⚡ Live Sandbox: AI Dev Tool: PostgreSQL Vector Search (pslz)

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```python
# AI Dev Tool: FastAPI Pydantic v2 Micro-SaaS Boilerplate
# =====================================================
# This is a production-ready boilerplate for building micro-SaaS applications
# using FastAPI, Pydantic v2, and PostgreSQL.

# Import required libraries
from fastapi import FastAPI, Depends
from fastapi.security import HTTPBearer, HTTPAuthorizationCredentials
from pydantic import BaseModel, Field
from pydantic.errors import ListError
from typing import List, Optional
from sqlalchemy import create_engine, Column, Integer, String
from sqlalchemy.ext.declarative import declarative_base
from sqlalchemy.orm import sessionmaker, Session
import uvicorn
import os

# Define the FastAPI application
app = FastAPI()

# Define the PostgreSQL database connection
SQLALCHEMY_DATABASE_URL = os.environ.get("DATABASE_URL")
engine = create_engine(SQLALCHEMY_DATABASE_URL)
SessionLocal = sessionmaker(autocommit=False, autoflush=False, bind=engine)

# Define the base class for models
Base = declarative_base()

# Define the User model
class User(Base):
    __tablename__ = "users"
    id = Column(Integer, primary_key=True, index=True)
    username = Column(String, unique=True,

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