WKB Asymptotics and Spectral Deformation in Semi-classical Limit

Abstract

Phase-integral method is applied to study semi-classical limit for the spectral problem related to dynamical system describing quantum mechanical evolution with space-time reflection symmetry. The main goal of the paper is to investigate the relationship between analytic properties of the corresponding potential and the structure of limiting spectral graph. Quantization conditions of Bohr-Sommerfeld type are derived specifying localization of the spectrum for the model problem with a one-parameter family of polynomial potentials treated as deformation of a linear one.

Appendix

Let us first specify the critical valueκ = κ₀ > 0 of deformation parameter at which topological type of limiting spectral graph Γ(κ) changes so that for κ > κ₀ an additional vertical edge [iμ, iK₋] arises outgoing from the node iμ = Γ₊ ∩Γ₋ where $\text {Re} \widehat {\Phi }(i\mu )$ = 0.

Assertion 1

Let t₀ be a unique real root of the polynomial 23t⁵ − 5(t³ + t²) − 4. Then one has

$$ K_{-}(\kappa) \gtrless \mu(\kappa) ,\quad \kappa \gtrless \kappa_{0} = \frac{1-{t_{0}^{2}}}{1+{t_{0}^{2}}/2} . $$

Proof

Due to the fact that given κ > 0 the quantity $\text {Re} \widehat {\Phi }(a+ib)$ increases in variable b for fixed a ∈ [0, Q_κ(1)] the critical value κ = κ₀ can be characterized in terms of function ${\Upsilon }(\kappa )= \text {Re} \widehat {\Phi }(iK_{-}(\kappa ))$ in the following way: Υ(κ₀) = 0 and Υ(κ)(κ − κ₀) < 0 if κ≠κ₀. To evaluate κ₀ it suffices to explicitly calculate the integral

$$ \begin{array}{@{}rcl@{}} \frac{(2\kappa)^{3/4}}{(1-\kappa)^{5/4}} \widehat{\Phi}(iK_{-}) &=& e^{i\pi/4} \frac{(2\kappa)^{3/4}}{(1-\kappa)^{5/4}}\!{\int}_{\xi(iK_{-}\!)}^{1}\!\!\sqrt{Q_{\kappa}(z)-iK_{-}} dz \\ &=& e^{i\pi/4}\!{\int}_{-i/\sqrt{3}}^{C(\kappa)}\sqrt{w^{3}+w+2i/3\sqrt{3}} dw = \frac2{\sqrt{3}} r^{3/2}e^{3i\varphi/2} \\ &&- \frac25 r^{5/2}e^{5i\varphi/2} - \frac45 \sqrt[4]{3} , \end{array} $$

where $w=C(\kappa )z, C(\kappa )=\sqrt {2\kappa /(1-\kappa )}, \varphi = \arctan \left (C(\kappa )\sqrt {3}/2\right ), r= 2/(\sqrt {3}\cos \nolimits \varphi ),$ so that

$$ \frac{(2\kappa)^{3/4}}{(1-\kappa)^{5/4}} {\Upsilon}(\kappa) = \frac{2^{3/2}}{3^{5/4}\cos^{3/2}\varphi} \cos\frac{3\varphi}2 - \frac{2^{5/2}}{3^{5/4}5 \cos^{5/2}\varphi} \cos\frac{5\varphi}2 - \frac25 3^{1/4} . $$

Thus equation Υ(κ) = 0 reduces to the form $5(\cos \nolimits ^{3}\varphi +\cos \nolimits ^{2}\varphi )+4=23\cos \nolimits ^{5}\varphi$ and since the polynomial 23t⁵ − 5(t³ + t²) − 4 has a unique real root t₀ ∈ [0.8613, 0.8614] it follows that Υ(κ) vanishes at point $\kappa =\kappa _{0}=(2-2{t_{0}^{2}})/(2+{t_{0}^{2}})\in [ 0.1881,0.1884]$ and moreover

$$ {\Upsilon}(\kappa) = \text{Re}\left( e^{i\pi/4}{\int}_{C(\kappa_{0})}^{C(\kappa)}\!\!\sqrt{w^{3}+w+2i/3\sqrt{3}} dw\right) \gtrless 0 ,\quad \kappa\gtrless\kappa_{0} . $$

