,dx$$

  is nonnegative on $C^{\infty}_{\mathrm{comp}}(\mathbb{R}^{d})$. Then either there is a continuous function $w(x)>0$ such that $\int_{\mathbb{R}^{d}}w|u|^{2}\,dx\leq\mathbf{a}[u]$ for any $u\in C^{\infty}_{\mathrm{comp}}(\mathbb{R}^{d})$ (one says that $\mathbf{a}[\cdot]$ has a weighted spectral gap), or there is a sequence $\varphi_{j}\in C^{\infty}_{\mathrm{comp}}(\mathbb{R}^{d})$ such that $\mathbf{a}[\varphi_{j}]\to 0$, $\varphi_{j}\to\varphi>0$ locally uniformly on $\mathbb{R}^{d}$ (then one says that $\mathbf{a}[\cdot]$ has a null state $\varphi$).

Let us mention that in the former case, when $\mathbf{a}[\cdot]$ has a weighted spectral gap, the operator $H$ is subcritical (that is, it admits a positive Green’s function), and that in the latter case, when $\mathbf{a}[\cdot]$ has a null state, $H$ is critical. This null state coincides with Agmon’s ground state, which can be characterized as a state with minimal growth at infinity from (Agm, 82, Definitions 4.1, 5.1). See Pin (88, 90); PT (06) for more details.

A null state, or Agmon’s ground state, corresponds to a virtual level at the bottom of the spectrum, in the following sense:

We now sketch our approach to virtual levels from BC (21). Let $\mathbf{X}$ be an infinite-dimensional complex Banach space and let $A\in\mathscr{C}(\mathbf{X})$ be a closed operator with dense domain $\mathfrak{D}(A)\subset\mathbf{X}$. We assume that there are some complex Banach spaces $\mathbf{E}$, $\mathbf{F}$ with embeddings $\mathbf{E}\hookrightarrow\mathbf{X}\hookrightarrow\mathbf{F}$. We will assume that the operator $A$ and the “regularizing” spaces $\mathbf{E}$ and $\mathbf{F}$ satisfy the following assumption.

We note that Assumption 3.0 is readily satisfied in the usual examples of differential operators. For convenience, from now on, we will assume that $\mathbf{E}\subset\mathbf{X}\subset\mathbf{F}$ (as vector spaces) and will omit $\imath$ and $\jmath$ in numerous relations.

Above, $\sigma_{\mathrm{ess}}(A)$ is F.  Browder’s essential spectrum (Bro, 61, Definition 11). It can be characterized as $\sigma(A)\setminus\sigma_{\mathrm{d}}(A)$, with the discrete spectrum $\sigma_{\mathrm{d}}(A)$ being the set of isolated points of $\sigma(A)$ with corresponding Riesz projectors having finite rank (see e.g. (BC, 19, Lemma III.125)). Let us emphasize that the existence of the limit (3.2) implicitly implies that there is $\delta>0$ such that $\varOmega\cap\sigma(A+B)\cap\mathbb{D}_{\delta}(z_{0})=\emptyset$.