Confining oscillator domain

Operator and energy domain

On a flat model of $d$ ordinary spatial directions within the one nested physical manifold, consider

$$H=-\Delta+\lambda|x|^2,\qquad \lambda>0,\qquad \mathcal H=L^2(\mathbb R^d,d^dx).$$

This is a specified scalar confinement operator. Its use for a particular sector requires the corresponding kinetic and binding dynamics. Define its energy through the positive quadratic form

$$q[u]=\int\left(|\nabla u|^2+\lambda|x|^2|u|^2\right)d^dx, \qquad \mathcal Q=H^1(\mathbb R^d)\cap\{u:|x|u\in L^2\}.$$

Here $H^1$ requires square-integrable weak first derivatives. The form norm is $(\|u\|_2^2+q[u])^{1/2}$. It is complete: a Cauchy sequence converges in $H^1$ and its products $|x|u$ converge in $L^2$; local convergence identifies the latter limit with $|x|$ times the former. The densely defined closed form therefore defines the positive self-adjoint Friedrichs operator $H$.

Compact resolvent

Suppose a sequence is bounded in this form norm by a common constant. For every $R>0$,

$$\int_{|x|>R}|u(x)|^2\,d^dx \leq \frac{q[u]}{\lambda R^2}.$$

On each finite ball, the Rellich compactness theorem supplies an $L^2$-convergent subsequence of the $H^1$-bounded restrictions. Choose a diagonal subsequence over increasing balls. The uniform tail estimate makes this subsequence Cauchy on all of $\mathbb R^d$: first make both tails small, then use convergence on the chosen ball. Thus $\mathcal Q\hookrightarrow L^2$ is compact.

The resolvent $(H+1)^{-1}$ maps bounded $L^2$ sets to bounded form-norm sets. Composing with that compact inclusion proves compactness of the resolvent. The spectrum consists of isolated eigenvalues of finite multiplicity tending to infinity, with a complete orthonormal eigenbasis. This is the confining-potential case of the spectral theorem discussed in Phan Thành Nam, Functional Analysis II, §2.5.

Explicit basis and general states

Set $w=\sqrt{\lambda}$. For a multi-index $\alpha=(\alpha_1,\ldots,\alpha_d)$ of nonnegative integers, write $|\alpha|=\sum_j\alpha_j=N$ and $\alpha!=\prod_j\alpha_j!$. The physicists' Hermite polynomials give

$$\phi_\alpha(x)= \frac{(w/\pi)^{d/4}}{\sqrt{2^N\alpha!}} \prod_{j=1}^d H_{\alpha_j}(\sqrt w\,x_j) e^{-w|x|^2/2}, \qquad H\phi_\alpha=(2N+d)w\,\phi_\alpha.$$

The one-dimensional Hermite differential equation proves the eigenvalue, and Hermite orthogonality proves normalization. Counting multi-indices with sum $N$ gives $\binom{N+d-1}{d-1}$ states at that energy. The spherical basis and radial overlaps are given in Normalized oscillator modes.

Every normalizable state has an $L^2$-convergent expansion $u=\sum_\alpha c_\alpha\phi_\alpha$ with $\sum_\alpha|c_\alpha|^2<\infty$. Writing $E_\alpha=(2|\alpha|+d)w$,

$$\begin{aligned} u\in\mathcal Q&\iff\sum_\alpha E_\alpha|c_\alpha|^2<\infty,\\ u\in D(H)&\iff\sum_\alpha E_\alpha^2|c_\alpha|^2<\infty. \end{aligned}$$

In the corresponding scaled time, unitary evolution is $u(t)=\sum_\alpha c_\alpha e^{-iE_\alpha t}\phi_\alpha$. It preserves the norm for every $L^2$ state, including superpositions. For example $(\phi_0+\phi_{(1,0,\ldots,0)})/\sqrt2$ has two energy components with relative phase $e^{-2iwt}$.

Gaussian tails and boundary terms

A fixed eigenfunction is a polynomial of degree at most $N$ times $e^{-w|x|^2/2}$. For $r\geq1$ its polynomial obeys a bound $Cr^N$. Using $\Omega_{d-1}=2\pi^{d/2}/\Gamma(d/2)$ gives, for $R\geq1$,

$$\begin{aligned} \int_{|x|>R}|\phi_\alpha|^2\,d^dx &\leq C^2\Omega_{d-1}\int_R^\infty r^{2N+d-1}e^{-wr^2}\,dr\\ &=\frac{C^2\Omega_{d-1}}{2w^{N+d/2}} \Gamma(N+d/2,wR^2). \end{aligned}$$

Here $\Gamma(s,z)=\int_z^\infty t^{s-1}e^{-t}\,dt$ is the upper incomplete gamma function. Polynomial derivatives have the same Gaussian envelope. Surface integrals of finite products of these eigenfunctions and their derivatives, with polynomially bounded coefficients, therefore tend to zero at infinity. This supplies explicit boundary control for finite oscillator-mode calculations.

For general energy-domain states, the form tail bound above still applies. Pointwise Gaussian decay or a particular boundary flux requires additional regularity and decay assumptions. The fibre-integrated stress calculation states the flux condition for a conserved source.

At level $N$, every normalized eigenstate has

$$\langle |x|^2\rangle=\frac{N+d/2}{w}, \qquad L=w^{-1/2}=\lambda^{-1/4}.$$

The diagonal one-coordinate identity $\langle x_j^2\rangle=(\alpha_j+1/2)/w$ proves this for the Cartesian basis; the terms in $|x|^2$ that change total level have zero matrix elements within a fixed eigenspace, proving the general case. The coefficient $\lambda$ has units of inverse length to the fourth in this scaled operator. Matching $L$ to a physical length requires the physical coefficients of the action.

Verification and related

spatial_profile_checks.py checks normalized Hermite modes, differential residuals, moments, superposition evolution, and independent tail quadratures. The compactness argument is analytic; finite numerical checks test its explicit examples.

• Gaussian contact profiles — uses the ground-state covariance in an overlap calculation.
• Interaction operator tools — navigation to the surrounding mathematical tools.