Our next goal is to establish existence and uniqueness of the triple point ${\Lambda } ={\Lambda } (\kappa )\in {\Gamma }_{+}^{(j)}, j=1,2,3,$ being a Y-shaped junction node of the limit spectral graph Γ = Γ(κ) in the case when κ ∈ [− 1,− 1/5). □

Assertion 2

For κ ∈ [− 1,− 1/5) there exists a unique intersection point Λ = Λ(κ) of level-lines $\text {Re} \widetilde {\Phi }(\lambda )=0$ and $\text {Re}\left (\widetilde {\Phi }(\lambda )-\widehat {\Phi }(\lambda )\right )=0$ which prove to be the graphs of monotonic functions b = f₂(a) and b = f₃(a) such that f₂(K₊) = f₃(Q_κ(1)) = 0 and f₂(Q_κ(1)) < 0, f₃(K₊) < 0.

Proof

By the same arguments as in the case of model potential Q_− 1(z) the segment $\widehat {\ell }=\{Q_{\kappa }(1)+ e^{-i\pi /6}\mathbb {R}_{+}\}\cap {\Pi }$ and the curve $\ell =\{Q_{\kappa }(r+e^{i\pi /8}\mathbb {R}_{+})\}\cap {\Pi }$ which proves to be the graph of increasing function are shown to lie below the level-lines $\text {Re}\left (\widetilde {\Phi }(\lambda )-\widehat {\Phi }(\lambda )\right )=0$ and $\text {Re} \widetilde {\Phi }(\lambda )=0$ respectively. Beforehand, let us verify that

$$ \partial \text{Re}\left( \widehat{\Phi}(\lambda)-\widetilde{\Phi}(\lambda)\right)\left/ \partial b = \text{Im}\left( \widetilde{\Phi} '(\lambda)-\widehat{\Phi} '(\lambda)\right) < 0\right. $$

in π and besides

$$ \partial \text{Re}\left( \widehat{\Phi}(\lambda)-\widetilde{\Phi}(\lambda)\right)\left/ \partial a = \text{Re}\left( \widehat{\Phi} '(\lambda)-\widetilde{\Phi}^{\prime}(\lambda)\right) < 0\right. $$

in the subdomain $\widetilde {\Pi }\subset {\Pi }$ located above the curve ℓ and the segment $\widehat {\ell }.$ To evaluate the imaginary part of the derivative

$$ \widetilde{\Phi} '(\lambda)-\widehat{\Phi} '(\lambda) = \frac{e^{3i\pi/4}}2{\int}_{1}^{\eta(\lambda,\kappa)}\!\!\!\!\frac{dz}{\sqrt{\lambda-Q_{\kappa}(z)}} ,$$

we will integrate over the path composed of the segment [1, η(a, κ)] and the arc of the level-line ReQ_κ(z) = a passing from η(a, κ) to η(λ, κ) so that the required sign-definiteness of $\text {Im}\left (\widetilde {\Phi } '(\lambda )-\widehat {\Phi } '(\lambda )\right )$ follows from the estimate $\arg \left (\left (\lambda -Q_{\kappa }(z)\right )^{-1/2}\!dz\right )\in (\pi /4,5\pi /4)$ valid along the integration path. As regards estimation of the real part for $\widehat {\Phi } '(\lambda )-\widetilde {\Phi } '(\lambda )$, we choose integration path with parametrization $z(s,\lambda )=\eta \left (\lambda (1-s) + Q_{\kappa }(1)s;\kappa \right ),$s ∈ [0, 1], so that

$$ \widehat{\Phi} '(\lambda)-\widetilde{\Phi} '(\lambda) = \frac{e^{-i\pi/4}}2{{\int}_{0}^{1}}\frac{\sqrt{\lambda-Q_{\kappa}(1)} ds} {\left( 6\kappa z(s,\lambda)^{2}+ (1-\kappa)\right)\sqrt{s}} $$

where $\text {Re}\sqrt {\lambda -Q_{\kappa }(1)}>0.$ In order to establish sign definiteness of $\text {Re}\left (\widehat {\Phi } '(\lambda )-\widetilde {\Phi } '(\lambda )\right )$ for $\lambda \in \widetilde {\Pi }$, we first note that $\arg \!\left (6\kappa \eta (\lambda ;\kappa )^{2}\!+(1-\kappa )\right )\in (\pi ,7\pi /6)$ where Reη(λ; κ) < 1 since the curve $Q_{\kappa }(1+i\mathbb {R}_{+})\cap {\Pi }$ lies below the segment $\widehat {\ell }$ and therefore $\arg \left (6\kappa z(s,\lambda )^{2}+ (1-\kappa )\right )\in (\pi ,7\pi /6)$ because $[ Q_{\kappa }(1),\lambda ]\subset \widetilde {\Pi }.$

Moreover, making use of the above integral representation of $\widehat {\Phi } '(\lambda )-\widetilde {\Phi } '(\lambda )$, one can straightforwardly verify that level-line $\text {Re}\left (\widetilde {\Phi }(\lambda )-\widehat {\Phi }(\lambda )\right )=0$ is lying above the semi-axis $Q_{\kappa }(1)+ e^{-i\pi /6}\mathbb {R}_{+}$ and hence its segment belonging to $\widetilde {\Pi }$ is the graph of monotonically decreasing function b = f₃(a) since

$$ f^{\prime}_3(a) = -\left( \partial{\text{Re}}\left( \widetilde{\Phi}(\lambda)-\widehat{\Phi}(\lambda)\right)\left/ \partial a\right)\left( \partial{\text{Re}}\left( \widetilde{\Phi}(\lambda)-\widehat{\Phi}(\lambda)\right.)\right/ \partial b\right)^{-1}\!< 0 . $$

The proof will be complete if we show that level-line $\text {Re} \widetilde {\Phi }(\lambda )=0$ is located in π above the curve ℓ being the graph of monotonically increasing function b = f₂(a). To this end, it clearly suffices to establish sign definiteness of partial derivatives

$$ \partial \text{Re} \widetilde{\Phi}(\lambda)\left/\partial a = \text{Re} \widetilde{\Phi} '(\lambda) < 0 ,\quad \partial \text{Re} \widetilde{\Phi}(\lambda)\right/\partial b = - \text{Im} \widetilde{\Phi} '(\lambda) > 0 $$

in π and the estimate $\text {Re} \widetilde {\Phi }(\lambda )<0$ which holds true for λ ∈ ℓ. To deal with the derivative

$$ \widetilde{\Phi} '(\lambda) = \frac{e^{3i\pi/4}}2{\int}_{\xi(\lambda;\kappa)}^{ \eta(\lambda\kappa)}\!\!\frac{dz}{\sqrt{\lambda-Q_{\kappa}(z)}}, $$

we choose integration path composed of the segment [ξ(a; κ), η(a; κ)] and the arcs of the level-line ReQ_κ(z) = a passing through the turning points ξ(λ; κ) and η(λ; κ). Monotonicity of $\text {Re} \widetilde {\Phi }(a+ib)$ with respect to each of the variables a and b follows now from the estimate $\arg \left (\left (\lambda -Q_{\kappa }(z)\right )^{-1/2}\!dz\right )\in (\pi /4,3\pi /4)$ valid along the integration path. In order to evaluate $\text {Re} \widetilde {\Phi } (\lambda )$ for λ ∈ ℓ, it makes sense to decompose the corresponding phase integral

$$ \widetilde{\Phi} (\lambda) = S(\xi,\widetilde{r};\lambda) + S(\widetilde{r},r;\lambda) + S(r,\eta;\lambda) $$

where ξ = ξ(λ; κ), η = η(λ; κ) and $\widetilde {r}=\{ \xi + i\mathbb {R}_{+}\}\cap \{ r+e^{i\pi /8}\mathbb {R}_{-}\}$ and consider the summands separately. Since curve ℓ is the graph of increasing function and lies below the line $K_{+}\!+ e^{i\pi /4}\mathbb {R}_{-}$, one has $\arg (Q_{\kappa }(z)-\lambda )\in (\pi /4,\pi /2)$ for z ∈ [r, η] and λ ∈ ℓ so that ReS(r, η; λ) < 0. Due to the fact that curve $\widetilde {\ell }=\{Q_{\kappa }(r+e^{i\pi /8}\mathbb {R}_{-})\}\cap {\Pi }$ is located above the line $K_{+}\!+ e^{i\pi /4}\mathbb {R}_{-}$, the turning point ξ = ξ(λ) for λ ∈ ℓ lies below the semi-axis $r+ e^{i\pi /8}\mathbb {R}_{+}$ and hence $\text {Re} S(\xi ,\widetilde {r};\lambda )<0.$ Finally, taking into account that curves ℓ and $\widetilde {\ell }$ are separated by the line $K_{+}\!+ e^{i\pi /4}\mathbb {R}_{-}$, we obtain the estimate $\pi /4<\arg \left (Q_{\kappa }(z)-\lambda \right )<5\pi /4$ valid for $z\in [ \widetilde {r},r]$ and λ ∈ ℓ which implies $\text {Re} S(\widetilde {r},r;\lambda )<0.$

Thus, we have established that level-lines $\text {Re} \widetilde {\Phi }(\lambda )=0$ and $\text {Re}\left (\widetilde {\Phi }(\lambda )-\widehat {\Phi }(\lambda )\right )=0$ possess a unique intersection Λ = Λ(κ) in π located above the curve ℓ and the segment $\widehat {\ell }$ and being a triple point for the edges ${\Gamma }_{+}^{(j)}, j=1,2,3,$ of the limit spectral graph Γ = Γ(κ). Moreover, numerical evaluation of the point ${\Lambda } ={\Lambda } (\kappa )\in {\Gamma }_{+}^{(j)}$ reveals the following regularity phenomenon of approximate conservation law type

$$ \frac{ \text{Re} {\Lambda} (\kappa)-Q_{\kappa}(1)}{K_{+}- \text{Re} {\Lambda} (\kappa)} = 2,1 \pm 0,02 $$

which is strongly confirmed by the results of computer experiments presented below. □

The three different topological types of limit spectral configuration appearing in the problem under consideration and specified by Theorems 1–3 are illustrated in Figs. 1–4. Figures 1 and 2 exhibit spectral locus for potentials z ± z³/3 which correspond to the values of deformation parameter κ = 1/7 and κ = − 1/5. Figures 3 and 4 show asymptotic distribution of eigenvalues in the cases of potentials z ± 3z³/4 corresponding to κ = 3/11 and κ = − 3/5.

Moreover, at Figs. 5 and 6, one can observe semi-classical location of the spectrum for the problem in question with potentials 5z/6 − z³ and 2z/3 − z³ which in fact belong to the family Q_κ(z) at points κ = − 3/2 and κ = − 3 from beyond the interval [− 1, 1].

Fig. 5

Spectral graph for potential Q(z) = 5z/6 − z³

Fig. 6

Spectral graph for potential Q(z) = 2z/3 − z³

Besides that making use of combined numerically-analytic method, we treat certain quadratic perturbations of potentials Q_κ(z) under which spectral symmetry is broken. Figures 7, 8, 9, 10, and 11 below depict spectral locus for the problem (1)–(2) with potentials of the form 2z³ + z²/2 − τz where parameter τ successively takes the values 2, 7/4, 3/2, 5/4, 1.

Fig. 7

Spectral graph for potential Q(z) = 2z³ + z²/2 − 2z

Fig. 8

Spectral graph for potential Q(z) = 2z³ + z²/2 − 7z/4

Fig. 9

Spectral graph for potential Q(z) = 2z³ + z²/2 − 3z/2

Fig. 10

Spectral graph for potential Q(z) = 2z³ + z²/2 − 5z/4

Fig. 11

Spectral graph for potential Q(z) = 2z³ + z²/2 − z

Finally, at Fig. 12, the asymptotic distribution of eigenvalues is plotted for the case of potential z³ + z²/4 + z.

Fig. 12

Spectral graph for potential Q(z) = z³ + z²/4 + z